Cosmochoron (EntityTopic, 12)

From Hi.gher. Space

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http://teamikaria.com/dl/Pmyg-el9wlGbaOueUDKBuTbIo5xIx8iFlG8rLaBYljwi-z1f.gif http://teamikaria.com/dl/4TMygnHLhLXtUc0oRXtHFapu60FMLK_ig1kN2fN5GSeSdbFz.gif
http://teamikaria.com/dl/Pmyg-el9wlGbaOueUDKBuTbIo5xIx8iFlG8rLaBYljwi-z1f.gif http://teamikaria.com/dl/4TMygnHLhLXtUc0oRXtHFapu60FMLK_ig1kN2fN5GSeSdbFz.gif
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Here are two perspective projections of the hecatonicosachoron into 3D. The highlighted center cell is closest to the 4D viewpoint. It is surrounded by 12 [[dodecahedra]] in a second layer. The right-hand image also shows the third layer of 47 dodecahedra. There are three more layers which are mirror images of the first three and cannot be seen as they are on the 4D "back" of the shape. Note that only the projection is rotating; the 4D shape itself is not.
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Here are two perspective projections of the hecatonicosachoron into 3D. The highlighted center cell is closest to the 4D viewpoint. It is surrounded by 12 [[dodecahedra]] in a second layer. The right-hand image also shows the third layer of 32 dodecahedra. After this is a fourth layer of 30 dodecahedra, followed by three more layers mirroring the three layers seen here (not seen here, because they lie on the 4D "back" of the shape).
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Note that only the projection is rotating; the 4D shape itself is not.
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{{Tetrashapes}}
{{Tetrashapes}}
[[Category:Regular polychora]]
[[Category:Regular polychora]]

Revision as of 20:27, 2 September 2008

The hecatonicosachoron, or 120-cell, is a 4D polytope bounded by 120 dodecahedra. Each dodecahedron lies on a ring of 10 dodecahedra along a great circle of the glome. There are 4 dodecahedra on such a ring between one dodecahedron and its antipodal dodecahedron.

Geometry

Equations

  • Variables:
l ⇒ length of the edges of the hecatonicosachoron
  • All points (x, y, z, w) that lie on the surcell of a hecatonicosachoron will satisfy the following equation:
Unknown
total edge length = 1200l
total surface area = 180l2sqrt(25+10sqrt(5))
surcell volume = 300l3(tan(3π10-1))2(tan(sin-1(2sin(π5-1))-1))
bulk = Unknown
Unknown

Projection

http://teamikaria.com/dl/Pmyg-el9wlGbaOueUDKBuTbIo5xIx8iFlG8rLaBYljwi-z1f.gif http://teamikaria.com/dl/4TMygnHLhLXtUc0oRXtHFapu60FMLK_ig1kN2fN5GSeSdbFz.gif

Here are two perspective projections of the hecatonicosachoron into 3D. The highlighted center cell is closest to the 4D viewpoint. It is surrounded by 12 dodecahedra in a second layer. The right-hand image also shows the third layer of 32 dodecahedra. After this is a fourth layer of 30 dodecahedra, followed by three more layers mirroring the three layers seen here (not seen here, because they lie on the 4D "back" of the shape).

Note that only the projection is rotating; the 4D shape itself is not.


Notable Tetrashapes
Regular: pyrochoronaerochorongeochoronxylochoronhydrochoroncosmochoron
Powertopes: triangular octagoltriatesquare octagoltriatehexagonal octagoltriateoctagonal octagoltriate
Circular: glomecubinderduocylinderspherindersphonecylindronediconeconinder
Torii: tigertorispherespheritorustorinderditorus