Pyrochoron (EntityTopic, 17)
From Hi.gher. Space
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{{STS Shape | {{STS Shape | ||
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| elements=5, 10, 10, 5 | | elements=5, 10, 10, 5 | ||
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The following diagram shows a perspective projection of the pentachoron. | The following diagram shows a perspective projection of the pentachoron. | ||
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The dotted line shows the far edge of the outer tetrahedron. The blue lines are inside the outer tetrahedron in this projection. The center of the projection where these blue lines meet is actually the apex of the pentachoron pointing away from us in the 4th direction. | The dotted line shows the far edge of the outer tetrahedron. The blue lines are inside the outer tetrahedron in this projection. The center of the projection where these blue lines meet is actually the apex of the pentachoron pointing away from us in the 4th direction. | ||
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The following diagrams illustrate 3 of these cells. | The following diagrams illustrate 3 of these cells. | ||
- | <blockquote> | + | <blockquote><[#img [hash FQ9W842388C0W3BEZVKWJ6WNQ5]]> <[#img [hash ZVAC8G6M7QYXEMWHKXWH0Z05YY]]> <[#img [hash FMZT4C97B0ZKWXTVV9KQYJ19YZ]]></blockquote> |
Note that if a 4D being were to look at an actual pentachoron, it would not be able to see all 5 cells at once. Rather, it would either see only the outer tetrahedron (if it looks at the “base” of the pentachoron), or the 4 inner cells (if it looks at the apex of the pentachoron). | Note that if a 4D being were to look at an actual pentachoron, it would not be able to see all 5 cells at once. Rather, it would either see only the outer tetrahedron (if it looks at the “base” of the pentachoron), or the 4 inner cells (if it looks at the apex of the pentachoron). | ||
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The next diagram shows the pentachoron viewed at from another angle. | The next diagram shows the pentachoron viewed at from another angle. | ||
- | <blockquote> | + | <blockquote><[#img [hash 72FDJK0D3DC1RVNNQJPZ4SWYJZ]]></blockquote> |
In this diagram, three of the pentachoron's cells are arranged around the central axis indicated by the blue line. The other two cells are the upper and lower halves of the outer shape, which is called a ''trigonal bipyramid''. All the cells appear somewhat deformed from a regular tetrahedron, because they are all being viewed at from an angle. Some of these cells are shown below: | In this diagram, three of the pentachoron's cells are arranged around the central axis indicated by the blue line. The other two cells are the upper and lower halves of the outer shape, which is called a ''trigonal bipyramid''. All the cells appear somewhat deformed from a regular tetrahedron, because they are all being viewed at from an angle. Some of these cells are shown below: | ||
- | <blockquote> | + | <blockquote><[#img [hash SY65T4MAPCPHM0BJ07DNCY9VBM]]> <[#img [hash 5FRGJGCZDK9HC4MB313460BZC4]]> <[#img [hash 7CAJA4JKK9SJ27RZ0HTAED8W6E]]></blockquote> |
Again, this projection represents two possible views of the pentachoron. From one view, the 4D being would see only the upper and lower tetrahedra. From the opposite view, it would see the 3 inner tetrahedra. It cannot see all 5 cells at once unless the pentachoron is transparent. | Again, this projection represents two possible views of the pentachoron. From one view, the 4D being would see only the upper and lower tetrahedra. From the opposite view, it would see the 3 inner tetrahedra. It cannot see all 5 cells at once unless the pentachoron is transparent. |
Revision as of 22:34, 8 March 2011
The pyrochoron, also known as the pentachoron and the 5-cell, is the four-dimensional simplex, and has the lowest possible element count of any flat, non-degenerate four-dimensional shape. It consists of 5 regular tetrahedra joined at their faces, folded into 4D to form a 4D volume. It is a special case of the pyramid where the base is a tetrahedron. There are 3 tetrahedra surrounding every edge.
Equations
- Variables:
l ⇒ length of the edges of the pentachoron
- All points (x, y, z, w) that lie on the surface of a pentachoron will satisfy the following equation:
Unknown
- The hypervolumes of a pentachoron are given by:
total edge length = 10l
total surface area = 32-15l22-1
surcell volume = 22-1l33-1
bulk = 52-1l496-1
- The realmic cross-sections (n) of a pentachoron are:
Unknown
Net
The net of a pentachoron is a tetrahedron surrounded by 4 more tetrahedra.
Projection
Cell-first / vertex-first projection
The following diagram shows a perspective projection of the pentachoron.
ExPar: [#img] is obsolete, use [#embed] instead
The dotted line shows the far edge of the outer tetrahedron. The blue lines are inside the outer tetrahedron in this projection. The center of the projection where these blue lines meet is actually the apex of the pentachoron pointing away from us in the 4th direction.
All 5 tetrahedral cells of the pentachoron are present in this diagram: the outer tetrahedron, and the 4 “inner” tetrahedra outlined by one triangular face of the outer tetrahedron and 3 of the blue lines each. Although they appear as slightly flattened tetrahedra, this is only because they are being viewed at from an angle. In actuality, they are perfectly regular tetrahedra.
The following diagrams illustrate 3 of these cells.
ExPar: [#img] is obsolete, use [#embed] instead ExPar: [#img] is obsolete, use [#embed] instead ExPar: [#img] is obsolete, use [#embed] instead
Note that if a 4D being were to look at an actual pentachoron, it would not be able to see all 5 cells at once. Rather, it would either see only the outer tetrahedron (if it looks at the “base” of the pentachoron), or the 4 inner cells (if it looks at the apex of the pentachoron).
Edge-first / face-first projection
The next diagram shows the pentachoron viewed at from another angle.
ExPar: [#img] is obsolete, use [#embed] instead
In this diagram, three of the pentachoron's cells are arranged around the central axis indicated by the blue line. The other two cells are the upper and lower halves of the outer shape, which is called a trigonal bipyramid. All the cells appear somewhat deformed from a regular tetrahedron, because they are all being viewed at from an angle. Some of these cells are shown below:
ExPar: [#img] is obsolete, use [#embed] instead ExPar: [#img] is obsolete, use [#embed] instead ExPar: [#img] is obsolete, use [#embed] instead
Again, this projection represents two possible views of the pentachoron. From one view, the 4D being would see only the upper and lower tetrahedra. From the opposite view, it would see the 3 inner tetrahedra. It cannot see all 5 cells at once unless the pentachoron is transparent.
Simplices |
triangle • tetrahedron • pyrochoron • pyroteron • pyropeton |
Notable Tetrashapes | |
Regular: | pyrochoron • aerochoron • geochoron • xylochoron • hydrochoron • cosmochoron |
Powertopes: | triangular octagoltriate • square octagoltriate • hexagonal octagoltriate • octagonal octagoltriate |
Circular: | glome • cubinder • duocylinder • spherinder • sphone • cylindrone • dicone • coninder |
Torii: | tiger • torisphere • spheritorus • torinder • ditorus |
27. [111]1 Triangular prismic pyramid | 28. 13 Pentachoron | 29. 1111 Duotrianglinder |
List of tapertopes |