I have developed some code to sort arrays of strings which contain a mixture of letters and numbers. A common problem in string sorts is that the natural alphabetical order is not the same as the order of digits, since we consider consecutive digits in the aggregate but consecutive letters individually. Hence, in alphabetical order "str10" precedes "str2", because 1 is before 2 alphabetically. What we normally want to do is to compare 10 with 2, which would cause us to say "str2" precedes "str10".

To accomplish this I first create linked lists of alphameric and numeric strings (one linked list for each string). I then compare the corresponding nodes of the two lists. Permission is granted to use and adapt this code on an "as is" basis with no warranty of any kind.

## Saturday, December 29, 2012

### Intuitive Sort Order for Alphanumeric Strings

## Friday, December 28, 2012

### Convective Heat Transfer on a Building Envelope (Wind Chill?)

I'd like to make a brief mathematical investigation into the concept of a wind chill factor and how one particular scheme common in popular culture (and even affecting some in industry) does not relate well to modelling of a building envelope. Wind chill is the temperature which a human being perceives accounting for the effects of wind and temperature together. The preceding sentence is awkward because it is trying to say too much at once, so let me try again:

h

which simplifies to

T

This leaves me with a problem. I want to have some type of reasonable value for the convective heat transfer coefficient ratio (h

(h

= (88 – (-59)) / (88 – (-31))

= 1.235

(I've used 88°F for human skin, but that temperature will be different depending on the ambient temperature. Oh, well. If our ratio is 1.1 or 4.7, the math is the same—just different numbers for the end result. The results will still show a difference from the "wind chill index" commonly talked about.)

Theoretically, the convective heat transfer coefficient is largely independent of temperature and area, but it will change based on the interaction of the fluid and the surface which will include things like geometry, orientation, surface roughness, moisture, etc. Unfortunately, temperature differences may affect the actual coefficient. They will affect the result if you consider radiation losses as well, since radiation transport is proportional to absolute temperature to the fourth power.

So, let's consider a surface which is maintained at a temperature of 10°F and is surrounded by an ambient temperature of -31°F and the wind speed is 15 mph. (This is not the temperature in the boundary layer. The point of the convective heat transfer equation is to deal with this phenomenon in a simplified way—essentially to circumvent it.) So, what's my "equivalent temperature in the absence of wind"?

T

= 10 – 1.235 (10 – (-31))

= -40.6 °F

So, the rate of heat loss from Surface X under the actual conditions is the same as the rate of heat loss for Surface X with no wind and a temperature of -40.6°F.

We obtained a different value than we did for the wind chill index applicable to human skin. The reason is straightforward: the rate of heat loss is proportional to the difference between the ambient air temperature and the surface temperature. Now, if it is -31°F outside, what temperature is the outside surface of a building? Perhaps -21°F? This makes quite a difference:

T

= -21 – 1.235 (-21 – (-31))

= -33.4 °F

So the effect is not zero, but it isn't much to talk about. If my h ratio is actually much higher, we will get a greater difference.

The thickness and properties of the building envelope materials may change the relationship between ambient air temperature and surface temperature. Such changes will change the rate of heat loss in proportion to the temperature difference. Is there a higher difference at glazing surfaces than concrete surfaces?

There's also a sticky point regarding solar heat gains. Such gains on the surface will increase the rate of heat loss

Note that in warm climate conditions (cooling conditions), heat on the outside surface migrates toward the interior and we must therefore consider it. It is no longer starting to go in only to double-back (as it were), but actually conducting through the envelope and so directly affecting interior conditions.

In closing, this article is

Note that ASHRAE Fundamentals chapter 26 (2009) gives effective R-values which can be used. The point of these values is to bypass all of the stuff I'm writing about in this article and account for convection and radiation losses at the surface under a 15 mph wind design condition. You don't actually need all that wind chill stuff—chapter 26 R-values already give a reasonable estimate of the wind effects, i.e., R-0.17 at 15 mph on the exterior instead of R-0.68 on the interior for still air at a wall.

A wind chill index (expressed in units equivalent(ish) to temperature measurements) is answering a question: What temperature T

Human beings, not buildings. Consider the following differences:

_{WC}(with wind speed = 0) will be perceived*by a human being*as equivalent to temperature T_{actual}(with wind speed = V_{actual})? (We are talking here more particularly about the exposed skin of a human being. Also, it isn't purely a matter of perception.)Human beings, not buildings. Consider the following differences:

**buildings (in cold climates) have a much lower surface temperature than humans****the amount (and temperature) of moisture on their surfaces is generally different and has a different internal transport mechanism**(brick has basically no moisture and that moisture is already cold, human skin normally has moisture—perspiration—which evaporates more rapidly in windy conditions than calm conditions causing heat loss, because evaporation requires energy)**unlike human beings, buildings do not have psychological comfort issues affecting their perception of temperature**(and, admittedly, cannot reasonably be considered to perceive anything)

The equation used today for wind chill index has some theoretical basis but in the end is empirically derived. That means they did experiments and found something that "works". It is a very pragmatic way to deal with complex issues, especially ones that include psychological/cognitive components. (I'm not "disrespecting" empirical equations, they're useful.)

So, where do I start? I would hard pressed with my current knowledge/understanding to deal with the evaporative cooling effects on human skin as compared with buildings, though such investigation might be meritorious. I will not attempt to compare the psychology of buildings with that of humans—I hope you understand. I will however seek to show how the concept of an "equivalent temperature in the absence of wind" yields different results depending on the surface temperature of the material in question. As such, although we will not compare human skin with face brick, we will compare warm Surface X with cool Surface X.

The convective heat transfer equation is simple enough:

q = h

where q is the heat energy (per unit time), h

The heat transfer coefficient varies with wind speed. Suppose we want to find an air temperature T

**Convective Heat Transfer**The convective heat transfer equation is simple enough:

q = h

_{c}A_{S}(T_{S}– T_{air})where q is the heat energy (per unit time), h

_{c}is the heat transfer coefficient, A_{S}is the surface area, T_{S}is the temperature of the surface and T_{air}is the ambient air temperature.The heat transfer coefficient varies with wind speed. Suppose we want to find an air temperature T

_{WC}(no wind, h_{NW}) which causes the same rate of heat loss as T_{actual}(wind speed = W_{actual}, h_{actual}). In that case, we need to relate:h

_{NW}A_{S}(T_{S}– T_{WC}) = q = h_{actual}A_{S}(T_{S}– T_{actual}),which simplifies to

T

_{WC}= T_{S}– (h_{actual}/ h_{NW})(T_{S}– T_{actual}).**A Little Problem**This leaves me with a problem. I want to have some type of reasonable value for the convective heat transfer coefficient ratio (h

_{actual}/ h_{NW}). How do I get that? I will engage in a little pragmatism and pretend that the wind chill equation will still give me a way to get an (h_{actual}/ h_{NW}) ratio. It probably gets me something close to reality, but the main point will stand regardless of the realism of this ratio. From the wind chill equation we get a 15 mph wind at -31°F being equivalent to no wind at -59°F. Here's what we get for the ratio for human skin (which I have to use since that's what the wind chill equation is based on):(h

_{actual}/ h_{NW}) = (T_{S}– T_{WC}) / (T_{S}– T_{actual})= (88 – (-59)) / (88 – (-31))

= 1.235

(I've used 88°F for human skin, but that temperature will be different depending on the ambient temperature. Oh, well. If our ratio is 1.1 or 4.7, the math is the same—just different numbers for the end result. The results will still show a difference from the "wind chill index" commonly talked about.)

Theoretically, the convective heat transfer coefficient is largely independent of temperature and area, but it will change based on the interaction of the fluid and the surface which will include things like geometry, orientation, surface roughness, moisture, etc. Unfortunately, temperature differences may affect the actual coefficient. They will affect the result if you consider radiation losses as well, since radiation transport is proportional to absolute temperature to the fourth power.

**Wind Chill Effect on Surfaces at Different Temperatures Than Human Skin?**So, let's consider a surface which is maintained at a temperature of 10°F and is surrounded by an ambient temperature of -31°F and the wind speed is 15 mph. (This is not the temperature in the boundary layer. The point of the convective heat transfer equation is to deal with this phenomenon in a simplified way—essentially to circumvent it.) So, what's my "equivalent temperature in the absence of wind"?

T

_{WC}= T_{S}– (h_{actual}/ h_{NW})(T_{S}– T_{actual})= 10 – 1.235 (10 – (-31))

= -40.6 °F

So, the rate of heat loss from Surface X under the actual conditions is the same as the rate of heat loss for Surface X with no wind and a temperature of -40.6°F.

We obtained a different value than we did for the wind chill index applicable to human skin. The reason is straightforward: the rate of heat loss is proportional to the difference between the ambient air temperature and the surface temperature. Now, if it is -31°F outside, what temperature is the outside surface of a building? Perhaps -21°F? This makes quite a difference:

T

_{WC}= T_{S}– (h_{actual}/ h_{NW})(T_{S}– T_{actual})= -21 – 1.235 (-21 – (-31))

= -33.4 °F

So the effect is not zero, but it isn't much to talk about. If my h ratio is actually much higher, we will get a greater difference.

**Considerations on Building Outside Surface Temperatures**The thickness and properties of the building envelope materials may change the relationship between ambient air temperature and surface temperature. Such changes will change the rate of heat loss in proportion to the temperature difference. Is there a higher difference at glazing surfaces than concrete surfaces?

There's also a sticky point regarding solar heat gains. Such gains on the surface will increase the rate of heat loss

*at the surface*. But watch out! This is mainly behaviour occurring at the surface. We're dealing with a cold climate situation. Radiant heat from the sun heats up the surface only to be removed by increased rates of convection and radiation loss. This does not help us analyse the heat losses from the interior of the building! My plain horse sense tells me that adding thermal energy to the exterior does not cause a net increase to the rate of heat loss from the building interior. At worst, the increased rate of convective (and radiation) loss will remove all of the radiant energy added. To deal thoroughly with this issue, we would either need to model both in parallel (allowing them to interact numerically with each other—for transient analysis) or use a "net direct solar gain" approach that deals with the interaction of convection and radiation (gains and losses) at the surface in an integrated way.Note that in warm climate conditions (cooling conditions), heat on the outside surface migrates toward the interior and we must therefore consider it. It is no longer starting to go in only to double-back (as it were), but actually conducting through the envelope and so directly affecting interior conditions.

**CAUTION: This is NOT a Final Answer**In closing, this article is

*not trying to present the final answer to this problem*. Rather I am attempting to expose the fallacy that the wind chill index (and corresponding formula) can be naively applied to energy modelling problems in building science. Furthermore, my criticism is not directed against nor does it address considerations of the effects of wind on air leakage analysis.Note that ASHRAE Fundamentals chapter 26 (2009) gives effective R-values which can be used. The point of these values is to bypass all of the stuff I'm writing about in this article and account for convection and radiation losses at the surface under a 15 mph wind design condition. You don't actually need all that wind chill stuff—chapter 26 R-values already give a reasonable estimate of the wind effects, i.e., R-0.17 at 15 mph on the exterior instead of R-0.68 on the interior for still air at a wall.

## Wednesday, December 26, 2012

### Degree of Saturation versus Relative Humidity

These two quantities are similar and as I worked through ASHRAE Handbook Fundamentals (2009) IP, chapter 1, I was initially baffled as to what the difference was. First, the definitions based on ASHRAE Fundamentals:

In relative humidity, imagine taking two samples, one of the air which you want to know the relative humidity of and one of saturated air at the same temperature and pressure. For example, we might take samples of 1 mole each. In the non-saturated air, we will have less water vapour than in the saturated air. On the other hand, we will have more dry air in the non-saturated sample than in the saturated sample. The same general relationship will be true of mass, but the masses and the mole fractions will relate differently due to differing molecular masses. Here's the bottom line: we compare the samples based on moles (which amounts to the number of molecules) not on a given mass or volume. When you're dealing with gases, moles is the way to go. In other words, our non-saturated and saturated samples relate by the equation:

where the items on the left hand side are the mole fractions of water vapour and dry air in the non-saturated sample and the items on the right hand side are for saturated air.

Understanding that we need to compare these quantities with a fixed number of total moles as in the previous equation is what makes ASHRAE's equation (14) work out. From the last given equation we have

Therefore,

It is worth noting that we only had to worry about this molar equality in developing this equation. When it comes to computing the degree of saturation from the relative humidity and vice versa, we don't have to think about that. The formula will do that for us. Also, we can still compute the degree of saturation directly from values off of a psychrometric chart (for example) by reading across to the right to get

So if you're getting different values for these quantities in your work, it's not (necessarily) a mistake or a misreading of the chart—they really are different.

**Relative humidity:**The mole fraction of water vapour in a sample divided by the mole fraction of water vapour in saturated air at the same temperature and pressure.**Degree of saturation:**The humidity ratio of water vapour in a sample divided by the humidity ratio of water vapour in saturated air at the same temperature and pressure.**Humidity ratio:**The humidity ratio is the mass of water vapour divided by the mass of dry air in a sample. This value can also be expressed in terms of molar fractions.

**Discussion of Differences**In relative humidity, imagine taking two samples, one of the air which you want to know the relative humidity of and one of saturated air at the same temperature and pressure. For example, we might take samples of 1 mole each. In the non-saturated air, we will have less water vapour than in the saturated air. On the other hand, we will have more dry air in the non-saturated sample than in the saturated sample. The same general relationship will be true of mass, but the masses and the mole fractions will relate differently due to differing molecular masses. Here's the bottom line: we compare the samples based on moles (which amounts to the number of molecules) not on a given mass or volume. When you're dealing with gases, moles is the way to go. In other words, our non-saturated and saturated samples relate by the equation:

where the items on the left hand side are the mole fractions of water vapour and dry air in the non-saturated sample and the items on the right hand side are for saturated air.

Understanding that we need to compare these quantities with a fixed number of total moles as in the previous equation is what makes ASHRAE's equation (14) work out. From the last given equation we have

Therefore,

It is worth noting that we only had to worry about this molar equality in developing this equation. When it comes to computing the degree of saturation from the relative humidity and vice versa, we don't have to think about that. The formula will do that for us. Also, we can still compute the degree of saturation directly from values off of a psychrometric chart (for example) by reading across to the right to get

*\(W\)*and finding \(W_S\) by finding where the dry bulb temperature intersects the saturation curve and reading across to the right. The definition of degree of saturation doesn't require us to account for the number of moles, only the relative humidity (and yes, if you're using charts you can read it off there as well). If you are familiar with reading psychrometric charts, you will know that reading*W*values is generally much easier than reading*RH*values.So if you're getting different values for these quantities in your work, it's not (necessarily) a mistake or a misreading of the chart—they really are different.

## Thursday, November 15, 2012

### Adding an Independent Section in Word Using VBA

If you are using .NET, this technique is still applicable. Here is the code we will be discussing (which is commented enough that you might not need to read anything else):

First, we insert a section break which starts a new page. We need to save the current range. If there is no selection, the current location is treated as a selection. If there is text selected, it will get replaced a bit later. We assign the current selection range to a variable to reference after we have changed the current selection. Changing the current selection will not change the range referred to by

After the second section break is added, we modify the range we saved a reference to, in this case, by adding some text.

If you had a document with nothing but the digits 12 in it and placed the cursor after the 1 but before the 2, after executing this code, page one would have 1 on it, page 2 would have Hello, Word! on it and page 3 would have 2 on it.

**Code Walk Through**First, we insert a section break which starts a new page. We need to save the current range. If there is no selection, the current location is treated as a selection. If there is text selected, it will get replaced a bit later. We assign the current selection range to a variable to reference after we have changed the current selection. Changing the current selection will not change the range referred to by

**r**. (It will continue to point to the same place in the document as when the assignment was made.) We now add a new section break—this should normally be done before we modify the current range. Otherwise, the modifications or additions will simply get pushed onto the next page since the selection is still at the beginning of the additions. If you were able to add stuff and move the current selection to the end of the new stuff, then you could add the section break without pushing the new stuff forward.After the second section break is added, we modify the range we saved a reference to, in this case, by adding some text.

If you had a document with nothing but the digits 12 in it and placed the cursor after the 1 but before the 2, after executing this code, page one would have 1 on it, page 2 would have Hello, Word! on it and page 3 would have 2 on it.

## Thursday, November 8, 2012

### Linear Interpolation Made Convenient in Maxima

(I usually put the load statement in a separate cell in wxMaxima so that I am not re-executing the load every time I correct my main code and rerun the cell. Minor point.)

Maxima can turn a set of points into a function for linear interpolation. For that matter, it can make a cubic spline out of the same points. I'm more interested in the linear interpolated function today—the other is for your reference. There are three main points of interest in the above code:

**linearinterpol()**takes points, sorts them by the first ordinate, and produces an expression.**linearinterpol()**produces expressions which use the characteristic function, rather than taking cases to make it transparent to other Maxima routines.- The use of the
*double single quote operator*('') and the*single quote operator*(') helps me to define the function f(x) such that x is a formal variable and not embedded as an expression. **linearinterpol()**produces an expression which extrapolates beyond the specified points (from negative infinity to positive infinity) and it is left to the user's discretion to determine the limits to which this extrapolation is reasonable or applicable.

**In a Nutshell**

**All you need is a list of 2D points. There is more than one way to specify these, but I find it easiest to use a list of lists. The main list elements are all lists of 2 representing x and y ordinates. The function sorts these points according to the x-coordinate before proceeding. You can specify the variable which will be used in the expression. Provided that you precede that variable with a single quote operator, you can use a variable that is elsewhere assigned a value. More on that below.**

**Transparent Output**

The characteristic function, which is often assigned the Greek letter chi ("χ"), returns either 0 or 1. It is a function of two variables, a set and a value. If the value is an element of the set, it returns 1, otherwise it returns 0. In the context of our linear interpolation expression, we have a linear expression for each interval in our set of points and each of these is multiplied by a characteristic function with a different set. Since the sets used for the characteristic functions form a partition of the real numbers, we won't have more than one of the terms of our overall expression from

The single quote operator (beside the key) is a way to tell Maxima, "Don't evaluate this thing." If I write x and x is assigned value, then Maxima will do what is normally very helpful to me, namely, make the substitution for me. If I don't want Maxima to do that, because I actually want the expression x, not the value that x stands for (because it has been assigned a value), I need to write 'x instead. In contrast, the double single quote operator ('') tells Maxima to replace the given expression with the value of the expression. This way, when I use the delayed assignment operator (:=) it does not delay the execution of the

Here is some sample input followed by the output. This is the best way to clarify in your mind the use of these operators:

x:2;

y:2*x;

z: 2*'x;

d(x) := z;

d(2);

d(3);

h(x) := ''(z);

h(2);

h(3);

(%o46) 2

(%o47) 4

(%o48) 2*x

(%o49) d(x):=z

(%o50) 2*x

(%o51) 2*x

(%o52) h(x):=2*x

(%o53) 4

(%o54) 6

The expression that

**linearinterpol()**evaluate to a non-zero value at a time. This expression can be integrated, differentiated, or combined with other expressions without having to take special cases. Take the monkey off your back, let Maxima do it.**Cool, Useful, but Initially Abstruse Operators**The single quote operator (beside the

**linearinterpol()**function, only the evaluation of the expression as a function of x is delayed. You may recall from my previous post that parameters in function declarations are formal. Even if x is elsewhere assigned a value, f(x) := some_expression_including_x + x, will still work intuitively.

Here is some sample input followed by the output. This is the best way to clarify in your mind the use of these operators:

x:2;

y:2*x;

z: 2*'x;

d(x) := z;

d(2);

d(3);

h(x) := ''(z);

h(2);

h(3);

(%o46) 2

(%o47) 4

(%o48) 2*x

(%o49) d(x):=z

(%o50) 2*x

(%o51) 2*x

(%o52) h(x):=2*x

(%o53) 4

(%o54) 6

**Extrapolation**The expression that

**linearinterpol()**produces does not protect you from yourself. If will produce a result from negative to positive infinity without telling you that you have gone beyond the boundaries of your initial data set. Actually, this is as it should be, in my opinion. The program gives you the freedom to ask all the what-ifs your little heart desires, but it is up to you to discern what is really true to life and applicable to the particular problem you are trying to solve—whether the extrapolation is truly justified by the data is rightly left as a matter for user judgement.
Labels:
estimation,
interpolation,
linear equations,
Maxima,
numerical

## Wednesday, October 31, 2012

### Local Variables, Function Parameters, and Blocks In Maxima

One of the things that was a real hang-up for me when I started programming in Maxima was its seeming lack of variable scoping rules. It made some irritating and non-intuitive things happen. But, because I was always in “get-it-done” mode, I didn’t stop to figure out if there was a better way. I would resort to adding letters to the name that I wanted to use. If my functions were going to use

Maxima’s primary scoping mechanism is the “block.” A block starts with

What Maxima actually does is not quite the same as in most programming languages, although the apparent runtime behaviour will be the same, if the

This behaviour is different from most other programming languages which use “dynamic scoping” in that there are no save and restore steps of the kind we are talking about here (in most other languages). In Maxima, variables have names at run-time and are referenced according to their name. In most languages, the name of a variable is solely a convenience for programmers. At run-time, values will move to various parts of the program memory space and the memory address values will fly. Just because the same name is used in the source code for two variable references, doesn’t mean the compiled code will have references to the same memory location. For example,

When it comes to function parameters, however, Maxima`s behaviour is not unlike most other languages. That is, the parameters are formal. They are not considered to constitute a variable declaration. When I write

f(x) := x^2;

x is not declared as a variable.

Even if I write

RotationalEquivalenceSet(n, bits) := block(

local(res, size),

res: {n},

size: 2^bits,

if n >= size then return({}),

n: 2*n,

if n >= size then n: n - size + 1,

do (

if elementp(n, res) then return(res),

res: union(res, {n}),

n: 2 * n,

if n >= size then n: n - size + 1

)

);

/* [wxMaxima: input start ] */

n: x^2;

RotationalEquivalenceSet(3,5);

n;

RotationalEquivalenceSet(5,5);

n;

/* [wxMaxima: input end ] */

(%o21) x^2

(%o22) {3,6,12,17,24}

(%o23) x^2

(%o24) {5,9,10,18,20}

(%o25) x^2

Note that I am using

**a**, I would instead name those**aa**, so that my variables in the “true” global realm wouldn’t be overwritten after the next function call. My convention would then be, “nice names in the global realm, ugly names in the local realm.” This is far from ideal. The fact is, Maxima defaults to global variables, but makes it easy to declare local variables.Maxima’s primary scoping mechanism is the “block.” A block starts with

**block(**and ends with**)**. Each statement within a block is separated by a comma (**,**). The first statement in your block should normally be**[v_1, v_2,…,v_n]**, where v_1, v_2, etc., are variables that you wish to be local (don’t literally type v_1, etc.). If you don’t want any local variables, then omit the local statement.What Maxima actually does is not quite the same as in most programming languages, although the apparent runtime behaviour will be the same, if the

**[v_1, v_2,…,v_n]**statement is used as above. Maxima saves the current values of the variables in the**[v_1, v_2,…,v_n]**statement and essentially “zeros out” (removes their properties) those variables. They become blank slates. When Maxima exits the**block()**in which the variable was declared as local, its current properties are removed and the saved values/properties are restored. As far as I know, Maxima does not support multi-threading. This is probably a good thing, as temporarily preventing a variable from “existing” (as itself, that is; a variable with the same name is not the same variable itself, philosophically) is problematic if that variable could be accessed at any time by a separate thread.This behaviour is different from most other programming languages which use “dynamic scoping” in that there are no save and restore steps of the kind we are talking about here (in most other languages). In Maxima, variables have names at run-time and are referenced according to their name. In most languages, the name of a variable is solely a convenience for programmers. At run-time, values will move to various parts of the program memory space and the memory address values will fly. Just because the same name is used in the source code for two variable references, doesn’t mean the compiled code will have references to the same memory location. For example,

**t**declared in**function1()**has nothing to do with**t**declared in**function2()**. These memory locations will be on the stack and exist only while each function is being executed. This, again, is what is true in most programming languages, not in Maxima.When it comes to function parameters, however, Maxima`s behaviour is not unlike most other languages. That is, the parameters are formal. They are not considered to constitute a variable declaration. When I write

f(x) := x^2;

x is not declared as a variable.

Even if I write

RotationalEquivalenceSet(n, bits) := block(

local(res, size),

res: {n},

size: 2^bits,

if n >= size then return({}),

n: 2*n,

if n >= size then n: n - size + 1,

do (

if elementp(n, res) then return(res),

res: union(res, {n}),

n: 2 * n,

if n >= size then n: n - size + 1

)

);

**n**is not declared as a variable. This is a very helpful thing. If**n**was assigned some value before the function call, nothing that I do inside this function call (which uses**n**as the name of a parameter) affects that value. The value of**n**will remain the same after this function has been invoked. Here is an input/output example of the behaviour:/* [wxMaxima: input start ] */

n: x^2;

RotationalEquivalenceSet(3,5);

n;

RotationalEquivalenceSet(5,5);

n;

/* [wxMaxima: input end ] */

(%o21) x^2

(%o22) {3,6,12,17,24}

(%o23) x^2

(%o24) {5,9,10,18,20}

(%o25) x^2

Note that I am using

**:=**to declare these functions. You may end up with different (and undesired) behaviour if you don’t use the “delayed evaluation” assignment statement. If you want the usual programming paradigm with your function declaration, use**:=**as per my sample code above.## Wednesday, October 17, 2012

### Making a Double Out of Parts in VB.NET

Recently I picked up on a piece of code I had started several months ago and didn't have time to finish. I was a bit stumped actually, though I've got it pretty well beat into shape now (maybe more on that later). I was making a VB.NET type Fraction using

4.445 × 10

An important finishing piece to solving this programming problem is the following function:

First, for a good summary of the double format, see Double-precision floating-point format (Wikipedia).

The

After making our biased exponent (

The biased exponent is now shifted into place—this, like the 1023 excess, falls into the category of "painful detail we want to encapsulate so we can forget about it." (This is not ambiguous like the too-large-mantissa.) In our final steps we "Or" the biased exponent and mantissa into place and use a call to

To make this function more robust, an exception class could be defined, either several different ones (one for each error) or a class with properties and enumerations to allow the caller to efficiently test for the particular condition that caused the "overflow". I note also that I have used OverflowException where an underflow has occurred. In program debugging, an accurate error message ("over" versus "under") can provide a clue to the person debugging the program.

**BigInteger**s as the base type. As such, the class would theoretically be able to represent any rational number. But what if I wanted to convert that result to a**double**? So, I started into the code necessary to do the job. I began with the easy part, the cases where the numerator and denominator can both be converted to**double**s. In that case, .NET already knows how to do the needed conversions. But if the numerator or denominator is too large to be converted to a double, the result of division (mathematically) may still be within the range of a**double**. For example,4.445 × 10

^{400}/ 4.222 × 10^{399}= 10.528185693983894An important finishing piece to solving this programming problem is the following function:

First, for a good summary of the double format, see Double-precision floating-point format (Wikipedia).

**Code Walkthrough**The

**double**type has 64 bits as does the**Int64**type. We begin with initializing our result variable to 0. Depending on whether the number is indicated as positive or not, we set the sign bit. The sign bit (bit 63, the most significant bit) needs to be set (1) to indicate a negative number. For simplicity of use, I wanted to let the caller work in terms of the exponent (base 2) rather than with a biased result. The 1023 "excess" or bias is one of those values you don't want to have to remember. This function obviates the need to be overly conscious of that value.After making our biased exponent (

**biasedExp**), we check that it is in range. We have exactly 11 bits to work with. If our result is greater than 11 bits (0x7FF in hex, 11111111111 in binary), then we complain to the caller with an OverflowException. If the result is negative, then it still does not fit into the 11 bits and we complain with an OverflowException in this case as well. The mantissa is required to be no more than 52 bits. That's 13 "nibbles" (4 bits, a single hex digit) or 6 1/2 bytes. If the mantissa has any bits set in the upper 12 bits, we complain. By calling the variable**mantissa**I have already declared my draconian intent with this function. I will not coddle the caller by shifting things that are in the wrong place to the right place. If the caller sends a mantissa which is too large, they may have done the shifting wrong or they may have incorrectly included the implicit 1 at the beginning of the "binary scientific notation." Either way, the caller has made an error and we cannot know which one it is. If we could know which error they made, it might make sense to correct it silently and move on. However, a wrong assumption here could be somebody's program's undoing. Therefore, we just complain and let the caller figure out what to do about it.The biased exponent is now shifted into place—this, like the 1023 excess, falls into the category of "painful detail we want to encapsulate so we can forget about it." (This is not ambiguous like the too-large-mantissa.) In our final steps we "Or" the biased exponent and mantissa into place and use a call to

**BitConverter.Int64BitsToDouble()**to ensure we have the proper format.**Room For Improvement**To make this function more robust, an exception class could be defined, either several different ones (one for each error) or a class with properties and enumerations to allow the caller to efficiently test for the particular condition that caused the "overflow". I note also that I have used OverflowException where an underflow has occurred. In program debugging, an accurate error message ("over" versus "under") can provide a clue to the person debugging the program.

## Tuesday, October 9, 2012

### How to make a .NET (Excel) Add-In Using Visual Basic Express 2010

Automation add-ins are easier than what follows and are the way to go if you just want a few custom functions. Actually, for efficiency's sake, you might want to just use VBA functions added to Module1 (must be Module1—no renaming to a more suitable name) and then save the workbook as an Excel Add-In (*.xlam). But, if it is user interaction or in some way making Excel "do stuff" (rather than just return a single result or an array of results), an add-in which implements the

The functions RegisterFunction and UnregisterFunction are invoked by utilities like regasm.exe which comes with the .NET Framework. I have created a batch file which I run in order to register the assembly. You will need to modify it to suit your assembly.

Note that once the Add-in is registered and added, you will not need to reregister it after every Build (while developing).

**IDTExtensibility2**and**IRibbonExtensibility**interfaces may be a good strategy. (The main alternative is to use VBA.)
Making an Excel Add-in in VB.NET involves the following (the ordering of the steps is not absolute):

- Ensure you have Office installed on the development machine and have the Primary Interop Assemblies (PIAs) installed in the Global Assembly Cache (GAC). You can download them from Microsoft. (The Microsoft installer will put the PIAs in the GAC.)
- Start a Class Library project in VB.NET.
- It's a good idea to generate your own strong name using the sn.exe tool (part of the .NET framework). You only need one for multiple projects. In a nutshell, you use the sn.exe tool to create a *.snk file which you will tell VB.NET about.
- let VB.NET know where your *.snk file is by going to Project/Properties..., under
**Signing**check "Sign the assembly" and choose a strong name key file. You will be able to browse to where your *.snk file is - if your add-in is intended to be widely distributed, and especially if you want to sell your add-in, you would be well-advised to learn more about this topic to ensure your end-users don't get warning messages that cause them to be (perhaps unduly) alarmed.
- Apparently, step 4 is unnecessary, but I did step 4, so here it is: In the project properties, go to
**Application**and click the button**Assembly Information...**. In this screen, check the**Make assembly COM-Visible**box. - Create a GUID. There are several ways but a program called guidgen.exe may already be on your computer which will do the job. Just search your computer for it. It is located under C:\Program Files\Microsoft SDKs\Windows\... on my computer.
- Copy and Paste the GUID into the
**Assembly Information...**dialog box referred to in step 4. - Encapsulate your project in a namespace either by typing it explicitly in the files or by specifying a root namespace in the project properties.
- Add references to needed PIAs, and extensibility (.NET tab)
- It is recommended that you add these by referencing the Interop Assembly on the .NET tab, not the COM tab. You may need to browse for this: Look under C:\Windows\assembly\gac_msil\Microsoft.Office.Interop.Excel\...
- Create a class named
**Connect**(seems that the name has to be Connect) which implements the**IDTExtensibility2**and**IRibbonExtensibility**interfaces. - Implements Extensibility.IDTExtensibility2, Microsoft.Office.Core.IRibbonExtensibility
- Implement the methods of the interfaces.
- Decorate the class with COM visibility specifications and include functions for creating and deleting custom registry settings.
- Register your assembly with regasm.exe.

Steps 8, 9, and 10 are implemented in the following code snippet, which has been parameterized somewhat.

The functions RegisterFunction and UnregisterFunction are invoked by utilities like regasm.exe which comes with the .NET Framework. I have created a batch file which I run in order to register the assembly. You will need to modify it to suit your assembly.

Note that once the Add-in is registered and added, you will not need to reregister it after every Build (while developing).

## Saturday, August 11, 2012

### DXFert: An Example VBA Add-In with Custom Ribbon

I implemented this project in Excel 2010. Only the Ribbon modifications are likely to be version dependent (unless you go way, way back). The form and macro in this project should work in any version of Excel.

To use the macro launchmyForm(control as IRibbonControl)without the Ribbon buttons, such as you would need to do in an earlier version of Excel, you will need to either change

launchmyForm(control As IRibbonControl) to launchmyForm()

or

write another macro that calls launchmyForm(someDummyObject).

You can download DXFert here to see an example (my previous post explains how to install an Excel Add-In in Excel 2010). In this post, I am going to walk through the building blocks of the project. Except for Excel, everything you need for this project is

**freely**available for download.**What You Will Need to Develop**

**Your Own Similar Project**

- Excel 2007 or later.
- DXF file format documentation from AutoDesk
- CustomUI Editor Tool from Open XML Developer
- A program to allow you to test your DXF output (if that's what you're going for), such as DWG TrueView.

**Overview**

You need to

- create a UserForm within a macro-enabled workbook (*.xlsm)
- write a one-liner macro that invokes the UserForm
- create a Ribbon interface which references the one-liner macro, and, finally,
- save the project as an Add-In (*.xlam) and activate it.

**Creating the UserForm**

If you don't have the developer tab available to you,...why ever not? Turn the silly thing on! (See here for instructions.) To learn about developing user forms in Excel generally, there are lots of examples on the web, but I'll just give you a few key pieces of code that were important in my project.

Here's the code I used to open a file SaveAs dialog to help the user indicate the name and path for the file that the results will be exported:

This should work in very early versions of Excel all the way up to the present (version 2010) and probably future versions as well. My main beef with it is there is a lack of control. I would like it to complain about the selection of a file which already exists, but if I really wanted to I could include that complaint in my macro code right before I follow through with the export (which I didn't do—but

*you*can, if you wish).

When you want to terminate the form (either by cancelling or after successful completion), you'll want use something like this:

You can learn a lot about the DXF format by looking at my code in the download. (Hint: If you have the Add-In added, you don't need to open up the file separately to see the code. Just go to the Visual Basic Editor (button on the developer tab) and locate the user form DXFout_Form. Right-click it and choose View Code.) See page 143 of the AutoDesk's DXF format documentation for a description of the fields in the polyline Entity.

The one-liner code which the Ribbon will call is pretty easy:

The Ribbon expects to pass this control parameter to the subroutines it calls, whether you need to use the parameter or not.

**Create the Ribbon Interface**

I overcomplicated this my first few tries at it. Which is what a lot of people on the internet do. If you want your add-in to have its own ribbon elements, you create those elements using CustomUI Editor (referred to above) within a regular *.xlsm file. When you are ready to finalize it as an Add-In you just save it as an add-in and activate it. The same custom UI that showed up in the original *.xlsm file, will then show up all the time. No fuss, no muss. No Workbook_AddinInstall() code, no SetCustomUI(longComplicatedXMLString), no nothing, it just works! You do have to look at a bit of XML code, mind you.

But first, close down your *.xlsm file and open up CustomUI. Open up your *.xlsm file from within CustomUI. Go to the Insert menu and select Office 2010 Custom Part and, likewise, Office 2007 Custom Part. Now it is time for something ugly. Here is the xml code I used in my project:

For 2010 (customUI14):

For 2007 (customUI):

If you look at the <mso:button> tag you'll notice the launchMyForm bit. That's the name of the procedure that is to be called. One parameter is passed, namely, the a reference to the ribbon control that called the subroutine.

**Save As an Add-In**

At any point, you can turn your *.xlsm project into an Add-In. (You can even continue editing it while it is an Add-In.) Go to File, Save As, and select Excel Add-In from the file

**type**options in the SaveAs dialog box. My previous post talks about installing and activating an Excel Add-In.

## Saturday, August 4, 2012

### DXFert: Creating a DXF file in Excel

DXF files are a means of providing interoperability between AutoCAD and other programs. AutoCAD can read and write *.dxf files and so can a number of other programs. It is fairly straightforward to write a program which outputs a DXF file—provided you just want to draw something simple. DXF files are a great option for auto-generating a bunch of drawings which can then be opened up in a CAD program and edited and saved in that program's native format.

I have written an Excel Add-In, DXFert (say, D-X-F-ert), in VBA which outputs a selection of points as either a 2D or 3D polyline. It has been tested in 2010, but hopefully works in many earlier versions. Here's how to install and use the add-in.

**How to Install DXFert**

- Download DXFert from my google shared document found here.
- Open Excel.
- Go to File → Options
- Select Add-Ins
- Click the
**Go...**button beside Manage |Excel Add-Ins|. - Click
**Browse...**and navigate to the download location of the add-in and select the file (DXFert.xlam). - Click the check box beside DXFert and click OK.

You should now see a DXF tab.

**How to Use DXFert**

Using DXFert is straightforward. First you'll need to create two or three columns of numbers which represent the x, y, (and z) coordinates of the vertices of the would-be polyline. Having so done:

- Optionally, pre-select the cells (headings, if any, not included) which contain the data you want to export as a polyline.
- Go to DXF → Export to DXF.
- In the dialog, select whether you want to export as 2D or 3D polyline.
- If you have not already selected the relevant cells, click the button beside the cell range input box and select the cells (when you have made your selection, you need to click the button on the right hand side of the window that came up for the cell reference selection to get back to the DXFert dialog). Alternatively, type the cell reference. You must choose a simple contiguous selection. I haven't added the smarts to the program to do anything more intelligent or convenient (so far anyway).
- Type a file name to save the exported file as or browse to a file name. (NOTE: DXFert is not careful about overwriting files so make sure you give your file a new name if you don't want to overwrite an existing file.) It is in your interest (normally) to make sure you give the file a .dxf extension (which the Browse... option enforces).
- Click Export. The file will be created silently and the dialog will close.

**How to View DXF Files**

Several programs open DXF files including several AutoDesk programs (like AutoCAD) and some free viewers. AutoDesk has their own free viewer program: DWG TrueView.

## Friday, August 3, 2012

### AutoDesk Design Review 2013 (0x2: Failed to download component.)

I said yes to updating my Design Review (2011) to the current version (2013) and got the above error message. The solution in my case was to download the MSI file. (Note: click on the language you want under the

After unzipping the .zip file you can just run the Setup.exe file. The principle difference is that the downloading has already taken place and the setup program will look at your local or network location for the relevant files to install.

Now, to use Design Review 2013 without AutoCAD installed, you need to install TrueView. If you have a version of AutoCAD installed on your computer which is temporarily without a current license (but it is installed on your computer), you'll need to explicitly tell Design Review to use TrueView.

To do so:

Within Design Review 2013, go to the main drop down menu (big "

→

→ Click on the

→ and check the box beside

→ Click

**Choose Your MSI**column after instruction 2 at the linked location.)After unzipping the .zip file you can just run the Setup.exe file. The principle difference is that the downloading has already taken place and the setup program will look at your local or network location for the relevant files to install.

Now, to use Design Review 2013 without AutoCAD installed, you need to install TrueView. If you have a version of AutoCAD installed on your computer which is temporarily without a current license (but it is installed on your computer), you'll need to explicitly tell Design Review to use TrueView.

To do so:

Within Design Review 2013, go to the main drop down menu (big "

**D**" in top-left)→

**Options**...(at the bottom middle right of drop down)→ Click on the

**DWG Viewing**tab→ and check the box beside

**Open DWG files using DWG TrueView**→ Click

**OK**.## Saturday, July 7, 2012

### Unfolded Approximation to a Hemisphere

The animated GIF featured below are a sort of slide show which give increasing levels of precision in polyhedral approximations to a hemisphere. The SVG one works properly in Google Chrome, but it may not work in all browsers. In theory, you could print these out, cut along the outline, and fold along the internal lines and you would have a polyhedral approximation to a sphere—if you use a lot of tape and your very best dexterity. More precisely, all of the corners would be coincident with the hemisphere being approximated. Notice that as the number of fold lines (latitudinal-ish lines) increases, for a given number of segments (longitudinal-ish divisions) the extent of the unfolded approximation approaches the grey line, which (probably) represents the limiting extent of these approximations. Specifically, this circle has radius of πr/2 or one fourth the circumference of the approximated sphere.

I wrote the code which generates this sequence of pictures in Maxima, a computer algebra system. The code is below and you can modify the range variable near the end to change the level of precision in the resulting pictures. The pattern for each range item is

The "segments" are longitudinal segments like an orange. The latitude divisions are the number of latitude divisions in each of the segments.

I wrote the code which generates this sequence of pictures in Maxima, a computer algebra system. The code is below and you can modify the range variable near the end to change the level of precision in the resulting pictures. The pattern for each range item is

`[number of segments, [starting latitude divisions, ending latitude divisions]]`The "segments" are longitudinal segments like an orange. The latitude divisions are the number of latitude divisions in each of the segments.

## Thursday, June 28, 2012

### Mathematics: Principles and Formulae

I gather that when most people think about mathematics, they think about two things, and in this order: numbers and formulas. In doing mathematics, most students are quick to add something else, though not by name: recipes – usually grossly oversimplified recipes. It is this tendency that runs them into so much trouble and many are the able teachers of mathematics that have tried to hand slap them (figuratively speaking) into using something better, namely,

A famous work of Sir Isaac Newton was titled, in Latin, Philosophiæ Naturalis Principia Mathematica ("Mathematical

A recipe is a sequence of steps or components with no (particular) reference to reasons. Computers do recipes - algorithms. Students that try to be computers (and nothing more) short-change themselves because there is already a much more efficient tool for that work than themselves. (And thus it is little wonder they complain about the pointlessness of the learning they are engaged in. Often of their own accord, they are not learning the really important lessons from mathematics that they should be.) Principles help me to justify, or at least partly understand, the steps of the recipe and apply it usefully. Principles are one of the key advantages you have over your calculator.

People try to simplify the setting up and solving of proportion problems. They do it this way: "cross-multiply and divide." They don't really know why. It is only a recipe to them. How do you know which numbers to multiply and divide? It increases the number of things to memorize and (I have observed) it distracts people from the real principles. Forget about cross-multiplying and dividing. If I know

Here are a few very generic principles to apply to problem solving:

**principles**.A famous work of Sir Isaac Newton was titled, in Latin, Philosophiæ Naturalis Principia Mathematica ("Mathematical

**Principles**of Natural Philosophy"). But you don't need to be as smart as Sir Isaac Newton to understand the importance of principles as superior to recipes. First of all, what's the difference?A recipe is a sequence of steps or components with no (particular) reference to reasons. Computers do recipes - algorithms. Students that try to be computers (and nothing more) short-change themselves because there is already a much more efficient tool for that work than themselves. (And thus it is little wonder they complain about the pointlessness of the learning they are engaged in. Often of their own accord, they are not learning the really important lessons from mathematics that they should be.) Principles help me to justify, or at least partly understand, the steps of the recipe and apply it usefully. Principles are one of the key advantages you have over your calculator.

People try to simplify the setting up and solving of proportion problems. They do it this way: "cross-multiply and divide." They don't really know why. It is only a recipe to them. How do you know which numbers to multiply and divide? It increases the number of things to memorize and (I have observed) it distracts people from the real principles. Forget about cross-multiplying and dividing. If I know

*a/b = c/d*, I do what I always do with equations: do "opposites" and do the same thing to both sides (subject to the constraint that the operation must be permissible on the values in the equation). If I want to isolate*a*, I have*a***divided by***b*, so I do the opposite of**divide by***b*, namely, I**multiply by***b*, and I do it to**both sides**. The end. (I would love to know why anybody thinks "cross-multiply and divide" is easier than just using basic principles *sarcasm*.) "Cross-multiple and divide" doesn't bridge well into solving other forms of equations, is unnecessary, gives little clue as to why it is supposed to work, and increases the number of things to memorize. How do you solve an equation?- Do "opposites."
- Do the same thing to both sides.
- Eliminate any apparent solutions which do not satisfy the original equations or the constraints of the problem (which may be implicit, such as, "negative time is not allowed").

Throw "cross-multiply and divide" and its ilk into the garbage bin.

Here are a few very generic principles to apply to problem solving:

**Don't ask "What should I do?", ask "What is true?"**You won't make very much ground figuring out what to do if you haven't established a few things that you know about the situation. This often starts with writing down some given information. Perhaps you print a few formulas and consider which variables you know and which you don't. Ask yourself, "Do I know any other equations that involve these unknowns? Are they valid in this situation?" Don't worry about whether an equation is going to give you the final answer, only concern yourself with whether it actually holds true in the situation. Thomas Edison didn't invent so many things by only trying things he knew would work. You also will write down formulas and find that they do not help you solve the problem – even though they are true and valid in the situation, they might not be useful.**Don't ask "Why can't I do this?", ask, "Can this method or statement be proven correct?"**People want to apply rules that are applicable to one type of operation to a completely different operation. They see that they are allowed to make the statement*a (b + c) = a b + a c*and want to know why they can't say log(*b + c*) = log(*b*) + log(*c*). This is one or both of two fundamental misunderstandings. Either it is assuming that the parenthesis have the same meaning in both cases (which means they don't know what a function is and certainly not what a log is) or they misunderstand the importance of proof. This later misunderstanding in particularly important. Often the student assumes that a similar appearance means a similar treatment is permitted. The fact is, the statement*a (b + c) = a b + a c*, where*a, b,*and*c*are numbers can be proven. Our acceptance of the statement is based on proof from simpler principles. We don't "just know" that that's true (although this example is among the more intuitive that I could have chosen). Similarly, we don't "just know" that the statement log(*b + c*) = log(*b*) + log(*c*) is true. If we can*prove*somehow that it is true, then we know we're allowed to make this expansion. On the other hand, if we can find a counter-example, we know it is not true. So here's a counter-example to the above (very silly) suggestion: log(10 + 10) = log 20 < 2, but log(10) + log(10) = 1 + 1 = 2; so they are clearly not equal.*This doesn't mean you need to*do*the proof, but you should be satisfied that a proof exists and that you are not assuming something which cannot be established.***Look for patterns.**Patterns have some kind of rule behind them. If you can determine the rule that makes the pattern, it may reduce work and memorization required. The rule will be more generic than the sampling that you noticed the pattern in – otherwise, it is a false pattern, because a counter-example exists. Don't assume the pattern is real. Test it, prove it, disprove it, as may be necessary.**Make analogies.**The main reason for teaching the principle of proportional triangles is not to teach students how to solve triangle problems – although there's good uses for that. Proportions are so common place that everyone should get comfortable with the archetypical example: proportional triangles. There are analogies between force and momentum, between torque and force, momentum and rotational inertia, fluid pressure and electricity. Some things are more intuitive to us than others and if an analogy exists between something you are familiar with and something you are not, it can help you with a "working understanding" of the unfamiliar concept.

## Monday, May 28, 2012

### "Next!": chapter 1

*Next:*

*(adj.) coming immediately after the time of speaking or writing (temporal)**(adj.) immediately adjacent (proximity)*

The first hundred whole numbers all decided to get their hair cut the same day at the same barber. Since they were already numbers, they were not required to take one. (It's considered offensive in their culture.) Mr. Ordinal, the owner and only barber on duty at that early hour of the day, announced that they would all be served in their natural order. One Hundred sighed.

I should tell you that odds and evens often have very different expectations and requirements for their hair cuts and so they are often serviced by barbers who specialize in one of the other. But Mr. Ordinal was a professional of the highest order and took each client in sequence. On the other hand, Mr. Ordinal had not counted on having such a prodigious quantity of clients in his waiting room and so he phoned up two of his specialist barbers to ask them to come in as soon as possible. He got their answering machine and only hoped that they didn't have negative feelings about coming into work on such short notice.

Now whole numbers are such naturals at following each other in turn that it was not necessary for Mr. Ordinal to announce the name of the next client. When he was done with One, he simply called out, "Next!" and Two's eyes lit up as he marched over to the chair. Mr. Ordinal was no stranger to hard work, but by the time he had finished with Fifteen, he was becoming anxious for help from his specialists. He exhaled loudly and, as if on cue, Maud and Steven stepped in.

"I'm so relieved to have you here!" Mr. Ordinal said with a sense of priority. "Now we can really get some work done." Much more quietly he added, "And you can take Seventeen, Maud. She can be such a prima donna..."

While Steven took Sixteen and Maud took Seventeen, Mr. Ordinal took 5 (minutes that is) to visit with Johnny, his nephew who had just walked in for his first day of work. He had offered to let his nephew manage the till that summer for minimum wage. He had reasoned to his nephew, saying, "It isn't a lot of money, I know, but it's a good way to learn to interact with a variety of people. It'll serve you well in the future."

"I had hoped you would be here at opening at 8:00 o'clock," Mr. Ordinal said. Johnny's slightly widened eyes plead ignorance for him. "Oh, well, we didn't really discuss hours properly yet. We'll figure it out before day's end." And so, Johnny began to man the till and call clients when it was their turn. Without being told, he quickly learned that the numbers didn't mind if you just said, "Next." They understood perfectly well what it meant.

But, mind you, there was now an added level of complexity. Mr. Ordinal could take everyone in sequence, but Steven and Maud specialized. But Johnny soon had the hang of it. If it was Mr. Ordinal's turn to take a new client, he would call, "Next," and the lowest number not already served or being served would step over to the chair. If Steven was ready for another client he would say, "Next even," and for Maud, "Next odd," and the predictable would happen. The lowest even or odd number, not already served or being served, would step into the available chair.

At last, Ninety Nine and One Hundred were getting their hair cut. Overall, Mr. Ordinal enjoyed his job, but to listen to these last two clients of the day complaining about their comrades was almost insufferable.

Ninety Nine would say, "You know the problem I always have with One is that when I call him a second time, he always get's this puzzled look on his face, as if maybe I might mean some other One."

"Well, what about Two? He's always moping about being the first loser," One Hundred said, derisively.

Closing up shop that evening, Johnny and his uncle swept and put things away and got ready to lock the doors. "Well then, will it be 8:00 o'clock tomorrow?" his uncle said, not unkindly.

"Yes, sir, uncle O," Johnny said with all the enthusiasm the wear of the long day would permit.

Mr. Ordinal seemed to think some sort of closing remark was necessary to end the day. "You know it was a long day, but I'll say this for our clients today, they were very orderly. It makes things go so much smoother." He took a deep breath and smiled, but strangely. "You should see what the

His uncle's last remark left Johnny a bit puzzled. It sounded on the surface like a knowing remark about the sorts of things that may happen throughout the summer. Or perhaps it was a way of saying, "You'll find out if you like this sort of work after you've had more experience." And yet, to hear Uncle Ordinal say it just so as he did, it seemed more cryptic than that.

"I'm so relieved to have you here!" Mr. Ordinal said with a sense of priority. "Now we can really get some work done." Much more quietly he added, "And you can take Seventeen, Maud. She can be such a prima donna..."

While Steven took Sixteen and Maud took Seventeen, Mr. Ordinal took 5 (minutes that is) to visit with Johnny, his nephew who had just walked in for his first day of work. He had offered to let his nephew manage the till that summer for minimum wage. He had reasoned to his nephew, saying, "It isn't a lot of money, I know, but it's a good way to learn to interact with a variety of people. It'll serve you well in the future."

"I had hoped you would be here at opening at 8:00 o'clock," Mr. Ordinal said. Johnny's slightly widened eyes plead ignorance for him. "Oh, well, we didn't really discuss hours properly yet. We'll figure it out before day's end." And so, Johnny began to man the till and call clients when it was their turn. Without being told, he quickly learned that the numbers didn't mind if you just said, "Next." They understood perfectly well what it meant.

But, mind you, there was now an added level of complexity. Mr. Ordinal could take everyone in sequence, but Steven and Maud specialized. But Johnny soon had the hang of it. If it was Mr. Ordinal's turn to take a new client, he would call, "Next," and the lowest number not already served or being served would step over to the chair. If Steven was ready for another client he would say, "Next even," and for Maud, "Next odd," and the predictable would happen. The lowest even or odd number, not already served or being served, would step into the available chair.

At last, Ninety Nine and One Hundred were getting their hair cut. Overall, Mr. Ordinal enjoyed his job, but to listen to these last two clients of the day complaining about their comrades was almost insufferable.

Ninety Nine would say, "You know the problem I always have with One is that when I call him a second time, he always get's this puzzled look on his face, as if maybe I might mean some other One."

"Well, what about Two? He's always moping about being the first loser," One Hundred said, derisively.

Closing up shop that evening, Johnny and his uncle swept and put things away and got ready to lock the doors. "Well then, will it be 8:00 o'clock tomorrow?" his uncle said, not unkindly.

"Yes, sir, uncle O," Johnny said with all the enthusiasm the wear of the long day would permit.

Mr. Ordinal seemed to think some sort of closing remark was necessary to end the day. "You know it was a long day, but I'll say this for our clients today, they were very orderly. It makes things go so much smoother." He took a deep breath and smiled, but strangely. "You should see what the

*days*are like."His uncle's last remark left Johnny a bit puzzled. It sounded on the surface like a knowing remark about the sorts of things that may happen throughout the summer. Or perhaps it was a way of saying, "You'll find out if you like this sort of work after you've had more experience." And yet, to hear Uncle Ordinal say it just so as he did, it seemed more cryptic than that.

## Sunday, May 20, 2012

### Herb Drying: Take 2

Last August/September I was sun-drying herbs, and started by putting them in my car in the sun while I worked. I gave a hint about another idea; that is, using reflection of sunlight and 6 mil poly to retain some of the heat. This way, you don't have the car intermittently occupied by silent, aromatic, green passengers all herb harvesting season. What I didn't tell you yet, is that I have already begun implementing such a plan and it works very well. It's actually not very expensive to build.

Before I show you the pictures I should tell you that you don't have to make a reflector as interesting as mine to make the plan work. Truthfully, I'm not making optimal use of my parabolic cylinder reflector. (I would need to track the sun to do so, which I have often wanted to do, but haven't made the time to research and source the needed parts for any of my unwritten, half-baked designs.) A three part reflector is probably more than adequate: orient the three parts so that you get (1) the mid to late morning sun, (2) the afternoon sun, and (3) the evening sun.

Options for reflectors include:

The more surface area you allow to be fully exposed, the better, which is why I am using three dishes when just one would do.

This is the unattractive monster which dries my herbs:

If I were picky, I would replace the hideous stained 6 mil poly. I could also use thinner poly which would be more translucent. In a way, this low translucence is good as it reduces the likelihood of UV reflection into people's eyes. From the solar cooking perspective, the low translucence is a limitation to your peak temperature and that is bad. But for drying herbs, you don't want it too warm anyway.

Here is my reflector and screen box.

The reflector is a parabolic cylinder covered with part of an "emergency blanket" (also called "space blankets"). It reflects a high percentage and broad spectrum of solar radiation. To stabilize the reflector, I drove a wood stake about a foot behind the reflector and screwed another scrap both to the back of the reflector and to the stake. The screen box is nothing special; I made it out of 2×2s (ripped 2×4s) and nylon screen. It simply allows air to flow through. My original idea was to put some of the derelict glass I have at my disposal over the screen box to protect from rain, bird droppings, and wind-blown debris organic or otherwise. But the reflector construction is susceptible to rain damage (the glue and Masonite in particular) and I wanted to benefit from a greenhouse effect as well as reflection. So, I poly over it. Here is the vegetation placed in my box:

If I have a lot of vegetation to deal with, I just spread it out as best I can, even without dishes inside the box. I usually give it a daily stir if I have time. This batch was almost done in one day, easily in two. Much larger batches have been done in three days. (If that sounds long, bear in mind that that's three days of paying hardly any attention to it.) Here I've tucked the poly under the box:

Note that the sides are somewhat open to allow moisture to escape. We actually have a few things in our favour to get the moisture out of those herbs:

I often put a bunch of herb in a sandwich bag, leaving it open, and scrunch it up inside the bag. This reduces loss of material and increases hygiene. Seal the bag when done, or transfer to another container.

Removing the dehydrated herbs from the dryer is best done prior to the setting in of dew in the evening - late afternoon or early evening. If you wait too long, equilibrium will demand the dry herbs to take on moisture which will require them to be re-dried. Just check how crinkly they are.

Before your next fill of the screen box it may be wise to vacuum it, especially if you are changing herbs. I find I have to vacuum occasionally since the plant material likes to hook on to the screen in places.

This would make an excellent project for someone looking to get their feet wet in the pursuit of solar cooking, especially if they have a garden and like to have summer herbs for free all winter long!

Before I show you the pictures I should tell you that you don't have to make a reflector as interesting as mine to make the plan work. Truthfully, I'm not making optimal use of my parabolic cylinder reflector. (I would need to track the sun to do so, which I have often wanted to do, but haven't made the time to research and source the needed parts for any of my unwritten, half-baked designs.) A three part reflector is probably more than adequate: orient the three parts so that you get (1) the mid to late morning sun, (2) the afternoon sun, and (3) the evening sun.

Options for reflectors include:

- shiny metal - excellent durability
- aluminium foil - inexpensive and easily obtainable, a little crinkly which results in poorer focus
- emergency blankets - slightly more expensive than foil, but smoother and shinier

I use option 3. My local Heritage Co-op grocery store (in Brandon, MB) often has them in stock along with a limited selection of other camping/survival supplies.

The last time I dried herbs, I took a few pictures. Here are some chives freshly picked and washed:

The more surface area you allow to be fully exposed, the better, which is why I am using three dishes when just one would do.

This is the unattractive monster which dries my herbs:

If I were picky, I would replace the hideous stained 6 mil poly. I could also use thinner poly which would be more translucent. In a way, this low translucence is good as it reduces the likelihood of UV reflection into people's eyes. From the solar cooking perspective, the low translucence is a limitation to your peak temperature and that is bad. But for drying herbs, you don't want it too warm anyway.

Here is my reflector and screen box.

The reflector is a parabolic cylinder covered with part of an "emergency blanket" (also called "space blankets"). It reflects a high percentage and broad spectrum of solar radiation. To stabilize the reflector, I drove a wood stake about a foot behind the reflector and screwed another scrap both to the back of the reflector and to the stake. The screen box is nothing special; I made it out of 2×2s (ripped 2×4s) and nylon screen. It simply allows air to flow through. My original idea was to put some of the derelict glass I have at my disposal over the screen box to protect from rain, bird droppings, and wind-blown debris organic or otherwise. But the reflector construction is susceptible to rain damage (the glue and Masonite in particular) and I wanted to benefit from a greenhouse effect as well as reflection. So, I poly over it. Here is the vegetation placed in my box:

If I have a lot of vegetation to deal with, I just spread it out as best I can, even without dishes inside the box. I usually give it a daily stir if I have time. This batch was almost done in one day, easily in two. Much larger batches have been done in three days. (If that sounds long, bear in mind that that's three days of paying hardly any attention to it.) Here I've tucked the poly under the box:

Note that the sides are somewhat open to allow moisture to escape. We actually have a few things in our favour to get the moisture out of those herbs:

- We are adding heat to the system which satisfies the energy requirements for evaporation.
- The temperature of the air inside the poly is increased which increases its capacity to hold moisture. By increasing the capacity of the air, it changes the equilibrium distribution of moisture (between the air inside the poly and the herbs). It's a bit like increasing the voltage of a circuit. The potential rate of evaporation is increased. (This, of course, would be largely useless without a means for the increased moisture content to exit the system.)
- As the air in the poly gets more moist, it wants to migrate. The moisture wants to go from moist to dry. It's like the tea I like to drink so much of. The contents of the tea leaves want to distribute themselves as evenly as possible. The same is true of moisture. (Although there's a sticky point with the relative capacities of the air inside and outside of the poly to hold moisture. But don't worry about that. Bottom line: we have vapour pressure taking moisture out through any opening it can find, provided the outdoor climate provides a suitable drying environment.)
- The ventilation can be increased by making sure there is an opening near the top, but in any case, higher than the herbs. (Determining the optimum opening sizes is probably non-trivial.) The hotter air inside the poly is less dense than that outside and so it rises up through the relatively cooler air outside. If there is no opening near the top, the effect of heat migration will be less effective for creating ventilation. Although the heat will still move from hot to cold and the hot air will still rise, the hot air at the top will transfer heat to the poly which will give off the heat on the outside air. So, if the openings were exclusively at the bottom of the system, the heat would still be transferred (at some rate) outside of the system, but the hot air itself (which contains our unwanted moisture) would not be exiting the system as a direct result of heating (although it may exit due to other factors or effects, such as the wind effect which creates pressure differentials).

Here is the finished product:

I often put a bunch of herb in a sandwich bag, leaving it open, and scrunch it up inside the bag. This reduces loss of material and increases hygiene. Seal the bag when done, or transfer to another container.

Removing the dehydrated herbs from the dryer is best done prior to the setting in of dew in the evening - late afternoon or early evening. If you wait too long, equilibrium will demand the dry herbs to take on moisture which will require them to be re-dried. Just check how crinkly they are.

Before your next fill of the screen box it may be wise to vacuum it, especially if you are changing herbs. I find I have to vacuum occasionally since the plant material likes to hook on to the screen in places.

This would make an excellent project for someone looking to get their feet wet in the pursuit of solar cooking, especially if they have a garden and like to have summer herbs for free all winter long!

## Friday, April 27, 2012

### Angle Between Intersecting Planes

In a practical sense, there are a few different ways that you can go about determining the angle between two planes. But before we get to specific methods and tools, let's try to lay the ground work
– let's try to understand what we really mean by an angle between two planes.

Two non-parallel planes intersect at a line. The angle between intersecting planes is measured perpendicularly to this line of intersection. In more detail, we can say, for planes P

We note in passing that there is no need for the two arms of the angle to have any certain length nor is there a need for the lengths to be constrained by any relationship concerning relative lengths – they may differ. Items 1 and 2 (above) should come as no surprise, but not everyone will immediately understand item 3. Indeed, if you consider only the two trivial cases – (i) two coincident planes (technically excluded from discussion here) or (ii) perpendicular planes – item 3 appears superfluous. To see that it is necessary, consider planes which meet at a 45° angle (as an example). Imagine putting a 45-45-90 triangle in place so that two of its edges are in contact with a plane each. Suppose you made one of those planes rest on the top of the triangle and allowed that plane to rotate freely around the line of intersection (L). Now, keeping the vertex of the triangle (the vertex on the line of intersection) in place and rotating the triangle around that point, you will find that the plane which is resting on the triangle will lift up – the angle between the planes will increase. By the time you have rotated the triangle so that the vertical edge is against the "resting" plane, the angle between the planes will be 90°.

Here is a video where I demonstrate this scenario:

In the above scenario we used the same triangle at different angles and changed the angle between the planes. Now let's leave the planes be and change the triangle. Let's draw the orthographic views of planes (represented as rectangles) intersecting at angle

I've been lazy in my drawing as the angle for β that I really want is in the plane of the triangle that's on an angle and not the angle of the orthographic projection of that triangle (which is what I have just drawn). But bear with me. The angle of intersection is given as

You can obtain a rise and run for β without a scientific calculator by using a little measurement and geometry. Suppose you are able to obtain a rise and run for each of angles

rise of β = (rise of θ) × (rise of

run of β = (hypotenuse of θ) × (run of

**Definition: Angle Between Intersecting Planes**Two non-parallel planes intersect at a line. The angle between intersecting planes is measured perpendicularly to this line of intersection. In more detail, we can say, for planes P

_{1}and P_{2}with line of intersection L:- The vertex of the angle between the planes is any given point on the line of intersection (that is, just pick one).
- One arm of the angle is coincident with P
_{1}and the other arm is coincident with P_{2.} - Both arms of the angle are perpendicular to L.

**Discussion of Definition**

We note in passing that there is no need for the two arms of the angle to have any certain length nor is there a need for the lengths to be constrained by any relationship concerning relative lengths – they may differ. Items 1 and 2 (above) should come as no surprise, but not everyone will immediately understand item 3. Indeed, if you consider only the two trivial cases – (i) two coincident planes (technically excluded from discussion here) or (ii) perpendicular planes – item 3 appears superfluous. To see that it is necessary, consider planes which meet at a 45° angle (as an example). Imagine putting a 45-45-90 triangle in place so that two of its edges are in contact with a plane each. Suppose you made one of those planes rest on the top of the triangle and allowed that plane to rotate freely around the line of intersection (L). Now, keeping the vertex of the triangle (the vertex on the line of intersection) in place and rotating the triangle around that point, you will find that the plane which is resting on the triangle will lift up – the angle between the planes will increase. By the time you have rotated the triangle so that the vertical edge is against the "resting" plane, the angle between the planes will be 90°.

Here is a video where I demonstrate this scenario:

**Perpendicular Angle Between Planes vs. Non-perpendicular Angle Between Planes**

**(Criteria 1, 2, and 3) vs. (Criteria 1 and 2)**

In the above scenario we used the same triangle at different angles and changed the angle between the planes. Now let's leave the planes be and change the triangle. Let's draw the orthographic views of planes (represented as rectangles) intersecting at angle

*i*, according to our criteria and draw two triangles in place: one at 90° to the line of intersection and one at θ to the line of intersection.

I've been lazy in my drawing as the angle for β that I really want is in the plane of the triangle that's on an angle and not the angle of the orthographic projection of that triangle (which is what I have just drawn). But bear with me. The angle of intersection is given as

*i*and we take the dimensions L and θ to be given. From this information we can find formulae for R, h, and β. Here are some formulae we can take directly off of the drawing:

\[\sin \theta = \frac{R}{L}\] \[\tan \beta = \frac{h}{L}\] \[\tan i = \frac{h}{R}\]

Manipulating these equations gives\[R = L \sin \theta \] \[h = L \sin \theta \tan i \] \[\tan \beta = \sin \theta \tan i \]

Observe that β ≠*i*.

**Calculations Without a Scientific Calculator**

You can obtain a rise and run for β without a scientific calculator by using a little measurement and geometry. Suppose you are able to obtain a rise and run for each of angles

*i*and θ. You might do this by measuring either a shop drawing or a physical situation. Choose the run arbitrarily and measure the rise which corresponds to it. The hypotenuse (for θ) can either be measured or calculated using the Pythagorean theorem. The formulae for the rise and run of β are simply

rise of β = (rise of θ) × (rise of

*i*)

run of β = (hypotenuse of θ) × (run of

*i*)

## Saturday, March 31, 2012

### Defining a User Coordinate System

In AutoCAD you have User Coordinate Systems (UCS) and the World Coordinate System (WCS). The WCS is the system which your model information is stored in. The UCS is what you draw in. When you first begin a drawing, the current UCS is the same as the WCS.

First, let's familiarize ourselves with how the WCS looks. I've drawn a pyramid in AutoCAD and shown it from several different angles:

The WCS is displayed in all four pictures (and, naturally, is exactly the same in each picture). One of the techniques for dealing with 3D when you are "doing the math" (perhaps in your own code) and want to make sure the math (or code) you're writing matches the geometry you're working with is known as "the right hand rule." (We're going to use this for some math coming up in a few paragraphs.) For each picture, look at the WCS and orient your

In math notation, we might write the relationship above as something like

Two of the best ways to change the UCS in AutoCAD are

To obtain consistent definitions for these axes we use the cross product (which I tried to sneak in earlier:

where

Here's how a UCS is defined based on three points. The first point (P

Depending on the use of your UCS (if you are programming one in your own software) you may wish to normalize the direction vectors. Normalized vectors (which have length equal to unity) have some useful properties which may save some computation in later calculations. Normalizing is straightforward and simply requires you to divide the value of each component of the vector by the current length of the vector.

First, let's familiarize ourselves with how the WCS looks. I've drawn a pyramid in AutoCAD and shown it from several different angles:

The WCS is displayed in all four pictures (and, naturally, is exactly the same in each picture). One of the techniques for dealing with 3D when you are "doing the math" (perhaps in your own code) and want to make sure the math (or code) you're writing matches the geometry you're working with is known as "the right hand rule." (We're going to use this for some math coming up in a few paragraphs.) For each picture, look at the WCS and orient your

**right**hand (which should be open face initially) so that your fingers are pointing in the direction of the x-axis and your palm is open toward the direction of the y-axis. (Don't think about your arm, just your hand. The arm will just go wherever it needs to go, though it may look funny to onlookers.) Curl your fingers into your palm and stick your thumb out. If you've done it correctly, your thumb should be pointing in the direction of the z-axis. If you can do this with all four pictures, you're well on your way to being able to understand the rest of the math in this post.In math notation, we might write the relationship above as something like

**z**=**x**×**y**. The relationships which apply to the WCS (and all UCSs) are**x**=**y**×**z****y**=**z**×**x****z**=**x**×**y**

Two of the best ways to change the UCS in AutoCAD are

- Rotate around a chosen axis: this is one of the more intuitive ways to adjust the current UCS when you don't have "pickable points" in the plane you want the new xy-plane to be in.
- Pick three points: this is a way to put the current UCS in line with a face or in line with somewhere you want to put a face. Lining up with an existing face of a 3D drawing only requires to you to pick three of the corners of the face. Depending on the order you pick those points in, you will get different directions for the axes,
*but*the xy-plane of the new UCS*will*coincide with the three points you pick.

To obtain consistent definitions for these axes we use the cross product (which I tried to sneak in earlier:

**z**=**x**×**y**). Here's the definition of the cross product:**×***a***b**= (*a*_{2}b_{3}- b_{2}*a*_{3},*a*_{3}b_{1}- b_{3}*a*_{1},*a*_{1}b_{2}-*a*_{2}b_{1}),where

**= (***a**a*_{1},*a*_{2},*a*_{3}) and**b**= (b_{1}, b_{2}, b_{3}). Think about these vectors as direction vectors as opposed to points. One of the important properties of the cross product is that it produces a vector which is perpendicular to both of the vectors that form the product. That is,**×***a***b**is perpendicular to**and to***a***b**. To determine which way**×***a***b**is pointing, point your fingers (using your right hand) in the direction of*with your palm facing in a direction such that you can curl your fingers toward***a****b**. Your thumb is pointing in the direction of**×***a***b**.Here's how a UCS is defined based on three points. The first point (P

_{0}) indicates the position of the new origin. The second point (P_{1}) defines the direction from the first point the x-axis goes in. So, the direction of the new x-axis is**x**= P_{1}− P_{0}. The third point (P_{2}) determines not only the plane, but also which side of the x-axis the y-axis will go (within the defined plane). At the same time, we are also defining, albeit indirectly, the direction of the z-axis by means of the right hand rule. We define our UCS accordingly:- Origin: P
_{0} - x-axis:
**x**= P_{1}− P_{0} - z-axis:
**z**=**x**× (P_{2}− P_{0}) - y-axis:
**y**=**z**×**x**

**y**. Also observe that the order in which you select the points affects which way your axes are oriented, even though it will not affect what plane is the xy-plane of your new UCS. Given any three points, there are three possible origins - so I need to specify which is the origin. With that selected, I have two options for which will indicate the direction of my x-axis. So, there are six possible UCSs given the same three points in different orders.Depending on the use of your UCS (if you are programming one in your own software) you may wish to normalize the direction vectors. Normalized vectors (which have length equal to unity) have some useful properties which may save some computation in later calculations. Normalizing is straightforward and simply requires you to divide the value of each component of the vector by the current length of the vector.

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