Hebesphenomegacorona

{{Infobox polyhedron

|image=Hebesphenomegacorona.png

|type=Johnson
{{math|sphenomegacorona – J{{sub|89}} – disphenocingulum}}

|faces=3x2+3x4 triangles
1+2 squares

|edges=33

|vertices=14

|symmetry={{math|C{{sub|2v}}}}

|vertex_config= {{math|4(3{{sup|2}}.4{{sup|2}})
2+2x2(3{{sup|5}})
4(3{{sup|4}}.4)}}

|properties=convex, elementary

|net=Johnson solid 89 net.png

}}

File:J89 hebesphenomegacorona.stl

In geometry, the hebesphenomegacorona is a Johnson solid with 18 equilateral triangles and 3 squares as its faces.

Properties

The hebesphenomegacorona is named by {{harvtxt|Johnson|1966}} in which he used the prefix hebespheno- referring to a blunt wedge-like complex formed by three adjacent lunes—a square with equilateral triangles attached on its opposite sides. The suffix -megacorona refers to a crownlike complex of 12 triangles.{{r|johnson}} By joining both complexes together, the result polyhedron has 18 equilateral triangles and 3 squares, making 21 faces.{{r|berman}}. All of its faces are regular polygons, categorizing the hebesphenomegacorona as a Johnson solid—a convex polyhedron in which all of its faces are regular polygons—enumerated as 89th Johnson solid $J_{89}$ .{{r|francis}} It is an elementary polyhedron, meaning it cannot be separated by a plane into two small regular-faced polyhedra.{{r|cromwell}}

The surface area of a hebesphenomegacorona with edge length $a$ can be determined by adding the area of its faces, 18 equilateral triangles and 3 squares

$\frac{6 + 9\sqrt{3}}{2}a^2 \approx 10.7942a^2,$

and its volume is $2.9129a^3$ .{{r|berman}}

Cartesian coordinates

Let $a \approx 0.21684$ be the second smallest positive root of the polynomial

$\begin{align} &26880x^{10} + 35328x^9 - 25600x^8 - 39680x^7 + 6112x^6 \\ &\quad {}+ 13696x^5 + 2128x^4 - 1808x^3 - 1119x^2 + 494x - 47 \end{align}$

Then, Cartesian coordinates of a hebesphenomegacorona with edge length 2 are given by the union of the orbits of the points

$\begin{align} &\left(1,1,2\sqrt{1-a^2}\right),\ \left(1+2a,1,0\right),\ \left(0,1+\sqrt{2}\sqrt{\frac{2a-1}{a-1}},-\frac{2a^2+a-1}{\sqrt{1-a^2}}\right),\ \left(1,0,-\sqrt{3-4a^2}\right), \\ &\left(0,\frac{\sqrt{2(3-4a^2)(1-2a)}+\sqrt{1+a}}{2(1-a)\sqrt{1+a}},\frac{(2a-1)\sqrt{3-4a^2}}{2(1-a)}-\frac{\sqrt{2(1-2a)}}{2(1-a)\sqrt{1+a}}\right) \end{align}$

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 =Hebesphenomegacorona| title = Hebesphenomegacorona}}

Category:Elementary polyhedron

Category:Johnson solids