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#REDIRECT[[tetrahedron#Regular tetrahedron]] |
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A '''regular tetrahedron''' is a tetrahedron in which all four faces are [[equilateral triangle]]s. It is one of the five regular [[Platonic solid]]s, which have been known since antiquity. |
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In a regular tetrahedron, all faces are the same size and shape (congruent) and all edges are the same length. |
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[[File:Tetrahedrons cannot fill space..PNG|thumb|300px|Five tetrahedra are laid flat on a plane, with the highest 3-dimensional points marked as 1, 2, 3, 4, and 5. These points are then attached to each other and [[Angular defect|a thin volume of empty space]] is left, where the five edge angles do not quite meet.]] |
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Regular tetrahedra alone do not [[Tessellation#Tessellations in higher dimensions|tessellate]] (fill space), but if alternated with [[regular octahedron|regular octahedra]] in the ratio of two tetrahedra to one octahedron, they form the [[alternated cubic honeycomb]], which is a tessellation. Some tetrahedra that are not regular, including the [[Schläfli orthoscheme]] and the [[Hill tetrahedron]], [https://demonstrations.wolfram.com/SpaceFillingTetrahedra/ can tessellate]. |
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The regular tetrahedron is self-dual, which means that its [[Dual polyhedron|dual]] is another regular tetrahedron. The [[Polyhedral compound|compound]] figure comprising two such dual tetrahedra form a [[stellated octahedron]] or stella octangula. |
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==Coordinates for a regular tetrahedron==<!--[[Tetrahedral angle]] redirects here--> |
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The following Cartesian coordinates define the four vertices of a tetrahedron with edge length 2, centered at the origin, and two level edges: |
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:<math>\left(\pm 1, 0, -\frac{1}{\sqrt{2}}\right) \quad \mbox{and} \quad \left(0, \pm 1, \frac{1}{\sqrt{2}}\right)</math> |
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Expressed symmetrically as 4 points on the [[unit sphere]], centroid at the origin, with lower face level, the vertices are: |
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<math>v_1 = \left(\sqrt{\frac{8}{9}},0,-\frac{1}{3}\right)</math> |
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<math>v_2 = \left(-\sqrt{\frac{2}{9}},\sqrt{\frac{2}{3}},-\frac{1}{3}\right)</math> |
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<math>v_3 = \left(-\sqrt{\frac{2}{9}},-\sqrt{\frac{2}{3}},-\frac{1}{3}\right)</math> |
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<math>v_4 = (0,0,1)</math> |
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with the edge length of <math>\sqrt{\frac{8}{3}}</math>. |
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Still another set of coordinates are based on an [[Alternation (geometry)|alternated]] [[cube]] or '''demicube''' with edge length 2. This form has [[Coxeter diagram]] {{CDD|node_h|4|node|3|node}} and [[Schläfli symbol]] h{4,3}. The tetrahedron in this case has edge length 2{{sqrt|2}}. Inverting these coordinates generates the dual tetrahedron, and the pair together form the stellated octahedron, whose vertices are those of the original cube. |
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:Tetrahedron: (1,1,1), (1,−1,−1), (−1,1,−1), (−1,−1,1) |
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:Dual tetrahedron: (−1,−1,−1), (−1,1,1), (1,−1,1), (1,1,−1) |
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[[File:Вписанный тетраэдр.svg|thumb|right|300px|Regular tetrahedron ABCD and its circumscribed sphere]] |
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===Angles and distances=== |
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For a regular tetrahedron of edge length ''a'': |
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{|class="wikitable" |
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|width=300|Face area |
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|align=center|<math>A_0=\frac{\sqrt{3}}{4}a^2\,</math> |
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|- |
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|[[area (mathematics)|Surface area]]<ref name="Cox">[[Harold Scott MacDonald Coxeter|Coxeter, Harold Scott MacDonald]]; ''[[Regular Polytopes (book)|Regular Polytopes]]'', Methuen and Co., 1948, Table I(i)</ref> |
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|align=center|<math>A=4\,A_0={\sqrt{3}}a^2\,</math> |
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|- |
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|Height of pyramid<ref>Köller, Jürgen, [http://www.mathematische-basteleien.de/tetrahedron.htm "Tetrahedron"], Mathematische Basteleien, 2001</ref> |
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|align=center|<math>h=\frac{\sqrt{6}}{3}a=\sqrt{\frac23}\,a\,</math> |
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|- |
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|Centroid to vertex distance |
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|align=center|<math>\frac34\,h = \frac{\sqrt{6}}{4}\,a = \sqrt{\frac{3}{8}}\,a\,</math> |
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|- |
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|Edge to opposite edge distance |
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|align=center|<math>l=\frac{1}{\sqrt{2}}\,a\,</math> |
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|- |
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|[[Volume]]<ref name="Cox" /> |
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|align=center|<math>V=\frac13 A_0h =\frac{\sqrt{2}}{12}a^3=\frac{a^3}{6\sqrt{2}}\,</math> |
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|- |
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|Face-vertex-edge angle |
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|align=center|<math>\arccos\left(\frac{1}{\sqrt{3}}\right) = \arctan\left(\sqrt{2}\right)\,</math><br>(approx. 54.7356°) |
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|- |
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|[[Table of polyhedron dihedral angles|Face-edge-face angle]], i.e., "dihedral angle"<ref name="Cox" /> |
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|align=center|<math>\arccos\left(\frac13\right) = \arctan\left(2\sqrt{2}\right)\,</math><br>(approx. 70.5288°) |
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|- |
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|Vertex-Center-Vertex angle,<ref name="pubs.acs.org">{{cite journal|doi=10.1021/ed022p145 | volume=22 | issue=3 | title=Valence angle of the tetrahedral carbon atom | year=1945 | journal=Journal of Chemical Education | page=145 | last1 = Brittin | first1 = W. E.| bibcode=1945JChEd..22..145B }}</ref> the angle between lines from the tetrahedron center to any two vertices. It is also the angle between [[Plateau's laws|Plateau borders]] at a vertex. In chemistry it is called the [[Tetrahedral molecular geometry|tetrahedral bond angle]]. This angle (in radians) is also the arclength of the geodesic segment on the unit sphere resulting from centrally projecting one edge of the tetrahedron to the sphere. |
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|align=center|<math>\arccos\left(-\frac13\right ) = 2\arctan\left(\sqrt{2}\right)\,</math><br>(approx. 109.4712°) |
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|- |
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|[[Solid angle]] at a vertex subtended by a face |
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|align=center|<math>\arccos\left(\frac{23}{27}\right)</math><br>(approx. 0.55129 [[steradian]]s)<br>(approx. 1809.8 [[square degree]]s) |
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|- |
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|Radius of [[circumsphere]]<ref name="Cox" /> |
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|align=center|<math>R=\frac{\sqrt{6}}{4}a=\sqrt{\frac38}\,a\,</math> |
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|- |
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|Radius of [[insphere]] that is tangent to faces<ref name="Cox" /> |
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|align=center|<math>r=\frac13R=\frac{a}{\sqrt{24}}\,</math> |
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|- |
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|Radius of [[midsphere]] that is tangent to edges<ref name="Cox" /> |
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|align=center|<math>r_\mathrm{M}=\sqrt{rR}=\frac{a}{\sqrt{8}}\,</math> |
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|- |
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|Radius of [[Exsphere (polyhedra)|exspheres]] |
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|align=center|<math>r_\mathrm{E}=\frac{a}{\sqrt{6}}\,</math> |
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|- |
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|Distance to exsphere center from the opposite vertex |
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|align=center|<math>d_\mathrm{VE}=\frac{\sqrt{6}}{2}a={\sqrt{\frac32}}a\,</math> |
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|} |
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With respect to the base plane the [[slope]] of a face (2{{sqrt|2}}) is twice that of an edge ({{sqrt|2}}), corresponding to the fact that the ''horizontal'' distance covered from the base to the [[Apex (geometry)|apex]] along an edge is twice that along the [[Median (geometry)|median]] of a face. In other words, if ''C'' is the [[centroid]] of the base, the distance from ''C'' to a vertex of the base is twice that from ''C'' to the midpoint of an edge of the base. This follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other (see [[centroid#Proof that the centroid of a triangle divides each median in the ratio 2:1|proof]]). |
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For a regular tetrahedron with side length ''a'', radius ''R'' of its circumscribing sphere, and distances ''d<sub>i</sub>'' from an arbitrary point in 3-space to its four vertices, we have<ref>Park, Poo-Sung. "Regular polytope distances", [[Forum Geometricorum]] 16, 2016, 227–232. http://forumgeom.fau.edu/FG2016volume16/FG201627.pdf</ref> |
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:<math>\begin{align}\frac{d_1^4 + d_2^4 + d_3^4 + d_4^4}{4} + \frac{16R^4}{9}&= \left(\frac{d_1^2 + d_2^2 + d_3^2 + d_4^2}{4} + \frac{2R^2}{3}\right)^2;\\ |
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4\left(a^4 + d_1^4 + d_2^4 + d_3^4 + d_4^4\right) &= \left(a^2 + d_1^2 + d_2^2 + d_3^2 + d_4^2\right)^2.\end{align}</math> |
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==Isometries of the regular tetrahedron== |
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[[Image:Symmetries of the tetrahedron.svg|thumb|400px|The proper rotations, (order-3 rotation on a vertex and face, and order-2 on two edges) and reflection plane (through two faces and one edge) in the symmetry group of the regular tetrahedron]] |
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The vertices of a [[cube]] can be grouped into two groups of four, each forming a regular tetrahedron (see above, and also [[:Image:Tetraeder animation with cube.gif|animation]], showing one of the two tetrahedra in the cube). The [[Symmetry in mathematics|symmetries]] of a regular tetrahedron correspond to half of those of a cube: those that map the tetrahedra to themselves, and not to each other. |
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The tetrahedron is the only Platonic solid that is not mapped to itself by [[point inversion]]. |
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The regular tetrahedron has 24 isometries, forming the [[symmetry group]] '''T<sub>d</sub>''', [3,3], (*332), isomorphic to the [[symmetric group]], ''S''<sub>4</sub>. They can be categorized as follows: |
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* '''T''', [3,3]<sup>+</sup>, (332) is isomorphic to [[alternating group]], ''A''<sub>4</sub> (the identity and 11 proper rotations) with the following [[conjugacy class]]es (in parentheses are given the permutations of the vertices, or correspondingly, the faces, and the [[Quaternions and spatial rotation|unit quaternion representation]]): |
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** identity (identity; 1) |
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** rotation about an axis through a vertex, perpendicular to the opposite plane, by an angle of ±120°: 4 axes, 2 per axis, together {{nowrap|8 ((1 2 3)}}, etc.; {{sfrac|1 ± ''i'' ± ''j'' ± ''k''|2}}) |
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** rotation by an angle of 180° such that an edge maps to the opposite edge: {{nowrap|3 ((1 2)(3 4)}}, etc.; {{nowrap|''i'', ''j'', ''k''}}) |
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* reflections in a plane perpendicular to an edge: 6 |
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* reflections in a plane combined with 90° rotation about an axis perpendicular to the plane: 3 axes, 2 per axis, together 6; equivalently, they are 90° rotations combined with inversion ('''x''' is mapped to −'''x'''): the rotations correspond to those of the cube about face-to-face axes |
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==Orthogonal projections of the regular tetrahedron== |
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The regular ''tetrahedron'' has two special [[orthogonal projection]]s, one centered on a vertex or equivalently on a face, and one centered on an edge. The first corresponds to the A<sub>2</sub> [[Coxeter plane]]. |
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{| class=wikitable |
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|+ [[Orthographic projection]] |
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!Centered by |
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!Face/vertex |
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!Edge |
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|- align=center |
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!Image |
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|[[File:3-simplex t0 A2.svg|100px]] |
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|[[File:3-simplex t0.svg|100px]] |
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|- align=center |
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!Projective<br>symmetry |
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![3] |
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![4] |
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|} |
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==Cross section of regular tetrahedron== |
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[[File:Regular_tetrahedron_square_cross_section.png|120px|thumb|A central cross section of a ''regular tetrahedron'' is a [[square]].]] |
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The two skew perpendicular opposite edges of a ''regular tetrahedron'' define a set of parallel planes. When one of these planes intersects the tetrahedron the resulting cross section is a [[rectangle]].<ref>[http://www.matematicasvisuales.com/english/html/geometry/space/sectetra.html Sections of a Tetrahedron]</ref> When the intersecting plane is near one of the edges the rectangle is long and skinny. When halfway between the two edges the intersection is a [[square]]. The aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly. If the tetrahedron is bisected on this plane, both halves become [[Wedge (geometry)|wedges]]. |
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[[File:Tetragonal disphenoid diagram.png|thumb|100px|left|A tetragonal disphenoid viewed orthogonally to the two green edges.]] |
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This property also applies for [[tetragonal disphenoid]]s when applied to the two special edge pairs. |
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{{Clear}} |
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==Spherical tiling== |
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The tetrahedron can also be represented as a [[spherical tiling]], and projected onto the plane via a [[stereographic projection]]. This projection is [[Conformal map|conformal]], preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane. |
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{|class=wikitable |
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|- align=center |
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|[[File:Uniform tiling 332-t2.png|160px]] |
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|[[File:Tetrahedron stereographic projection.svg|160px]] |
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|- |
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![[Orthographic projection]] |
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!colspan=1|[[Stereographic projection]] |
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|} |
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== Helical stacking == |
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[[File:600-cell tet ring.png|thumb|A single 30-tetrahedron ring [[Boerdijk–Coxeter helix]] within the [[600-cell]], seen in stereographic projection]] |
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Regular tetrahedra can be stacked face-to-face in a chiral aperiodic chain called the [[Boerdijk–Coxeter helix]]. In [[Four-dimensional space|four dimensions]], all the convex [[regular 4-polytope]]s with tetrahedral cells (the [[5-cell#Boerdijk–Coxeter helix|5-cell]], [[16-cell#Boerdijk–Coxeter helix|16-cell]] and [[600-cell#Union of two tori|600-cell]]) can be constructed as tilings of the [[3-sphere]] by these chains, which become periodic in the three-dimensional space of the 4-polytope's boundary surface. |
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{{Polyhedra}} |
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{{Convex polyhedron navigator}} |
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{{Polytopes}} |
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[[Category:Platonic solids]] |
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Revision as of 14:02, 19 August 2021
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