Disphenocingulum

{{Infobox polyhedron

|image=Disphenocingulum.png

|type=Johnson
{{math|hebesphenomegacorona – J{{sub|90}} – bilunabirotunda}}

|faces=20 triangles
4 squares

|edges=38

|vertices=16

|symmetry={{math|D{{sub|2d}}}}

|vertex_config={{math|4(3{{sup|2}}.4{{sup|2}})
4(3{{sup|5}})
8(3{{sup|4}}.4)}}

|properties=convex, elementary

|net=Johnson solid 90 net.png

}}

File:J90 disphenocingulum.stl

In geometry, the disphenocingulum is a Johnson solid with 20 equilateral triangles and 4 squares as its faces.

Properties

The disphenocingulum is named by {{harvtxt|Johnson|1966}}. The prefix dispheno- refers to two wedgelike complexes, each formed by two adjacent lunes—a figure of two equilateral triangles at the opposite sides of a square. The suffix -cingulum, literally 'belt', refers to a band of 12 triangles joining the two wedges.{{r|johnson}} The resulting polyhedron has 20 equilateral triangles and 4 squares, making 24 faces.{{r|berman}}. All of the faces are regular, categorizing the disphenocingulum as a Johnson solid—a convex polyhedron in which all of its faces are regular polygon—enumerated as 90th Johnson solid $J_{90}$ .{{r|francis}}. It is an elementary polyhedron, meaning that it cannot be separated by a plane into two small regular-faced polyhedra.{{r|cromwell}}

The surface area of a disphenocingulum with edge length $a$ can be determined by adding all of its faces, the area of 20 equilateral triangles and 4 squares $(4 + 5\sqrt{3})a^2 \approx 12.6603a^2$ , and its volume is $3.7776a^3$ .{{r|berman}}

Cartesian coordinates

Let $a \approx 0.76713$ be the second smallest positive root of the polynomial

$\begin{align} &256x^{12} - 512x^{11} - 1664x^{10} + 3712x^9 + 1552x^8 - 6592x^7 \\ &\quad{} + 1248x^6 + 4352x^5 - 2024x^4 - 944x^3 + 672x^2 - 24x - 23 \end{align}$

and $h = \sqrt{2+8a-8a^2}$ and $c = \sqrt{1-a^2}$ . Then, the Cartesian coordinates of a disphenocingulum with edge length 2 are given by the union of the orbits of the points

$\left(1,2a,\frac{h}{2}\right),\ \left(1,0,2c+\frac{h}{2}\right),\ \left(1+\frac{\sqrt{3-4a^2}}{c},0,2c-\frac{1}{c}+\frac{h}{2}\right)$

under the action of the group generated by reflections about the xz-plane and the yz-plane.{{r|timofeenko}}

References

{{reflist|refs=

{{cite journal

| last = Berman | first = M.

| doi = 10.1016/0016-0032(71)90071-8

| journal = Journal of the Franklin Institute

| mr = 290245

| pages = 329–352

| title = Regular-faced convex polyhedra

| volume = 291

| year = 1971| issue = 5

}}

{{cite book

| last = Cromwell | first = P. R.

| title = Polyhedra

| year = 1997

| url = https://archive.org/details/polyhedra0000crom/page/87/mode/1up

| publisher = Cambridge University Press

| isbn = 978-0-521-66405-9

| page = 86–87, 89

}}

{{cite journal

| last = Francis | first = D.

| title = Johnson solids & their acronyms

| journal = Word Ways

| date = August 2013

| volume = 46 | issue = 3 | page = 177

| url = https://go.gale.com/ps/i.do?id=GALE%7CA340298118

}}

{{cite journal

| last = Johnson | first = N. W. | author-link = Norman Johnson (mathematician)

| title = Convex polyhedra with regular faces

| journal = Canadian Journal of Mathematics

| volume = 18 | pages = 169–200

| year = 1966

| doi = 10.4153/cjm-1966-021-8|mr=0185507

| zbl = 0132.14603

| s2cid = 122006114

| doi-access = free

}}

{{cite journal

| last = Timofeenko | first = A. V.

| year = 2009

| title = The non-Platonic and non-Archimedean noncomposite polyhedra

| journal = Journal of Mathematical Science

| volume = 162 | issue = 5 | pages = 717

| doi = 10.1007/s10958-009-9655-0

| s2cid = 120114341

}}

External links

{{Mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid | urlname =Disphenocingulum| title = Disphenocingulum}}

Category:Elementary polyhedron

Category:Johnson solids