What is it?
This geocoin contains a circular slide rule. The slide rule can be used to for calculations. It might be surprising, but the simplest operations, additions and subtractions, cannot be performed on this slide rule. This actually holds for most slide rules. But this slide rule can be used to do multiplication and division and even to calculate sine and cosine functions.
Sierpinski’s triangle
The triangle in the middle of the rotating disk of the coin, shows Sierpinski’s triangle. This is will not help you in any calculations but shows a part of an infinite repetition in which a triangle is regularly divided in smaller triangles and so on and so forth. More information, drawings and animations of Sierpinski triangle can be found at Wikipedia.
Technetium
In the most central triangle you see some circles. These illustrate the electron configuration of the element Technetium (Tc). There are 5 circles, showing the 5 orbitals for the electrons around the nucleus of a Technetium atom. Technetium is lightest element, atomic number 43, which has no stables isotopes. Some isotopes have a half-life of less than hour while the most stable ones have a half life of more than a million years. When an atom Technetium decays, it emits an electron (beta-decay) or a neutrino (electron capture). These emission rays are shown on the coin by rays you see in Sierpinski triangle. Again no help in doing calculations, just an/the element of the coin designer 
. More information on Technetium and more information on the isotopes and their decay.
Back side of the coin: Smith chart
On the other side of the coin you see all kinds of curves drawn. This is a Smith chart. A Smith chart is used in electrical engineering to determine things like impedances, reflections, noise figures etc. of transmission lines. The Smith chart on the coin doesn’t have any numbers on it, so it cannot be used, unless you know the numbers by heart . Anyway, more information on Smith charts and how to use them can be found on the Internet.
How to use this coin?
Multiplication
First the multiplication. For this we’ll use the outer fixed scale of the coin and the outermost scale on the rotating disk.
Now, let’s start with an example: the calculation of 3×2. Align the 1 of the inner scale (rotating disk) to the 2 of the outer scale. The answer to 3×2 can now be found be looking at where the 3 of the inner scale lines up with the outer scale. It turns out the 3 of the inner scale aligns with ... drum roll ... 6 on the outer scale! So, there you have it, 3×2 is 6. Now also look at the number of the outer scale which aligns with the 4 on the inner scale... It is the 8, so 4×2 is 8. Just be shifting the 1 of the inner scale to the 2 of the outer scale, we simply read all the multiples of 2. Isn’t this a great tool...
Now before you get all excited about this cool calculator, it is good to know there are limitations and you actually also need to think a little yourself when using the circular slide rule. For example, let’s try to work out 6×4. To do this we need to put the 1 of the inner scale to the 4 of the outer scale and then check where the 6 of the inner scale aligns with the outer scale. The result is 2.4 (note that there are ten marks between 2 and 2.5, so by each mark the increment is 0.05, so the 8-th mark equals to 2.4). Now wait a minute 6×4 equals 24 and not 2.4, what happened? Well this where the you-need- to-think-yourself-part comes in. A limitation of the slide rule is that you need to know the order of magnitude of the outcome. Or to put simple, you need to know the position of the decimal point in the outcome. So the slide rule tells you it is 6×4 is 2.4, it is up to you to realize that the correct answer is ten times larger, so 24.
Okay, now that the basics are known, let’s try calculating with some larger numbers. We are going to calculate 25×23. Here we have a small problem, normally you would move the 1 of the inner scale to the 23 on the outer scale, but there is no 23 on the outer scale. Now remember that it is up to you to keep track of the position of the decimal point, so to calculate 25×23 we will calculate 2.5 × 2.3. However we need to remember that the result needs to be multiplied by 100, since we divided both 25 and 23 by ten.
Now to calculate we move the 1 of the inner scale to the 2.3 of the outer scale and we find our wanted result where the 2.5 of the inner scale align with the outer scale. What’s the result? Well it is between 5.7 and 5.8. So the slide rule tells us the result of 25×23 is somewhere between 570 and 580. Using your own brain power (or an electronic calculator ) gives the result of 25×23, which is 575. So here we are limited by the accuracy of the scales on the slide rule or our own ability to read the scale. Anyway we do not get the exact answer.
Division
Of course when multiplication can be done with the help of this coin, division must be possible as well!
Let’s start simple again and try to calculate 6/3. Align the 3 of the inner scale to the 6 of the outer scale. Check where the 1 of the inner scale aligns with the outer scale, this is at 2, the correct answer. Why is this correct? We know that the outcome of 6/3, multiplied by 3 has to be 6 again. So, what times 3 equals 6? Remember from the multiplication that the 1 of the inner scale was put at 2 and then the 3 of the inner scale aligned with the outer scale at 3×2. So to find 6/3 we indeed need to put the 3 of the inner scale to the 6 of the outer scale and read where the 1 of the inner scale aligns with the outer scale, exactly what we did in the division. The position in which the slide rule is after the division, will directly show you the multiplication you would need to do, to verify the division.
Sine function
For finding the sine of a certain angle, all that is needed is the rotating part of the coin. The rotation part contains two scales. The outer one which ranges from 1 to 9 and an inner one that ranges from 0.11 to 1.4 . The inner scale contains the angle from 0.11 radians to 1.4 (pi/2) radians. For those of you more common to angles expressed in degrees: it means from 6.3 to 90 degrees.
Now how to find, for example, the sine of 0.5 radians (28.6 degrees)? Look up the value of 0.5 on the inner scale and read out the corresponding value on the outer scale. The result is 4.8. Again we have to think a little bit ourselves because the sine function is bound by -1 and 1, so the result can’t be 4.8. Well it is that decimal point again, given the limits of the sine function the result has to be 0.48 and not 4.8. So keep in mind you need to divide the result found on the outer scale by ten to get the correct value.
Let’s try one more to get the hang of it, what is sin(0.7)? Looking at 0.7 on the inner scale, shows that it lines up with 6.5 on the outer scale, so sin(0.7) = 0.65. This of course is not an exact answer since we are limited by the accuracy of the scales on the coin.
Now you might wonder what about angles smaller 0.11 radians? How to determine the sine of those angles? Well that is very simple: for angles smaller 0.11 radians, sin(x) = x, or in words: the sine of an angle smaller than 0.11 radians is equal to that angle.
And how about angles larger than pi/2 radians? You can use these rules that are a result of the symmetry of the sine function:
- For an angle x between pi/2 and pi, lookup pi-x.
- For an angle x between pi and 3pi/2, lookup 3pi/2-x and add a minus sign to the result.
- For an angle x between 3pi/2 and 2pi, lookup 2pi-x and add a minus sign to the result.
- For an angle x smaller than 0, add multiples of 2pi until you are in the range for the first 3 rules.
- For an angle x larger than 2pi, subtract multiples of 2pi until you are in the range for the first 3 rules.
Cosine function
The cosine function can also determined using this coin. For this the relation cos(x) = sin(x+pi/2) is used. Suppose we need to know cos(1), this is equal to sin(1+pi/2). The angle 1+pi/2 is larger than pi/2 and smaller than pi, so according to the rules given in the section on the sine function, we need to lookup pi-(1+pi/2) = pi/2 -1. First we calculate pi/2 as shown in the section on divisions, align the 2 of the outer rotating scale to the pi on the fixed scale and read the result where the 1 of the outer rotating scale aligns with the fixed scale. The result is somewhere between 1.55 and 1.60, let’s say 1.57. Now we need to subtract 1 from 1.57, this you have to do yourself, the slide rule cannot really help here, the result is 0.57. In the next step we look up 0.57 on the inner scale of the rotating disk and find that it aligns with 5.4 on the outer scale of the rotating disk. Remember from the section on the sine function we still need to divide this result by ten, given us the final result of 0.54. So cos(1) = 0.54!
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<a href="http://www.geocaching.com/seek/cache_details.aspx?wp=GC2QCKN"><img src="http://img.geocaching.com/cache/large/245225e6-3a4c-4a3a-8f0b-cf93350edae7.jpg" width="86" height="50" alt="Technetium's Slide Rule Geocoin Cache (GC2QCKN)" border="0"/></a> |