Zef Damen Non ruler-and-compass constructions (2)

Non ruler-and-compass construction of odd-numbered regular polygons

From the previous page, this picture remains, showing an interesting characteristic of a nonagon (regular 9-sided polygon). One half of a sector of the nonagon is covered up exactly by 5 lines of equal length, head to tail, between one (half) sector's border and the other. Starting with one side of the 18-gon (regular 18-sided polygon), a number of circles (4) has been constructed, with that line as the radius and their centers on those borders. The question arises: is this only true for a nonagon, or does it hold for other polygons as well?

Well, let's try a heptagon (with 7 sides). The picture shows clearly, that the heptagon has a similar characteristic.
(This is not new! See for instance the page about heptagons on Eric Weisstein's brilliant World of Mathematics! There, we see the same figure, and it's called matchstick construction).

And a pentagon? Yes, a pentagon (5 sides) also does obey the matchstick construction rule. These are polygons with an odd number of sides; what about polygons with an even number of sides?

Let's take 8: does an octagon also show the same feature? No, from the picture can be seen, that the octagon fails. The last circle should pass through the basic center, but it is much to large. The circle that fits is much smaller (dashed circle).

So, let's try to find out, whether the construction with a number of lines of equal length can be generalised for any odd-numbered regular polygon, but not for even-numbered ones.

A good way to generalise is to start counting. How many lines build up a 5-, 7-, 9-gon?
number of sides     number of lines
          5                    3
          7                    4
          9                    5
          n                1/2(n + 1)
Here, we immediately get a clue, that only odd numbers are involved. With even numbers, we would end up with a fractal number for the number of lines.

Next, let's see what happens, if we take an "n-gon" (a regular polygon with n sides). The figure shows the zig-zag lines of the matchstick construction, covering up half of the sector. So, top-angle at is half that of the n-gon's sector, or:
     at = 1/2(360°/n) = 180°/n(1)
Since the sector is an isosceles triangle, it follows:
     a1 = 1/2(180° - at) = 90° - 1/2at(2)
For the next angles, we can derive:
     a2 = at(3)
     a3 = a1 - a2 = 90° - 1/2at - at = 90° - 3/2at (4)
     a4 = 180° - 2a3 = 180° - 180° + 3at = 3at(5)
     a5 = 180° - a4 - a1 = 180° - 3at - 90° + 1/2at = 90° - 5/2at (6)
     a6 = 180° - 2a5 = 180° - 180° + 5at = 5at(7)
     a7 = 180° - a6 - a3 = 180° - 5at - 90° + 3/2at = 90° - 7/2at (8)
     a8 = 180° - 2a7 = 180° - 180° + 7at = 7at(9)
     a9 = 180° - a8 - a5 = 180° - 7at - 90° + 5/2at = 90° - 9/2at (10)
We now have an iteration of two steps. If we look carefully what happens, we see that every additional "matchstick" involves two new angles, and consequently adds two new formula's to the series. The series stops, when the right number of lines (sticks) has been reached, according to the list above. With two lines, we have: a1 = at and n = 3 (equilateral triangle), with three lines: a3 = at and n = 5 (pentagon), with four lines: a5 = at and n = 7 (heptagon), and so on. It is also clear, that the even-numbered angles cannot become equal to the top-angle, since they are a multiple of it (apart from a2, but in this case, a1 = a2 = 60°). If we generalise the odd-numbered angle for the n-gon, we get:

     at = 90° - (n - 2)/2at(11)
which is equivalent to:
     at = 90° - n/2at + 2/2at(12)
     n/2at = 90°(13)
     at = 180°/n(14)
and this is indeed what we started with: the correct top-angle of the (half) sector of the n-gon (see (1)).

A consequence of all this is, that we are now able to define a new construction device for any odd-numbered regular polygon! It can be conceived of consisting of two legs, pivoted at one end, with small sleeves along their lengths, and a number of arms of equal length, pivoted haid to tail. The pivots of the arms slide through the sleeves of the legs alternatively. From the top of the two legs, the arms are inserted into the sleeves one-by-one, one leg alternated by the other. When the right number of arms is inserted for the desired number of sides for the polygon, the last arm's pivot is brought precisely in-line with the legs' pivot. There is one complicating factor: the first (farthest) arm must remain perpendicular to the (virtual) connecting line between its midpoint and the legs' pivot.

If all this is correctly carried out, the angle between both legs is equal to the top-angle of one sector of the regular polygon with double the number of sides, a 2n-gon. Combined with a circle of desired size, it can be easily used to make the construction of the n-gon complete. In the age of computers, this device is of course a very cumbersome one, but nevertheless a mechanical device fitting nicely into the ruler-and-compass spirit of ancient Greece.

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Copyright © 2002, Zef Damen, The Netherlands


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