Truncated dodecadodecahedron

File:Truncated dodecadodecahedron.stl

In geometry, the truncated dodecadodecahedron (or stellatruncated dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U₅₉. It is given a Schläfli symbol {{math|t_0,1,2{{mset|{{frac|5|3}},5}}.}} It has 54 faces (30 squares, 12 decagons, and 12 decagrams), 180 edges, and 120 vertices.{{Cite web|url=https://www.mathconsult.ch/static/unipoly/59.html|title=59: truncated dodecadodecahedron|last=Maeder|first=Roman|date=|website=MathConsult|archive-url=|archive-date=|access-date=}} The central region of the polyhedron is connected to the exterior via 20 small triangular holes.

The name truncated dodecadodecahedron is somewhat misleading: truncation of the dodecadodecahedron would produce rectangular faces rather than squares, and the pentagram faces of the dodecadodecahedron would turn into truncated pentagrams rather than decagrams. However, it is the quasitruncation of the dodecadodecahedron, as defined by {{harvtxt|Coxeter|Longuet-Higgins|Miller|1954}}.{{citation

| last1 = Coxeter | first1 = H. S. M. | author1-link = Harold Scott MacDonald Coxeter

| last2 = Longuet-Higgins | first2 = M. S. | author2-link = Michael S. Longuet-Higgins

| last3 = Miller | first3 = J. C. P. | author3-link = J. C. P. Miller

| journal = Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences

| jstor = 91532

| mr = 0062446

| pages = 401–450

| title = Uniform polyhedra

| volume = 246

| year = 1954 | issue = 916 | doi=10.1098/rsta.1954.0003| bibcode = 1954RSPTA.246..401C}}. See especially the description as a quasitruncation on p. 411 and the photograph of a model of its skeleton in Fig. 114, Plate IV. For this reason, it is also known as the quasitruncated dodecadodecahedron.Wenninger writes "quasitruncated dodecahedron", but this appears to be a mistake. {{citation|contribution=98 Quasitruncated dodecahedron|pages=152–153|first=Magnus J.|last=Wenninger|authorlink=Magnus Wenninger|title=Polyhedron Models|publisher=Cambridge University Press|year=1971}}. Coxeter et al. credit its discovery to a paper published in 1881 by Austrian mathematician Johann Pitsch.{{citation

| last = Pitsch | first = Johann

| journal = Zeitschrift für das Realschulwesen

| pages = 9–24, 72–89, 216

| title = Über halbreguläre Sternpolyeder

| volume = 6

| year = 1881}}. According to {{harvtxt|Coxeter|Longuet-Higgins|Miller|1954}}, the truncated dodecadodecahedron appears as no. XII on p.86.

Cartesian coordinates

Cartesian coordinates for the vertices of a truncated dodecadodecahedron are all the triples of numbers obtained by circular shifts and sign changes from the following points (where $\varphi = \tfrac{1 + \sqrt{5}}{2}$ is the golden ratio):

$\begin{array}{lcr}$

\Bigl(1,& 1,& 3\Bigr), \\

\Bigl(\frac{1}{\varphi},& \frac{1}{\varphi^2},& 2\varphi\Bigr), \\

\Bigl(\varphi,& \frac{2}{\varphi},& \varphi^2\Bigr), \\

\Bigl(\varphi^2,& \frac{1}{\varphi^2},& 2\Bigr), \\

\Bigl(\sqrt{5},& 1,& \sqrt{5}\Bigr).

\end{array}

Each of these five points has eight possible sign patterns and three possible circular shifts, giving a total of 120 different points.

As a Cayley graph

The truncated dodecadodecahedron forms a Cayley graph for the symmetric group on five elements, as generated by two group members: one that swaps the first two elements of a five-tuple, and one that performs a circular shift operation on the last four elements. That is, the 120 vertices of the polyhedron may be placed in one-to-one correspondence with the 5! permutations on five elements, in such a way that the three neighbors of each vertex are the three permutations formed from it by swapping the first two elements or circularly shifting (in either direction) the last four elements.{{citation

| last = Eppstein | first = David | title = Graph Drawing | authorlink = David Eppstein

| editor1-last = Tollis | editor1-first = Ioannis G.

| editor2-last = Patrignani | editor2-first = Marizio

| arxiv = 0709.4087

| contribution = The topology of bendless three-dimensional orthogonal graph drawing

| doi = 10.1007/978-3-642-00219-9_9

| location = Heraklion, Crete

| pages = 78–89

| publisher = Springer-Verlag

| series = Lecture Notes in Computer Science

| date = 2009 | volume = 5417

| isbn = 978-3-642-00218-2 }}.

Related polyhedra

= Medial disdyakis triacontahedron=

File:Medial disdyakis triacontahedron.stl

The medial disdyakis triacontahedron is a nonconvex isohedral polyhedron. It is the dual of the uniform truncated dodecadodecahedron.

References

{{Citation | last1=Wenninger | first1=Magnus | author1-link=Magnus Wenninger | title=Dual Models | publisher=Cambridge University Press | isbn=978-0-521-54325-5 |mr=730208 | year=1983 | doi=10.1017/CBO9780511569371}}

External links

{{mathworld | urlname = TruncatedDodecadodecahedron| title = Truncated dodecadodecahedron}}
{{mathworld | urlname = MedialDisdyakisTriacontahedron| title =Medial disdyakis triacontahedron}}

Category:Uniform polyhedra