Acronym ...
Name (N-g, dual N-g, para N-p, gyro N-p) lace simplex
Circumradius ...
Lace city
in approx. ASCII-art 
xNo x                     
(1)                       
                     xNo o
                     (3)  
                          
               oNx o      
               (4)        
   oNx x                  
   (2)
Face vector 6n, 26n, 44n+6, 36n+13, 14n+13, 2n+6
Especially tritra ope (N=3)  
Confer
general polytopal classes:
segmentopeta   lace simplices  

For non-convex cases use N = n/d.


Incidence matrix according to Dynkin symbol

xoxoNoxox xxoo&#x   → height(1,2) = height(3,4) = sqrt[(1+2 cos(π/N))/(2+2 cos(π/N))]
                      height(1,3) = height(2,4) = sqrt(3)/2 = 0.866025
                      height(1,4) = height(2,3) = sqrt[(1+3 cos(π/N))/(4+4 cos(π/N))]

o...No... o...    & | 4n  * |  2  1  2  1  2  0  0 | 1  2  3  2  2  1  2  1  2  4  2 0  0 | 1 1  3 1  2 1  2  1  3  3  4  2  1  2  2 0 | 1 1 1 2  3  3 1  2  1  3  2 | 2 1 1  3
..o.N..o. ..o.    & |  * 2n |  0  0  0  2  4  2  2 | 0  0  0  0  4  1  2  4  2  4  8 1  3 | 0 0  0 2  2 2  1  2  4  2  2  6  6  4  4 1 | 0 1 1 2  2  1 4  3  3  6  2 | 1 2 2  3
--------------------+-------+----------------------+--------------------------------------+--------------------------------------------+-----------------------------+---------
x... .... ....    & |  2  0 | 4n  *  *  *  *  *  * | 1  1  1  0  1  0  1  0  0  0  0 0  0 | 1 1  1 1  1 1  1  0  1  1  0  1  0  0  0 0 | 1 1 1 2  1  1 1  1  0  1  0 | 2 1 1  1
.... .... x...    & |  2  0 |  * 2n  *  *  *  *  * | 0  2  0  2  0  1  0  0  2  0  0 0  0 | 1 0  3 0  2 0  2  1  0  0  4  0  0  2  0 0 | 1 1 1 0  3  3 0  2  1  0  2 | 2 1 0  3
oo..Noo.. oo..&#x   |  2  0 |  *  * 4n  *  *  *  * | 0  0  2  1  0  0  0  0  0  2  0 0  0 | 0 1  2 0  0 0  0  0  2  2  2  0  0  0  1 0 | 1 0 0 2  2  2 0  0  0  2  1 | 2 0 1  2
o.o.No.o. o.o.&#x & |  1  1 |  *  *  * 4n  *  *  * | 0  0  0  0  2  1  0  0  0  2  2 0  0 | 0 0  0 1  2 0  0  0  2  1  2  2  1  2  2 0 | 0 1 0 1  2  1 1  2  1  3  2 | 1 1 1  3
o..oNo..o o..o&#x & |  1  1 |  *  *  *  * 8n  *  * | 0  0  0  0  0  0  1  1  1  1  1 0  0 | 0 0  0 0  0 1  1  1  1  1  1  1  1  1  1 0 | 0 0 1 1  1  1 1  1  1  2  1 | 1 1 1  2
..x. .... ....    & |  0  2 |  *  *  *  *  * 2n  * | 0  0  0  0  2  0  0  2  0  0  0 1  1 | 0 0  0 2  1 2  0  1  2  0  0  2  2  0  0 1 | 0 1 1 2  1  0 4  1  1  2  0 | 1 2 2  1
..ooN..oo ..oo&#x   |  0  2 |  *  *  *  *  *  * 2n | 0  0  0  0  0  0  0  0  0  0  4 0  2 | 0 0  0 0  0 0  0  0  0  0  0  4  4  2  2 1 | 0 0 0 0  0  0 4  2  2  4  1 | 0 2 2  2
--------------------+-------+----------------------+--------------------------------------+--------------------------------------------+-----------------------------+---------
x...No... ....    & |  n  0 |  n  0  0  0  0  0  0 | 4  *  *  *  *  *  *  *  *  *  * *  * | 1 1  0 1  0 1  0  0  0  0  0  0  0  0  0 0 | 1 1 1 2  0  0 1  0  0  0  0 | 2 1 1  0
x... .... x...    & |  4  0 |  2  2  0  0  0  0  0 | * 2n  *  *  *  *  *  *  *  *  * *  * | 1 0  1 0  1 0  1  0  0  0  0  0  0  0  0 0 | 1 1 1 0  1  1 0  1  0  0  0 | 2 1 0  1
xo.. .... ....&#x & |  3  0 |  1  0  2  0  0  0  0 | *  * 4n  *  *  *  *  *  *  *  * *  * | 0 1  1 0  0 0  0  0  1  1  0  0  0  0  0 0 | 1 0 0 2  1  1 0  0  0  1  0 | 2 0 1  1
.... .... xx..&#x   |  4  0 |  0  2  2  0  0  0  0 | *  *  * 2n  *  *  *  *  *  *  * *  * | 0 0  2 0  0 0  0  0  0  0  2  0  0  0  0 0 | 1 0 0 0  2  2 0  0  0  0  1 | 2 0 0  2
x.x. .... ....&#x & |  2  2 |  1  0  0  2  0  1  0 | *  *  *  * 4n  *  *  *  *  *  * *  * | 0 0  0 1  1 0  0  0  1  0  0  1  0  0  0 0 | 0 1 0 1  1  0 1  1  0  1  0 | 1 1 1  1
.... .... x.o.&#x & |  2  1 |  0  1  0  2  0  0  0 | *  *  *  *  * 2n  *  *  *  *  * *  * | 0 0  0 0  2 0  0  0  0  0  2  0  0  2  0 0 | 0 1 0 0  2  1 0  2  1  0  2 | 1 1 0  3
x..o .... ....&#x & |  2  1 |  1  0  0  0  2  0  0 | *  *  *  *  *  * 4n  *  *  *  * *  * | 0 0  0 0  0 1  1  0  0  1  0  1  0  0  0 0 | 0 0 1 1  0  1 1  1  0  1  0 | 1 1 1  1
.... o..x ....&#x & |  1  2 |  0  0  0  0  2  1  0 | *  *  *  *  *  *  * 4n  *  *  * *  * | 0 0  0 0  0 1  0  1  1  0  0  0  1  0  0 0 | 0 0 1 1  1  0 1  0  1  1  0 | 1 1 1  1
.... .... x..o&#x & |  2  1 |  0  1  0  0  2  0  0 | *  *  *  *  *  *  *  * 4n  *  * *  * | 0 0  0 0  0 0  1  1  0  0  1  0  0  1  0 0 | 0 0 1 0  1  1 0  1  1  0  1 | 1 1 0  2
ooo.Nooo. ooo.&#x & |  2  1 |  0  0  1  1  1  0  0 | *  *  *  *  *  *  *  *  * 8n  * *  * | 0 0  0 0  0 0  0  0  1  1  1  0  0  0  1 0 | 0 0 0 1  1  1 0  0  0  2  1 | 1 0 1  2
o.ooNo.oo o.oo&#x & |  1  2 |  0  0  0  1  1  0  1 | *  *  *  *  *  *  *  *  *  * 8n *  * | 0 0  0 0  0 0  0  0  0  0  0  1  1  1  1 0 | 0 0 0 0  0  0 1  1  1  2  1 | 0 1 1  2
..x.N..o. ....    & |  0  n |  0  0  0  0  0  n  0 | *  *  *  *  *  *  *  *  *  *  * 2  * | 0 0  0 2  0 2  0  0  0  0  0  0  0  0  0 1 | 0 1 1 2  0  0 4  0  0  0  0 | 1 2 2  0
..xo .... ....&#x & |  0  3 |  0  0  0  0  0  1  2 | *  *  *  *  *  *  *  *  *  *  * * 2n | 0 0  0 0  0 0  0  0  0  0  0  2  2  0  0 1 | 0 0 0 0  0  0 4  1  1  2  0 | 0 2 2  1
--------------------+-------+----------------------+--------------------------------------+--------------------------------------------+-----------------------------+---------
x...No... x...    & ♦ 2n  0 | 2n  n  0  0  0  0  0 | 2  n  0  0  0  0  0  0  0  0  0 0  0 | 2 *  * *  * *  *  *  *  *  *  *  *  *  * * | 1 1 1 0  0  0 0  0  0  0  0 | 2 1 0  0
xo..Nox.. ....&#x   ♦ 2n  0 | 2n  0 2n  0  0  0  0 | 2  0 2n  0  0  0  0  0  0  0  0 0  0 | * 2  * *  * *  *  *  *  *  *  *  *  *  * * | 1 0 0 2  0  0 0  0  0  0  0 | 2 0 1  0
xo.. .... xx..&#x & ♦  6  0 |  2  3  4  0  0  0  0 | 0  1  2  2  0  0  0  0  0  0  0 0  0 | * * 2n *  * *  *  *  *  *  *  *  *  *  * * | 1 0 0 0  1  1 0  0  0  0  0 | 2 0 0  1
x.x.No.o. ....&#x & ♦  n  n |  n  0  0  n  0  n  0 | 1  0  0  0  n  0  0  0  0  0  0 1  0 | * *  * 4  * *  *  *  *  *  *  *  *  *  * * | 0 1 0 1  0  0 1  0  0  0  0 | 1 1 1  0
x.x. .... x.o.&#x & ♦  4  2 |  2  2  0  4  0  1  0 | 0  1  0  0  2  2  0  0  0  0  0 0  0 | * *  * * 2n *  *  *  *  *  *  *  *  *  * * | 0 1 0 0  1  0 0  1  0  0  0 | 1 1 0  1
x..oNo..x ....&#x & ♦  n  n |  n  0  0  0 2n  n  0 | 1  0  0  0  0  0  n  n  0  0  0 1  0 | * *  * *  * 4  *  *  *  *  *  *  *  *  * * | 0 0 1 1  0  0 1  0  0  0  0 | 1 1 1  0
x..o .... x..o&#x & ♦  4  1 |  2  2  0  0  4  0  0 | 0  1  0  0  0  0  2  0  2  0  0 0  0 | * *  * *  * * 2n  *  *  *  *  *  *  *  * * | 0 0 1 0  0  1 0  1  0  0  0 | 1 1 0  1
.... o..x x..o&#x & ♦  2  2 |  0  1  0  0  4  1  0 | 0  0  0  0  0  0  0  2  2  0  0 0  0 | * *  * *  * *  * 2n  *  *  *  *  *  *  * * | 0 0 1 0  1  0 0  0  1  0  0 | 1 1 0  1
xox. .... ....&#x & ♦  3  2 |  1  0  2  2  2  1  0 | 0  0  1  0  1  0  0  1  0  2  0 0  0 | * *  * *  * *  *  * 4n  *  *  *  *  *  * * | 0 0 0 1  1  0 0  0  0  1  0 | 1 0 1  1
.... oxo. ....&#x & ♦  3  1 |  1  0  2  1  2  0  0 | 0  0  1  0  0  0  1  0  0  2  0 0  0 | * *  * *  * *  *  *  * 4n  *  *  *  *  * * | 0 0 0 1  0  1 0  0  0  1  0 | 1 0 1  1
.... .... xxo.&#x & ♦  4  1 |  0  2  2  2  2  0  0 | 0  0  0  1  0  1  0  0  1  2  0 0  0 | * *  * *  * *  *  *  *  * 4n  *  *  *  * * | 0 0 0 0  1  1 0  0  0  0  1 | 1 0 0  2
x.xo .... ....&#x & ♦  2  3 |  1  0  0  2  2  1  2 | 0  0  0  0  1  0  1  0  0  0  2 0  1 | * *  * *  * *  *  *  *  *  * 4n  *  *  * * | 0 0 0 0  0  0 1  1  0  1  0 | 0 1 1  1
.... o.ox ....&#x & ♦  1  3 |  0  0  0  1  2  1  2 | 0  0  0  0  0  0  0  1  0  0  2 0  1 | * *  * *  * *  *  *  *  *  *  * 4n  *  * * | 0 0 0 0  0  0 1  0  1  1  0 | 0 1 1  1
.... .... x.oo&#x & ♦  2  2 |  0  1  0  2  2  0  1 | 0  0  0  0  0  1  0  0  1  0  2 0  0 | * *  * *  * *  *  *  *  *  *  *  * 4n  * * | 0 0 0 0  0  0 0  1  1  0  1 | 0 1 0  2
ooooNoooo oooo&#x   ♦  2  2 |  0  0  1  2  2  0  1 | 0  0  0  0  0  0  0  0  0  2  2 0  0 | * *  * *  * *  *  *  *  *  *  *  *  * 4n * | 0 0 0 0  0  0 0  0  0  2  1 | 0 0 1  2
..xoN..ox ....&#x   ♦  0 2n |  0  0  0  0  0 2n 2n | 0  0  0  0  0  0  0  0  0  0  0 2 2n | * *  * *  * *  *  *  *  *  *  *  *  *  * 1 | 0 0 0 0  0  0 4  0  0  0  0 | 0 2 2  0
--------------------+-------+----------------------+--------------------------------------+--------------------------------------------+-----------------------------+---------
xo..Nox.. xx..&#x   ♦ 4n  0 | 4n 2n 4n  0  0  0  0 | 4 2n 4n 2n  0  0  0  0  0  0  0 0  0 | 2 2 2n 0  0 0  0  0  0  0  0  0  0  0  0 0 | 1 * * *  *  * *  *  *  *  * | 2 0 0  0
x.x.No.o. x.o.&#x & ♦ 2n  n | 2n  n  0 2n  0  n  0 | 2  n  0  0 2n  n  0  0  0  0  0 1  0 | 1 0  0 2  n 0  0  0  0  0  0  0  0  0  0 0 | * 2 * *  *  * *  *  *  *  * | 1 1 0  0
x..oNo..x x..o&#x & ♦ 2n  n | 2n  n  0  0 4n  n  0 | 2  n  0  0  0  0 2n 2n 2n  0  0 1  0 | 1 0  0 0  0 2  n  n  0  0  0  0  0  0  0 0 | * * 2 *  *  * *  *  *  *  * | 1 1 0  0
xox.Noxo. ....&#x & ♦ 2n  n | 2n  0 2n  n 2n  n  0 | 2  0 2n  0  n  0  n  n  0 2n  0 1  0 | 0 1  0 1  0 1  0  0  n  n  0  0  0  0  0 0 | * * * 4  *  * *  *  *  *  * | 1 0 1  0
xox. .... xxo.&#x & ♦  6  2 |  2  3  4  4  4  1  0 | 0  1  2  2  2  2  0  2  2  4  0 0  0 | 0 0  1 0  1 0  0  1  2  0  2  0  0  0  0 0 | * * * * 2n  * *  *  *  *  * | 1 0 0  1
.... oxo. xxo.&#x & ♦  6  1 |  2  3  4  2  4  0  0 | 0  1  2  2  0  1  2  0  2  4  0 0  0 | 0 0  1 0  0 0  1  0  0  2  2  0  0  0  0 0 | * * * *  * 2n *  *  *  *  * | 1 0 0  1
x.xoNo.ox ....&#x & ♦  n 2n |  n  0  0  n 2n 2n 2n | 1  0  0  0  n  0  n  n  0  0 2n 2 2n | 0 0  0 1  0 1  0  0  0  0  0  n  n  0  0 1 | * * * *  *  * 4  *  *  *  * | 0 1 1  0
x.xo .... x.oo&#x & ♦  4  3 |  2  2  0  4  4  1  2 | 0  1  0  0  2  2  2  0  2  0  4 0  1 | 0 0  0 0  1 0  1  0  0  0  0  2  0  2  0 0 | * * * *  *  * * 2n  *  *  * | 0 1 0  1
.... o.ox x.oo&#x & ♦  2  3 |  0  1  0  2  4  1  2 | 0  0  0  0  0  1  0  2  2  0  4 0  1 | 0 0  0 0  0 0  0  1  0  0  0  0  2  2  0 0 | * * * *  *  * *  * 2n  *  * | 0 1 0  1
xoxo .... ....&#x & ♦  3  3 |  1  0  2  3  4  1  2 | 0  0  1  0  1  0  1  1  0  4  4 0  1 | 0 0  0 0  0 0  0  0  1  1  0  1  1  0  2 0 | * * * *  *  * *  *  * 4n  * | 0 0 1  1
.... .... xxoo&#x   ♦  4  2 |  0  2  2  4  4  0  1 | 0  0  0  1  0  2  0  0  2  4  4 0  0 | 0 0  0 0  0 0  0  0  0  0  2  0  0  2  2 0 | * * * *  *  * *  *  *  * 2n | 0 0 0  2
--------------------+-------+----------------------+--------------------------------------+--------------------------------------------+-----------------------------+---------
xox.Noxo. xxo.&#x & ♦ 4n  n | 4n 2n 4n 2n 4n  n  0 | 4 2n 4n 2n 2n  n 2n 2n 2n 4n  0 1  0 | 2 2 2n 2  n 2  n  n 2n 2n 2n  0  0  0  0 0 | 1 1 1 2  n  n 0  0  0  0  0 | 2 * *  *
x.xoNo.ox x.oo&#x & ♦ 2n 2n | 2n  n  0 2n 4n 2n 2n | 2  n  0  0 2n  n 2n 2n 2n  0 4n 2 2n | 1 0  0 2  n 2  n  n  0  0  0 2n 2n 2n  0 1 | 0 1 1 0  0  0 2  n  n  0  0 | * 2 *  *
xoxoNoxox ....&#x   ♦ 2n 2n | 2n  0 2n 2n 4n 2n 2n | 2  0 2n  0 2n  0 2n 2n  0 4n 4n 2 2n | 0 1  0 2  0 2  0  0 2n 2n  0 2n 2n  0 2n 1 | 0 0 0 2  0  0 2  0  0 2n  0 | * * 2  *
xoxo .... xxoo&#x & ♦  6  3 |  2  3  4  6  8  1  2 | 0  1  2  2  2  3  2  2  4  8  8 0  1 | 0 0  1 0  1 0  1  1  2  2  4  2  2  4  4 0 | 0 0 0 0  1  1 0  1  1  2  2 | * * * 2n

© 2004-2024
top of page