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Compound of five tetrahedra
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<h2 id="as-a-compound">As a compound</h2> <p>It can be constructed by arranging five <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a> in <a href="/facts/Rotational_icosahedral_symmetry/pDlwAy23">rotational icosahedral symmetry</a> (I), as colored in the upper right model. It is one of <a href="/facts/Polytope_compound/L13pItNN">five regular compounds</a> which can be constructed from identical <a href="/facts/Platonic_solid/jraREXFD">Platonic solids</a>.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p><p>It shares the same <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a> as a <a href="/facts/Regular_dodecahedron/xClYqhcp">regular dodecahedron</a>. </p><p>There are two <a href="/facts/Chirality_(mathematics)/FtKNlHTb">enantiomorphous</a> forms (the same figure but having opposite chirality) of this compound polyhedron. Both forms together create the reflection symmetric <a href="/facts/Compound_of_ten_tetrahedra/z9LgGAPL">compound of ten tetrahedra</a>. </p><p>It has a <a href="/facts/Density_(polyhedra)/LLQmbxty">density</a> of higher than 1. </p> <table><tbody><tr><td>As a <a href="/facts/Spherical_tiling/jaglysFR">spherical tiling</a></td><td>Transparent Models(Animation)</td><td>Five interlocked tetrahedra</td></tr></tbody></table> <h2 id="as-a-stellation">As a stellation</h2> <p>It can also be obtained by <a href="/facts/Stellation/T7NY6CWv">stellating</a> the <a href="/facts/Icosahedron/wWbpnDYp">icosahedron</a>, and is given as <a href="/facts/List_of_Wenninger_polyhedron_models/hwApnNpt">Wenninger model index 24</a>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <table><tbody><tr><th><a href="/facts/Stellation_diagram/8b5PuPkH">Stellation diagram</a></th><th><a href="/facts/Stellation/T7NY6CWv">Stellation</a> core</th><th><a href="/facts/Convex_hull/D4Id3JDS">Convex hull</a></th></tr><tr><td></td><td><a href="/facts/Stellations_of_icosahedron/syuwcVfg">Icosahedron</a></td><td><a href="/facts/Dodecahedron/7r519eCl">Dodecahedron</a></td></tr></tbody></table> <h2 id="as-a-faceting">As a faceting</h2> <p>It is a <a href="/facts/Faceting/Yte7F7bh">faceting</a> of a dodecahedron, as shown at left. </p> <h2 id="group-theory">Group theory</h2> <p>The compound of five tetrahedra is a geometric illustration of the notion of <a href="/facts/Group_action_(mathematics)/Vh9VtUUw">orbits and stabilizers</a>, as follows. </p><p>The symmetry group of the compound is the (rotational) <a href="/facts/Icosahedral_group/pDlwAy23">icosahedral group</a> <i>I</i> of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) <a href="/facts/Tetrahedral_group/hIVlf9pu">tetrahedral group</a> <i>T</i> of order 12, and the orbit space <i>I</i>/<i>T</i> (of order 60/12 = 5) is naturally identified with the 5 tetrahedra – the coset <i>gT</i> corresponds to which tetrahedron <i>g</i> sends the chosen tetrahedron to. </p> <h2 id="an-unusual-dual-property">An unusual dual property</h2> <p>This compound is unusual, in that the <a href="/facts/Dual_polyhedron/rkIHbmAf">dual</a> figure is the <a href="/facts/Enantiomorph/FtKNlHTb">enantiomorph</a> of the original. If the faces are twisted to the right, then the vertices are twisted to the left. When we <a href="/facts/Dual_polyhedron/rkIHbmAf">dualise</a>, the faces dualise to right-twisted vertices and the vertices dualise to left-twisted faces, giving the chiral twin. Figures with this property are extremely rare. </p> <h2 id="in-4-dimensional-space">In 4-dimensional space</h2> <p>The compound of five tetrahedra is related to the regular <a href="/facts/5-cell/Rh6lwDcc">5-cell</a>, the 4-simplex <a href="/facts/Regular_4-polytope/lFf8IH1C">regular 4-polytope</a>, which is also composed of 5 regular tetrahedra. In the 5-cell the tetrahedra are bonded face-to-face such that each triangular face is shared by two tetrahedral cells. </p><p>The compound of five tetrahedra occurs embedded in <a href="/facts/4-dimensional_space/vFQd0M7T">4-dimensional space</a>, inscribed in the 120 dodecahedral cells of the <a href="/facts/120-cell/W4nroBjJ">120-cell</a>. The 120-cell is the <a href="/facts/120-cell/W4nroBjJ">largest and most comprehensive regular 4-polytope</a>; the regular 5-cell is the smallest and most elemental. The 120-cell contains inscribed within itself instances of every other regular 4-polytope.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> In each of the 120-cell's dodecahedral cells, there are two inscribed instances of the compound of 5 tetrahedra (in other words, one instance of the <a href="/facts/Compound_of_ten_tetrahedra/z9LgGAPL">compound of ten tetrahedra</a>). The 5 tetrahedra of each compound of five occur as cells of another regular 4-polytope inscribed within the 120-cell, the <a href="/facts/600-cell/tNBtDUe9">600-cell</a>, which has 600 regular tetrahedra as its cells. The 120-cell is a compound of 5 disjoint 600-cells, and each of its dodecahedral cells is a <a href="/facts/120-cell/W4nroBjJ">compound of 5 tetrahedral cells</a>, one cell from each of the 5 disjoint 600-cells. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Compound_of_ten_tetrahedra/z9LgGAPL">Compound of ten tetrahedra</a></li> <li><a href="/facts/Compound_of_five_cubes/PNIAfKTN">Compound of five cubes</a></li> <li><a href="/facts/Compound_of_five_truncated_tetrahedra/PlMQP1Ic">Compound of five truncated tetrahedra</a></li></ul> <h2 id="citations">Citations</h2> <ul><li><a href="/facts/Magnus_Wenninger/2QeSYKu2">Wenninger, Magnus</a> (1974). <i>Polyhedron Models</i>. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-09859-9.</li> <li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">Coxeter, H.S.M.</a> (1973) [1948]. <a href="/facts/Regular_Polytopes_(book)/jomuRNha"><i>Regular Polytopes</i></a> (3rd ed.). New York: Dover.</li> <li><a href="/facts/Harold_Scott_MacDonald_Coxeter/D2l5jWGG">Coxeter, Harold Scott MacDonald</a>; Du Val, P.; Flather, H. T.; Petrie, J. F. (1999) [1938]. <i><a href="/facts/The_Fifty-Nine_Icosahedra/syuwcVfg">The Fifty-Nine Icosahedra</a></i> (3rd ed.). Tarquin. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-899618-32-3. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0676126">0676126</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Tetrahedron5-Compound.html">"Tetrahedron 5-Compound"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="http://bulatov.org/metal/five_tetrahedra.html">Metal Sculpture of Five Tetrahedra Compound</a></li> <li><a href="/facts/VRML/EkNXun7G">VRML</a> model: <a href="https://web.archive.org/web/20081119051953/http://www.interocitors.com/polyhedra/UCs/05__5_Tetrahedra.wrl">[1]</a></li> <li><a href="http://demonstrations.wolfram.com/CompoundsOf5And10Tetrahedra/">Compounds of 5 and 10 Tetrahedra</a> by Sándor Kabai, <a href="/facts/The_Wolfram_Demonstrations_Project/l0sh4gln">The Wolfram Demonstrations Project</a>.</li> <li>Klitzing, Richard. <a href="https://bendwavy.org/klitzing/dimensions/../incmats/ki.htm">"3D compound"</a>.</li></ul> <table><tbody><tr><td colspan="10">Notable <a href="/facts/The_Fifty-Nine_Icosahedra/syuwcVfg">stellations of the icosahedron</a></td></tr><tr><td><a href="/facts/Platonic_solid/jraREXFD">Regular</a></td><td colspan="3"><a href="/facts/Dual_uniform_polyhedron/eSEanSNt">Uniform duals</a></td><td colspan="3"><a href="/facts/Compound_polyhedron/L13pItNN">Regular compounds</a></td><td><a href="/facts/Kepler%25E2%2580%2593Poinsot_polyhedron/mzUUgBYS">Regular star</a></td><td colspan="2">Others</td></tr><tr><th><a href="/facts/Regular_icosahedron/1U93vmXV">(Convex) icosahedron</a></th><th><a href="/facts/Small_triambic_icosahedron/1LU1h5dU">Small triambic icosahedron</a></th><th><a href="/facts/Medial_triambic_icosahedron/F4txMCXY">Medial triambic icosahedron</a></th><th><a href="/facts/Great_triambic_icosahedron/F4txMCXY">Great triambic icosahedron</a></th><th><a href="/facts/Compound_of_five_octahedra/8M825cd5">Compound of five octahedra</a></th><th>Compound of five tetrahedra</th><th><a href="/facts/Compound_of_ten_tetrahedra/z9LgGAPL">Compound of ten tetrahedra</a></th><th><a href="/facts/Great_icosahedron/jaAk3a5Q">Great icosahedron</a></th><th><a href="/facts/Excavated_dodecahedron/lf3pYPTs">Excavated dodecahedron</a></th><th><a href="/facts/Final_stellation_of_the_icosahedron/UMKwLhvr">Final stellation</a></th></tr><tr><td></td><td></td><td colspan="2"></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td colspan="2"></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td colspan="10">The stellation process on the icosahedron creates a number of related <a href="/facts/Polyhedron/UC0p9FTF">polyhedra</a> and <a href="/facts/Compound_polyhedron/L13pItNN">compounds</a> with <a href="/facts/Icosahedral_symmetry/pDlwAy23">icosahedral symmetry</a>.</td></tr></tbody></table> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Coxeter 1973, p. 98. - Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Coxeter 1973, pp. 47–50, §3.6 The five regular compounds. - Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Coxeter 1973, pp. 96–104, §6.2 Stellating the Platonic solids. - Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Coxeter 1973, p. 269, Compounds; "It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions." - Coxeter, H.S.M. (1973) [1948]. Regular Polytopes (3rd ed.). New York: Dover. <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>