Geoteron (EntityTopic, 20)

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{{Shape|Pentacube|''No image''|5|10, 40, 80, 80, 32|0|{[[Square|4,]][[Cube|3,]][[Tesseract|3,]]3}|N/A|[[Line (object)|E]][[Square|E]][[Cube|E]][[Tesseract|E]]E|11111 xyzwφ|[[Pentachoron]], edge 2|N/A|[[Icosidodecateron]]|45|[xyzwφ]|50|pure}}
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<[#ontology [kind topic] [cats 5D Hypercube]]>
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{{STS Shape
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| image=<[#embed [hash T0AB6F1XQAXC4F5VQ1QQG132B3] [width 180]]><br>[[Petrie polygon]]
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| dim=5
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| elements=10, 40, 80, 80, 32
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| genus=0
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| ssc=[xyzwφ]
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| ssc2=K5c1
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| pv_square=1
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| extra={{STS Tapertope
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| order=5, 0
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| notation=11111
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| index=36
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}}{{STS Toratope
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| expand=[[Penteract|11111]]
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| notation=IIIII
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| index=9a
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}}{{STS Bracketope
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| index=31
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}}{{STS Polytope
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| altern=[[Demipenteract]]
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| dual=[[Aeroteron]]
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}}{{STS Uniform polytope
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| vfigure=[[Pyrochoron]], edge 2
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| vlayout=(([[Square|4]][[Cube|<sup>3</sup>]])[[Tesseract|<sup>4</sup>]])<sup>5</sup>
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| dual=[[Icosidodecateron]]
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}}}}
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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]].
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== Geometry ==
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== Equations ==
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A '''pentacube''', also known as a [[regular]] ''decateron'' is a special case of the [[prism]] where the base is a [[tesseract]].
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=== Equations ===
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*Variables:
*Variables:
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<blockquote>''l'' ⇒ length of the edges of the pentacube</blockquote>
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<blockquote>''l'' ⇒ length of the edges of the penteract</blockquote>
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*All points (''x'', ''y'', ''z'', ''w'', ''φ'') that lie on the [[surteron]] of a pentacube will satisfy the following equation:
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*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>
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*The [[hypervolume]]s of a pentacube are given by:
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*The [[hypervolume]]s of a penteract are given by:
<blockquote>total edge length = 80''l''<br>
<blockquote>total edge length = 80''l''<br>
total surface area = 80''l''<sup>2</sup><br>
total surface area = 80''l''<sup>2</sup><br>
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pentavolume = ''l''<sup>5</sup></blockquote>
pentavolume = ''l''<sup>5</sup></blockquote>
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*The [[flunic]] [[cross-section]]s (''n'') of a pentacube are:
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*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>
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=== Net ===
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== Net ==
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The net of a pentacube is a tesseract surrounded by eight more tesseracts, with one more tesseract added to one of these.
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The net of a penteract is a tesseract surrounded by eight more tesseracts, with one more tesseract added to one of these.
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<[#polytope [id 90]]>
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{{Hypercubes|5}}
{{Pentashapes}}
{{Pentashapes}}
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{{Rotope Nav|44|45|46|((II)(II))<br>Tiger|IIIII<br>Pentacube|IIII'<br>Tesseric pyramid|tera}}
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{{Tapertope Nav|35|36|37|2111<br>Tesserinder|11111<br>Penteract|4<sup>1</sup><br>Glone|tera}}
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{{Bracketope Nav|49|50|51|(<xy><zw>)<br>Doubly-narrow duocrind|[xyzwφ]<br>Pentacube|[<xy>zwφ]<br>Narrow pentacube|tera}}
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{{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}}
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[[Category:Regular polytera]]
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{{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
total edge length = 80l
total surface area = 80l2
total surcell volume = 40l3
surteron bulk = 10l4
pentavolume = l5
[!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

#TXIDVaEa4aC1aH4.1aTypeName
0 Va = point ;
1 Ea 2 = digon ;
2 4a 44 = square ;
3 C1a 8126 = cube ;
4 H4.1a 1632248 = base of prism: geochoron ;
5 H5.1a 3280804010 = 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
Retrieved from "http://hi.gher.space/wiki/Geoteron"