Geoteron (EntityTopic, 20)
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| - | {{Shape| | + | <[#ontology [kind topic] [cats 5D Hypercube]]> |
| + | {{STS Shape | ||
| + | | image=<[#embed [hash T0AB6F1XQAXC4F5VQ1QQG132B3] [width 180]]><br>[[Petrie polygon]] | ||
| + | | dim=5 | ||
| + | | elements=10, 40, 80, 80, 32 | ||
| + | | genus=0 | ||
| + | | ssc=[xyzwφ] | ||
| + | | ssc2=K5c1 | ||
| + | | pv_square=1 | ||
| + | | extra={{STS Tapertope | ||
| + | | order=5, 0 | ||
| + | | notation=11111 | ||
| + | | index=36 | ||
| + | }}{{STS Toratope | ||
| + | | expand=[[Penteract|11111]] | ||
| + | | notation=IIIII | ||
| + | | index=9a | ||
| + | }}{{STS Bracketope | ||
| + | | index=31 | ||
| + | }}{{STS Polytope | ||
| + | | altern=[[Demipenteract]] | ||
| + | | dual=[[Aeroteron]] | ||
| + | }}{{STS Uniform polytope | ||
| + | | vfigure=[[Pyrochoron]], edge 2 | ||
| + | | vlayout=(([[Square|4]][[Cube|<sup>3</sup>]])[[Tesseract|<sup>4</sup>]])<sup>5</sup> | ||
| + | | dual=[[Icosidodecateron]] | ||
| + | }}}} | ||
| + | The '''geoteron''', also known as the '''penteract''', the '''pentacube''' and the [[regular]] '''decateron''' is the five-dimensional [[hypercube]]. It is a special case of the [[prism]] where the base is a [[geochoron]]. | ||
| - | + | == Equations == | |
| - | + | ||
| - | + | ||
| - | + | ||
*Variables: | *Variables: | ||
| - | <blockquote>''l'' ⇒ length of the edges of the | + | <blockquote>''l'' ⇒ length of the edges of the penteract</blockquote> |
| - | *All points (''x'', ''y'', ''z'', ''w'', ''φ'') that lie on the [[surteron]] of a | + | *All points (''x'', ''y'', ''z'', ''w'', ''φ'') that lie on the [[surteron]] of a penteract will satisfy the following equation: |
<blockquote>''Unknown''</blockquote> | <blockquote>''Unknown''</blockquote> | ||
| - | *The [[hypervolume]]s of a | + | *The [[hypervolume]]s of a penteract are given by: |
| - | <blockquote>total edge length = | + | <blockquote>total edge length = 80''l''<br> |
| - | total surface area = | + | total surface area = 80''l''<sup>2</sup><br> |
| - | total surcell volume = | + | total surcell volume = 40''l''<sup>3</sup><br> |
| - | surteron bulk = | + | surteron bulk = 10''l''<sup>4</sup><br> |
| - | pentavolume = l<sup>5</sup></blockquote> | + | pentavolume = ''l''<sup>5</sup></blockquote> |
| - | *The [[flunic]] [[cross-section]]s (''n'') of a | + | *The [[flunic]] [[cross-section]]s (''n'') of a penteract are: |
<blockquote>[!x, !y, !z, !w, !φ] ⇒ tesseract of side (''l'')</blockquote> | <blockquote>[!x, !y, !z, !w, !φ] ⇒ tesseract of side (''l'')</blockquote> | ||
| - | + | == Net == | |
| - | The net of a | + | The net of a penteract is a tesseract surrounded by eight more tesseracts, with one more tesseract added to one of these. |
| + | |||
| + | <[#polytope [id 90]]> | ||
| - | {{ | + | {{Hypercubes|5}} |
| - | {{ | + | {{Pentashapes}} |
| + | {{Tapertope Nav|35|36|37|2111<br>Tesserinder|11111<br>Penteract|4<sup>1</sup><br>Glone|tera}} | ||
| + | {{Toratope Nav A|8|9|10|((II)I)I<br>Torinder|(((II)I)I)<br>Ditorus|IIIII<br>Penteract|(IIIII)<br>Pentasphere|(II)III<br>Tesserinder|((II)III)<br>Toratesserinder|tera}} | ||
| + | {{Bracketope Nav|30|31|32|([II][II])<br>Duocrind|[IIIII]<br>Geoteron|<nowiki><IIIII></nowiki><br>Aeroteron|tera}} | ||
Latest revision as of 21:12, 14 March 2014
The geoteron, also known as the penteract, the pentacube and the regular decateron is the five-dimensional hypercube. It is a special case of the prism where the base is a geochoron.
Equations
- Variables:
l ⇒ length of the edges of the penteract
- All points (x, y, z, w, φ) that lie on the surteron of a penteract will satisfy the following equation:
Unknown
- The hypervolumes of a penteract are given by:
total edge length = 80l
total surface area = 80l2
total surcell volume = 40l3
surteron bulk = 10l4
pentavolume = l5
- The flunic cross-sections (n) of a penteract are:
[!x, !y, !z, !w, !φ] ⇒ tesseract of side (l)
Net
The net of a penteract is a tesseract surrounded by eight more tesseracts, with one more tesseract added to one of these.
Incidence matrix
Dual: aeroteron
| # | TXID | Va | Ea | 4a | C1a | H4.1a | Type | Name |
|---|---|---|---|---|---|---|---|---|
| 0 | Va | = point | ; | |||||
| 1 | Ea | 2 | = digon | ; | ||||
| 2 | 4a | 4 | 4 | = square | ; | |||
| 3 | C1a | 8 | 12 | 6 | = cube | ; | ||
| 4 | H4.1a | 16 | 32 | 24 | 8 | = base of prism: geochoron | ; | |
| 5 | H5.1a | 32 | 80 | 80 | 40 | 10 | = geoteron | ; |
Usage as facets
This polytope does not currently appear as facets in any higher-dimensional polytopes in the database.
| Hypercubes |
| point • digon • square • cube • geochoron • geoteron • geopeton |
| Notable Pentashapes | |
| Flat: | pyroteron • aeroteron • geoteron |
| Curved: | tritorus • pentasphere • glone • cylspherinder • tesserinder |
| 35. 2111 Tesserinder | 36. 11111 Penteract | 37. 41 Glone |
| List of tapertopes | ||
| 8a. ((II)I)I Torinder | 8b. (((II)I)I) Ditorus | 9a. IIIII Penteract | 9b. (IIIII) Pentasphere | 10a. (II)III Tesserinder | 10b. ((II)III) Toratesserinder |
| List of toratopes | |||||
| 30. ([II][II]) Duocrind | 31. [IIIII] Geoteron | 32. <IIIII> Aeroteron |
| List of bracketopes | ||
