Take a polygon with area S1 and pick a number r in [0,1/2]. Take vertex A that connects sides AB and AC and add points M and N on these sides so that AM/AB=AN/AC=r. Cut corner A along MN. Cut all other corners the same way.
After repeating these steps infinite times we will get a figure with an area S2. Let's F(r)=S2/S1. It's clear that F(0)=1 and F(½)=0.
Questions:
(a) What is this function for square?
(b) What is this function for equilateral triangle?
(c) Is it possible to get a circle from a square or from an equilateral triangle this way?
(d) Is it possible that this function is universal for all triangles, or for all rectangles, or for all polygons?
If the final shape is to be a circle, it would seem that each step of the process should be a regular polygon. If that is in fact the case, then it would be impossible to form a circle, as the same x cannot be used at each step and maintain regularity. The reasoning is as follows:
If the side is taken as 1, the remaining portion of the side will be 1 - 2*x. The chord will be 2 * x * sin(alpha/2), where alpha is the angle of the polygon, alpha = 180 - 360/n degrees. We need to equate the chord with the remaining portion of the side. This program evaluates this for various sided regular polygons:
DEFDBL A-Z
pi = ATN(1) * 4
FOR n = 3 TO 30
alpha = 180 - 360 / n
x = 1 / 3
DO
ppx = px
px = x
x = .5 - x * SIN((alpha / 2) * pi / 180)
x = (x + px) / 2
LOOP UNTIL ppx = x
PRINT USING "## ### ##.##########"; n; 2 * n; x
NEXT
The table resulting is:
n1 n2 x
3 6 0.3333333333
4 8 0.2928932188
5 10 0.2763932023
6 12 0.2679491924
7 14 0.2630237709
8 16 0.2598915325
9 18 0.2577728010
10 20 0.2562714077
11 22 0.2551680495
12 24 0.2543330950
13 26 0.2536858245
14 28 0.2531737978
15 30 0.2527617251
16 32 0.2524251391
17 34 0.2521466378
18 36 0.2519135666
19 38 0.2517165428
20 40 0.2515484897
21 42 0.2514039874
22 44 0.2512788281
23 46 0.2511697041
24 48 0.2510739864
25 50 0.2509895638
26 52 0.2509147254
27 54 0.2508480725
28 56 0.2507884526
29 58 0.2507349094
30 60 0.2506866438
where n1 is the number of sides of the starting regular polygon and n2 is twice that, the number of sides of the resulting regular polygon. So for example, to transform an equilateral triangle into a regular hexagon, x needs to be 1/3. But to change a regular hexagon into a regular dodecagon requires x to be 0.2679491924. And then to change that to a regular 24-gon, requires x to be 0.2543330950. The above table results are monotonically decreasing, so no matter what regular polygon you start with, the series will require ever decreasing x values.
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Posted by Charlie
on 2007-02-27 15:58:44 |
comment on part c
