- published: 11 Feb 2011
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In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices truncated. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM6 for a 6-dimensional half measure polytope.
Coxeter named this polytope as 131 from its Coxeter diagram, with a ring on one of the 1-length branches, . It can named similarly by a 3-dimensional exponential Schläfli symbol or {3,33,1}.
Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:
with an odd number of plus signs.
There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:
The 6-demicube, 131 is third in a dimensional series of uniform polytopes, expressed by Coxeter as k31 series. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each progressive uniform polytope is constructed from the previous as its vertex figure.
Tears in my ears
I can’t stop crying...
Tears in my ears
I can’t stop...
Tears in my ears
I can’t stop crying...
Tears in my ears
I can’t stop crying...
I found a new stain
The deepest i know
I wonder for how long
I’ll be stucked on the snow
Last time i had no chance to refuse
One day I’ll show you
I’m not so confused...
Last hopings of night
I just couldn’t jump over the dike
Whatever can’t stop it
Don't be just what you’re like
There are no last chances
So i hardly ever had one
Any further
No lasting
Just being always wrong
I’m so sorry after all
Now I see I have no future without making up to you
I left the fire on again and no one went for help but
Every other seem so far away from you
So why...?
I’m not so confused...
Last time i had no chance to refuse
One day I’ll show you