Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Snub square antiprism
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
{{short description|85th Johnson solid (26 faces)}} {{Infobox polyhedron | image = snub_square_antiprism.png | type = [[Johnson solid|Johnson]]<br>{{math|[[snub disphenoid|''J''{{sub|84}}]] β '''''J''{{sub|85}}''' β [[sphenocorona|''J''{{sub|86}}]]}} | faces = 24 [[triangle]]s<br>2 [[square (geometry)|square]]s | edges = 40 | vertices = 16 | symmetry = <math> D_{4d} </math> | vertex_config = <math> 8 \times 3^5 + 8 \times 3^4 \times 4 </math> | properties = [[convex set|convex]] | net = Johnson solid 85 net.png }} [[File:J85 snub square antiprism.stl|thumb|3D model of a snub square antiprism]] In [[geometry]], the '''snub square antiprism''' is the [[Johnson solid]] that can be constructed by [[Snub (geometry)|snubbing]] the [[square antiprism]]. It is one of the elementary Johnson solids that do not arise from "cut and paste" manipulations of the [[Platonic solid|Platonic]] and [[Archimedean solid|Archimedean]] solids, although it is a relative of the [[icosahedron]] that has fourfold symmetry instead of threefold. == Construction and properties == The [[Snub (geometry)|snub]] is the process of constructing polyhedra by cutting loose the edge's faces, twisting them, and then attaching [[Equilateral triangle|equilateral triangles]] to their edges.{{r|holme}} As the name suggested, the snub square antiprism is constructed by snubbing the [[square antiprism]],{{r|johnson}} and this construction results in 24 equilateral triangles and 2 squares as its faces.{{r|berman}} The [[Johnson solid]]s are the convex polyhedra whose faces are regular, and the snub square antiprism is one of them, enumerated as <math> J_{85} </math>, the 85th Johnson solid.{{r|francis}} <!--It can also be constructed as a square [[Johnson solid#Snub antiprisms|gyrobianticupolae]], connecting two [[anticupola]]e with gyrated orientations.--> Let <math> k \approx 0.82354 </math> be the positive root of the [[cubic polynomial]] <math display="block"> 9x^3+3\sqrt{3}\left(5-\sqrt{2}\right)x^2-3\left(5-2\sqrt{2}\right)x-17\sqrt{3}+7\sqrt{6}. </math> Furthermore, let <math> h \approx 1.35374 </math> be defined by <math display="block"> h = \frac{\sqrt{2}+8+2\sqrt{3}k-3\left(2+\sqrt{2}\right)k^2}{4\sqrt{3-3k^2}}. </math> Then, [[Cartesian coordinate system|Cartesian coordinates]] of a snub square antiprism with edge length 2 are given by the union of the orbits of the points <math display="block"> (1,1,h),\,\left(1+\sqrt{3}k,0,h-\sqrt{3-3k^2}\right) </math> under the action of the [[Symmetry group|group]] generated by a rotation around the {{nowrap|1=<math> z </math>-}}axis by 90Β° and by a rotation by 180Β° around a straight line perpendicular to the {{nowrap|1=<math> z </math>-}}axis and making an angle of 22.5Β° with the {{nowrap|1=<math> x </math>-}}axis.{{r|timofeenko}} It has the [[Dihedral symmetry in three dimensions|three-dimensional symmetry]] of [[dihedral group]] <math> D_{4d} </math> of order 16.{{r|johnson}} The surface area and volume of a snub square antiprism with edge length <math> a </math> can be calculated as:{{r|berman}} <math display="block"> \begin{align} A = \left(2+6\sqrt{3}\right)a^2 &\approx 12.392a^2, \\ V &\approx 3.602 a^3. \end{align} </math> ==References== {{reflist|refs= <ref name="berman">{{cite journal | last = Berman | first = Martin | doi = 10.1016/0016-0032(71)90071-8 | journal = Journal of the Franklin Institute | mr = 290245 | pages = 329β352 | title = Regular-faced convex polyhedra | volume = 291 | year = 1971 | issue = 5 }}</ref> <ref name="francis">{{cite journal | last = Francis | first = Darryl | title = Johnson solids & their acronyms | year = 2013 | journal = Word Ways | volume = 46 | issue = 3 | page = 177 | url = https://go.gale.com/ps/i.do?id=GALE%7CA340298118 }}</ref> <ref name="holme">{{cite book | last = Holme | first = Audun | year = 2010 | title = Geometry: Our Cultural Heritage | publisher = Springer | url = https://books.google.com/books?id=zXwQGo8jyHUC&pg=PA99 | page = 99 | isbn = 978-3-642-14441-7 | doi = 10.1007/978-3-642-14441-7 }}</ref> <ref name="johnson">{{cite journal | last = Johnson | first = Norman W. | author-link = Norman Johnson (mathematician) | doi = 10.4153/cjm-1966-021-8 | journal = [[Canadian Journal of Mathematics]] | mr = 0185507 | pages = 169β200 | title = Convex polyhedra with regular faces | volume = 18 | year = 1966 | zbl = 0132.14603 }}</ref> <ref name="timofeenko">{{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 = 725 | doi = 10.1007/s10958-009-9655-0 | s2cid = 120114341 }}</ref> }} ==External links== *{{MathWorld2|urlname2=JohnsonSolid|title2=Johnson solid|urlname=SnubSquareAntiprism|title=Snub square antiprism}} [[Category:Johnson solids]] {{Johnson solids}}
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)
Pages transcluded onto the current version of this page
(
help
)
:
Template:Infobox
(
edit
)
Template:Infobox polyhedron
(
edit
)
Template:Johnson solids
(
edit
)
Template:MathWorld2
(
edit
)
Template:Nowrap
(
edit
)
Template:R
(
edit
)
Template:Reflist
(
edit
)
Template:Short description
(
edit
)
Template:Template other
(
edit
)