Hans
Walser, [20090715b]
Regular K-gon
and trapezoids
1 Example
Starting
by a regular yellow heptagon (
) in the unit circle we add squares on every side. Then we
proceed as indicated in the following figure.
Heptagon
and squares
We get
red isosceles trapezoids between the white squares. In the first ring we see
triangles, bat we count them as special trapezoids with upper side zero. We
would like to compare the areas of the trapezoids.
2
Modification
We can
modify the figure by transforming the white squares into rhombuses. This does
not change shape or size of the red trapezoids.
Collapsing
the squares
3
Some calculations
We use
the notations of the following figure.
Notations
We
get:
Setting
we can establish
the recursion formula:
For
the area 
we get:
Since
we want to compare the areas of the red polygons, we introduce the relative
area:
4
Special cases
The
tables indicate for different K the numerical
values.
4.1
K = 1
We get
.
4.2
K = 2
|
n |
a[n] |
|
0 |
0 |
|
1 |
2.0 |
|
2 |
4.0 |
|
3 |
10.0 |
|
4 |
24.0 |
|
5 |
58.0 |
|
6 |
140.0 |
|
7 |
338.0 |
|
8 |
816.0 |
We
have ŇflatÓ trapezoids. But the squares are interesting.
Squares
only
If we
add only the next square above, we get a rectangle with nearly the shape of the
European standard paper shape DIN A.
Close
to European standard shape DIN A
Indeed
we have:
|
n |
q[n] |
|
0 |
1.0 |
|
1 |
1.5 |
|
2 |
1.4 |
|
3 |
1.4166667 |
|
4 |
1.4137931 |
|
5 |
1.4142857 |
|
6 |
1.4142012 |
|
7 |
1.4142157 |
|
8 |
1.4142132 |
|
9 |
1.4142136 |
|
10 |
1.4142136 |
4.3
K = 3
K = 3
|
n |
a[n] |
phi[n] |
psi[n] |
|
0 |
0 |
1.299038106 |
1.0 |
|
1 |
1.732050808 |
6.495190528 |
5.0 |
|
2 |
3.0 |
31.17691454 |
24.0 |
|
3 |
6.92820323 |
149.3893822 |
115.0 |
|
4 |
15.0 |
715.7699962 |
551.0 |
|
5 |
32.90896534 |
3429.460599 |
2640.0 |
|
6 |
72.0 |
16431.533 |
12649.0 |
|
7 |
157.6166235 |
78728.20439 |
60605.0 |
|
8 |
345.0 |
377209.489 |
290376.0 |
We
see, that 
are integer
numbers, but not the 
. The case
|
n |
a[n] |
phi[n] |
psi[n] |
|
1 |
1.732050808 |
6.495190528 |
5.0 |
is subject of [Deshpande 2009].
We have the recursion formula:
4.4
K = 4
K = 4
The
Figure on the left fits into a square lattice, but not so the figure on the
right.
|
n |
a[n] |
phi[n] |
psi[n] |
|
0 |
0 |
1.0 |
1.0 |
|
1 |
1.414213562 |
4.0 |
4.0 |
|
2 |
2.0 |
15.0 |
15.0 |
|
3 |
4.242640687 |
56.0 |
56.0 |
|
4 |
8.0 |
209.0 |
209.0 |
|
5 |
15.55634919 |
780.0 |
780.0 |
|
6 |
30.0 |
2911.0 |
2911.0 |
|
7 |
57.98275606 |
10864.0 |
10864.0 |
|
8 |
112.0 |
40545.0 |
40545.0 |
Both
the 
and the 
are integers. We
have the recursion formula:
4.5
K = 5
K = 5
|
n |
a[n] |
phi[n] |
psi[n] |
|
0 |
0 |
0.6571638901 |
1.0 |
|
1 |
1.175570505 |
2.22250594 |
3.381966011 |
|
2 |
1.381966011 |
6.85927566 |
10.4376941 |
|
3 |
2.800168986 |
20.9753312 |
31.91796068 |
|
4 |
4.673762079 |
64.07858154 |
97.50776405 |
|
5 |
8.294505831 |
195.7362536 |
297.8499832 |
|
6 |
14.42453848 |
597.8947754 |
909.8107555 |
|
7 |
25.25156781 |
1826.323555 |
2779.099069 |
|
8 |
44.1095368 |
5578.669413 |
8489.007836 |
No
integer numbers. Probably there is the golden section in it, but I do not see
it. But there is a very interesting recursion formula:
4.6
K = 6
K = 6
The
figure on the right fits into a regular triangular lattice. The figure on the
left not, since squares and regular triangles donŐt like each other.
|
n |
a[n] |
phi[n] |
psi[n] |
|
0 |
0 |
0.4330127019 |
1.0 |
|
1 |
1.0 |
1.299038106 |
3.0 |
|
2 |
1.0 |
3.464101615 |
8.0 |
|
3 |
2.0 |
9.09326674 |
21.0 |
|
4 |
3.0 |
23.8156986 |
55.0 |
|
5 |
5.0 |
62.35382907 |
144.0 |
|
6 |
8.0 |
163.2457886 |
377.0 |
|
7 |
13.0 |
427.3835368 |
987.0 |
|
8 |
21.0 |
1118.904822 |
2584.0 |
The 
are the
Fibonacci numbers (
), and the 
are every second
Fibonacci number (
).We have the recursion formula:
4.7
K = 7
K = 7
|
n |
a[n] |
phi[n] |
psi[n] |
|
0 |
0 |
0.2943675264 |
1.0 |
|
1 |
0.8677674782 |
0.8103998041 |
2.753020396 |
|
2 |
0.7530203963 |
1.936679664 |
6.579121302 |
|
3 |
1.521214089 |
4.521318811 |
15.35943474 |
|
4 |
2.07308051 |
10.51060324 |
35.70571581 |
|
5 |
3.320165935 |
24.41458629 |
82.93912915 |
|
6 |
4.95421253 |
56.70325077 |
192.6273984 |
|
7 |
7.619270449 |
131.6906196 |
447.3680275 |
|
8 |
11.56596763 |
305.8437111 |
1038.985906 |
Awful
numbers. We have again the interesting recursion formula:
4.8
K = 8
K = 8
|
n |
a[n] |
phi[n] |
psi[n] |
|
0 |
0 |
0.2071067812 |
1.0 |
|
1 |
0.7653668647 |
0.5355339059 |
2.585786438 |
|
2 |
0.5857864376 |
1.17766953 |
5.686291501 |
|
3 |
1.213708394 |
2.509667992 |
12.11774901 |
|
4 |
1.514718626 |
5.311795927 |
25.64761953 |
|
5 |
2.373023839 |
11.22550187 |
54.20151774 |
|
6 |
3.330952442 |
23.71495458 |
114.5059299 |
|
7 |
4.922424466 |
50.09630604 |
241.8863629 |
|
8 |
7.098413022 |
105.8233942 |
510.9605468 |
We
have again the interesting recursion formula:
5
Summary
Cases
with integer numbers only for K = 2, 3, 4, 6.
In the
general case there is the conjecture:
If
anybody has time to prove it, I would be glad to hear about.
References
[Deshpande 2009] Deshpande, M. N. : Proof Without
Words: Beyond Extriangles. MATHEMATICS MAGAZINE. Vol. 82, No. 3, June 2009, p.
208.