D4.16 (EntityTopic, 13)

From Hi.gher. Space

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I left the octahedra colored, even though they are on the far side of this projection, so that it's easier to orient yourself w.r.t. the polychoron in this 4D viewpoint.
I left the octahedra colored, even though they are on the far side of this projection, so that it's easier to orient yourself w.r.t. the polychoron in this 4D viewpoint.
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<[#polytope [id 142]]>

Revision as of 01:41, 15 March 2016

Most of the content on this page is taken directly from the forum thread where it was discovered.

D4.16 is a CRF polytope discovered by Klitzing and constructed by User:Quickfur in March 2016.

Its symbol is: xfox-2-oxfo-3-ooox&#xt

It has 3 mibdies and 2 tetrahedra on one side, 1 trigonal prism, 2 octahedra, 3 square pyramids, and 6 tetrahedra on the other side, for a total of 17 cells. It also has 17 vertices, the same as the cell count.

Coordinates

The coordinates are:

# x2o3o
<±1, 0, 0, -phi^2/√3>

# f2x3o
<±phi, ±1, -1/√3, 0>
<±phi, 0, 2/√3, 0>

# o2f3o
<0, ±phi, -phi/√3, phi/√3>
<0, 0, 2*phi/√3, phi/√3>

# x2o3x
<±1, ±1, 1/√3, phi^2/√3>
<±1, 0, -2/√3, phi^2/√3>

Images

Here's a look at the 3 mibdies surrounding an edge:

(image)

The 2 tetrahedra that fill in the gaps between the mibdies should be obvious.

Now here's the far side, which has a most interesting configuration of cells:

(image)

I highlighted the trigonal prism which lies antipodal to the edge shared by the 3 mibdies. We see a most interesting pattern of tetrahedron-square pyramid-tetrahedron interfacing each mibdi to the trigonal prism.

Now let's take a look at the side view, which is quite interesting:

(image)

Here, I turned off visibility clipping so that you can see the entire structure of the polychoron. The two octahedral cells are highlighted in green. They happen to project concentrically from this particular 4D viewpoint. The square pyramids aren't very obvious, but you can find them if you first look for the triangular prism (foreshortened to be quite squished) at the bottom of the projection.

Here's another side view, showing how two mibdies are folded together with a square pyramid sandwiched between them:

(image)

I left the octahedra colored, even though they are on the far side of this projection, so that it's easier to orient yourself w.r.t. the polychoron in this 4D viewpoint.

Incidence matrix

Dual: D4.16 dual

#TXIDVaVbVcVdEaEbEcEdEeEfEgEh5a3a3b3c3d3e3f3g3h4aC1aC2aC3aC4aC5aC2bTypeName
0 Va = point ; in digon
1 Vb = point ; outermost
2 Vc = point ; tips
3 Vd = point ; in trip
4 Ea 2000 = digon ; digon
5 Eb 1100 = digon ; digon to outermost
6 Ec 0110 = digon ; outermost to tips
7 Ed 0101 = digon ; outermost to trip
8 Ee 0011 = digon ; tips to trip
9 Ef 0200 = digon ; between outermost
10 Eg 0002 = digon ; ends of trip
11 Eh 0002 = digon ; sweep of trip
12 5a 221012200000 = pentagon ;
13 3a 120002000100 = triangle ; digon to outermost
14 3b 030000000300 = triangle ; equatorial caps
15 3c 020100020100 = triangle ; equatorial cap-petals
16 3d 011100111000 = triangle ; equatorial sweep-petals
17 3e 001200002001 = triangle ; equatorial sweeps
18 3f 010200020010 = triangle ; cap guard
19 3g 001200002010 = triangle ; sweep guard
20 3h 000300000030 = triangle ; ends of trip
21 4a 000400000022 = square ; sweep of trip
22 C1a 2422144442012202420000 = metabidiminished icosahedron ;
23 C2a 1300030003000310000000 = tetrahedron ; cap tets
24 C3a 0006000000630000000023 = triangular prism ;
25 C4a 0303000603300013003010 = octahedron ;
26 C5a 0014000040220000020201 = square pyramid ;
27 C2b 0112001220100000201100 = tetrahedron ; sweep tets
28 H4.1a 2636166121266336261266623321236 = D4.16 ;

Usage as facets

This polytope does not currently appear as facets in any higher-dimensional polytopes in the database.