Movatterモバイル変換

⭳  ⭳

The height formula given below shows that only2 < n/d < 6 is possible. The maximal height would be obtained atn/d = 2 with upright latteral triangles but then degenerate base,the other extremal valuen/d = 6 would generate a height of zero.

Incidence matrix according toDynkin symbol

xx ox-n/d-oo&#x   (2<n/d<6)   → height = sqrt(1-[1/4 *sin^2(π d/n)])(line||{n/d}-p)o. o.-n/d-o.    | 2  * | 1  n 0  0 | n  n 0 0 | n 1 0.o .o-n/d-.o    | * 2n | 0  1 1  2 | 1  2 2 1 | 2 1 1----------------+------+-----------+----------+------x. ..     ..    | 2  0 |1  * *  * | n  0 0 0 | n 0 0oo oo-n/d-oo&#x | 1  1 | * 2n *  * | 1  2 0 0 | 2 1 0.x ..     ..    | 0  2 | *  * n  * | 1  0 2 0 | 2 0 1.. .x     ..    | 0  2 | *  * * 2n | 0  1 1 1 | 1 1 1----------------+------+-----------+----------+------xx ..     ..&#x | 2  2 | 1  2 1  0 | n  * * * | 2 0 0.. ox     ..&#x | 1  2 | 0  2 0  1 | * 2n * * | 1 1 0.x .x     ..    | 0  4 | 0  0 2  2 | *  * n * | 1 0 1.. .x-n/d-.o    | 0  n | 0  0 0  n | *  * * 2 | 0 1 1----------------+------+-----------+----------+------xx ox     ..&#x♦ 2  4 | 1  4 2  2 | 2  2 1 0 | n * *.. ox-n/d-oo&#x♦ 1  n | 0  n 0  n | 0  n 0 1 | * 2 *.x .x-n/d-.o♦ 0 2n | 0  0 n 2n | 0  0 n 2 | * *1

{n/d}-py||{n/d}-py   (2<n/d<6)   → height = 1  1 * * * | n 1 0 0 0 0 | n n 0 0 0 0 | 1 n 0 0  top-tip  * n * * | 1 0 2 1 0 0 | 2 1 1 0 0 0 | 1 2 1 0  top-base  * * 1 * | 0 1 0 0 n 0 | 0 n 0 0 n 0 | 0 n 0 1  bottom-tip  * * * n | 0 0 0 1 1 2 | 0 1 0 2 2 1 | 0 2 1 1  bottom-base----------+-------------+-------------+--------  1 1 0 0 | n * * * * * | 2 1 0 0 0 0 | 1 2 0 0  1 0 1 0 | * 1 * * * * | 0 n 0 0 0 0 | 0 n 0 0  0 2 0 0 | * * n * * * | 1 0 1 1 0 0 | 1 1 1 0  0 1 0 1 | * * * n * * | 0 1 0 2 0 0 | 0 2 1 0  0 0 1 1 | * * * * n * | 0 1 0 0 2 0 | 0 2 0 1  0 0 0 2 | * * * * * n | 0 0 0 1 1 1 | 0 1 1 1----------+-------------+-------------+--------  1 2 0 0 | 2 0 1 0 0 0 | n * * * * * | 1 1 0 0  1 1 1 1 | 1 1 0 1 1 0 | * n * * * * | 0 2 0 0  0 n 0 0 | 0 0 n 0 0 0 | * * 1 * * * | 1 0 1 0  0 2 0 2 | 0 0 1 2 0 1 | * * * n * * | 0 1 1 0  0 0 1 2 | 0 0 0 0 2 1 | * * * * n * | 0 1 0 1  0 0 0 n | 0 0 0 0 0 n | * * * * * 1 | 0 0 1 1----------+-------------+-------------+--------♦ 1 n 0 0 | n 0 n 0 0 0 | n 0 1 0 0 0 |1 * * *♦ 1 2 1 2 | 2 1 1 2 2 1 | 1 2 0 1 1 0 | * n * *♦ 0 n 0 n | 0 0 n n 0 n | 0 0 1 n 0 1 | * * 1 *♦ 0 0 1 n | 0 0 0 0 n n | 0 0 0 0 n 1 | * * *1

oxxo-n/d-oooo&#xr   (2<n/d<6)   → height(1,2) = height(3,4) = sqrt(1-[1/4 *sin^2(π/n)])                                height(1,4) = height(2,3) = 1( (pt|| {n/d})||  (pt|| {n/d}) )o...-n/d-o...     |1 * * * | n 1 0 0 0 0 | n n 0 0 0 0 | 1 n 0 0.o..-n/d-.o..     | * n * * | 1 0 2 1 0 0 | 2 1 1 2 0 0 | 1 2 1 0..o.-n/d-..o.     | * * n * | 0 0 0 1 2 1 | 0 1 0 2 1 2 | 0 2 1 1...o-n/d-...o     | * * *1 | 0 1 0 0 0 n | 0 n 0 0 0 n | 0 n 0 1------------------+---------+-------------+-------------+--------oo..-n/d-oo..&#x  | 1 1 0 0 | n * * * * * | 2 1 0 0 0 0 | 1 2 0 0o..o-n/d-o..o&#x  | 1 0 0 1 | * 1 * * * * | 0 n 0 0 0 0 | 0 n 0 0.x..     ....     | 0 2 0 0 | * * n * * * | 1 0 1 1 0 0 | 1 1 1 0.oo.-n/d-.oo.&#x  | 0 1 1 0 | * * * n * * | 0 1 0 2 0 0 | 0 2 1 0..x.     ....     | 0 0 2 0 | * * * * n * | 0 0 0 1 1 1 | 0 1 1 1..oo-n/d-..oo&#x  | 0 0 1 1 | * * * * * n | 0 1 0 0 0 2 | 0 2 0 1------------------+---------+-------------+-------------+--------ox..     ....&#x  | 1 2 0 0 | 2 0 1 0 0 0 | n * * * * * | 1 1 0 0oooo-n/d-oooo&#xr | 1 1 1 1 | 1 1 0 1 0 1 | * n * * * * | 0 2 0 0.x..-n/d-.o..     | 0 n 0 0 | 0 0 n 0 0 0 | * *1 * * * | 1 0 1 0.xx.     ....&#x  | 0 2 2 0 | 0 0 1 2 1 0 | * * * n * * | 0 1 1 0..x.-n/d-..o.     | 0 0 n 0 | 0 0 0 0 n 0 | * * * *1 * | 0 0 1 1..xo     ....&#x  | 0 0 2 1 | 0 0 0 0 1 2 | * * * * * n | 0 1 0 1------------------+---------+-------------+-------------+--------ox..-n/d-oo..&#x♦ 1 n 0 0 | n 0 n 0 0 0 | n 0 1 0 0 0 | 1 * * *oxxo     ....&#xr♦ 1 2 2 1 | 2 1 1 2 1 2 | 1 2 0 1 0 1 | * n * *.xx.-n/d-.oo.&#x♦ 0 n n 0 | 0 0 n n n 0 | 0 0 1 n 1 0 | * * 1 *..xo-n/d-..oo&#x♦ 0 0 n 1 | 0 0 0 0 n n | 0 0 0 0 1 n | * * * 1

o(xo)x-n/d-o(oo)o&#xt   (2<n/d<6)(pt|| ({n/d}|| pt)|| para {n/d})o(..).-n/d-o(..).     |1 * * * | n 1 0 0 0 0 | n n 0 0 0 0 | 1 n 0 0.(o.).-n/d-.(o.).     | * n * * | 1 0 2 1 0 0 | 2 1 1 2 0 0 | 1 2 1 0.(.o).-n/d-.(.o).     | * *1 * | 0 1 0 0 n 0 | 0 n 0 0 n 0 | 0 n 0 1.(..)o-n/d-.(..)o     | * * * n | 0 0 0 1 1 2 | 0 1 0 2 2 1 | 0 2 1 1----------------------+---------+-------------+-------------+--------o(o.).-n/d-o(o.).&#x  | 1 1 0 0 | n * * * * * | 2 1 0 0 0 0 | 1 2 0 0o(.o).-n/d-o(.o).&#x  | 1 0 1 0 | * 1 * * * * | 0 n 0 0 0 0 | 0 n 0 0.(x.).     .(..).     | 0 2 0 0 | * * n * * * | 1 0 1 1 0 0 | 1 1 1 0.(o.)o-n/d-.(o.)o&#x  | 0 1 0 1 | * * * n * * | 0 1 0 2 0 0 | 0 2 1 0.(.o)o-n/d-.(.o)o&#x  | 0 0 1 1 | * * * * n * | 0 1 0 0 2 0 | 0 2 0 1.(..)x     .(..).     | 0 0 0 2 | * * * * * n | 0 0 0 1 1 1 | 0 1 1 1----------------------+---------+-------------+-------------+--------o(x.).     .(..).&#x  | 1 2 0 0 | 2 0 1 0 0 0 | n * * * * * | 1 1 0 0o(oo)o-n/d-o(oo)o&#xt | 1 1 1 1 | 1 1 0 1 1 0 | * n * * * * | 0 2 0 0.(x.).-n/d-.(o.).     | 0 n 0 0 | 0 0 n 0 0 0 | * *1 * * * | 1 0 1 0.(x.)x     .(..).&#x  | 0 2 0 2 | 0 0 1 2 0 1 | * * * n * * | 0 1 1 0.(.o)x     .(..).&#x  | 0 0 1 2 | 0 0 0 0 2 1 | * * * * n * | 0 1 0 1.(..)x-n/d-.(..)o     | 0 0 0 n | 0 0 0 0 0 n | * * * * *1 | 0 0 1 1----------------------+---------+-------------+-------------+--------o(x.).-n/d-o(o.).&#x♦ 1 n 0 0 | n 0 n 0 0 0 | n 0 1 0 0 0 | 1 * * *o(xo)x     .(..).&#xt♦ 1 2 1 2 | 2 1 1 2 2 1 | 1 2 0 1 1 0 | * n * *.(x.)x-n/d-.(o.)o&#x♦ 0 n 0 n | 0 0 n n 0 n | 0 0 1 n 0 1 | * * 1 *.(.o)x-n/d-.(.o)o&#x♦ 0 0 1 n | 0 0 0 0 n n | 0 0 0 0 n 1 | * * * 1