Tetragonal Antiwedge

Author

Eric W. Weisstein
September 30, 2022

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Polyhedra/TetragonalAntiwedge.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/TetragonalAntiwedge.html.

©2022 Wolfram Research, Inc. except for portions noted otherwise

Graphics

Graphics3D[{Opacity[.5],PolyhedronData["TetragonalAntiwedge","GraphicsComplex"]},Boxed->False]

PolyhedronData["TetragonalAntiwedge","Region"]

Graphics3D[{FaceForm[Green,Red],PolyhedronData["TetragonalAntiwedge","GraphicsComplex"]},Boxed->False]

Solid and Dual

poly=PolyhedronData["TetragonalAntiwedge","Polyhedron"];
dual=poly/.Polyhedron[v_,f_]:>Polyhedron[-v,Reverse/@f]

Graphics3D[{{Opacity[.3],Green,poly,Red,dual},Opacity[.8],Yellow,PolyhedronData["TetragonalAntiwedge","Midsphere"]},Boxed->False,Method->{"ShrinkWrap"->True},ImageSize->250]

Net

PolyhedronData["TetragonalAntiwedge","Net"]

PolyhedronData["TetragonalAntiwedge","Net","Colored"]

Definition

Michon

http://nbarth.net/notes/src/notes-calc-raw/others/X-numericana/polyhedra.htm#hexahedra

Wikipedia

https://en.wikipedia.org/wiki/Hexahedron#Convex,_Others

Properties

HiddenWireframe[PolyhedronData["TetragonalAntiwedge","Polyhedron"]]

PolyhedronData["TetragonalAntiwedge",{"V","E","F"}]

VertexDegree[PolyhedronData["TetragonalAntiwedge","SkeletonGraph"]]//Sort

PolyhedronData["TetragonalAntiwedge","EdgeLengths"]

Properties

Initialization

<<MathWorld`Polyhedra`

p=PolyhedronData[pname="TetragonalAntiwedge","Polyhedron"]

vol=PolyhedronData[pname,"Volume"]

Centroid

PolyhedronData[p,"Centroid"]

(xbar=Centroid[PolyhedronData[p,"GraphicsComplex"],vol])//Timing

Circumsphere

PolyhedronData[p,"Circumsphere"]

PolyhedronCircumsphere[PolyhedronData[p,"Vertices"]]

EdgeLengths

PolyhedronData[p,"Edges","Length"]

N[%]

Edges

PolyhedronData[p,"EdgeCount"]

PolyhedronData[p,"Edges","Count"]

PolyhedronData[p,"Edges"]

Length[%]

Faces

PolyhedronData[p,"Faces","Count"]

PolyhedronData[p,"FaceCount"]

PolyhedronData[p,"Faces"]

Length[%]

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]//Timing

MyInsphere[p,Debug->True,"Simplification"->RootReduce]//Timing

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

((NIntegrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))

(i=((Integrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"])))//Timing

Divergence theorem

(rxy=MeanCylindricalRadius[p,"Simplification"->RootReduce,Debug->True])//Timing

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

((NIntegrate[(x^2)+(y^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))

(i=((Integrate[(x^2)+(y^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//FullSimplify)//Timing

N[i]

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))

(i=((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"])))//Timing

N[i]

MeanSquareSphericalRadius

PolyhedronData[pname,"MeanSquareSphericalRadius"]

((NIntegrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))

(i=((Integrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/(PolyhedronData[pname,"Volume"]))//FullSimplify)//Timing

N[i]

Mesh

PolyhedronData[p,"MeshRegion"]

Midsphere

PolyhedronData[pname,"Midsphere"]

(midsphere=Midsphere[p//N])//Timing

Graphics3D[{Red,Opacity[.5],p,Opacity[.4],Yellow,midsphere},Boxed->False]

ToAlgebraicRoot[midsphere[[2]]]

Sqrt[1+Sqrt[GoldenRatio]]/2//RootReduce

FullSimplify[(NumericalApproximation[Root, (RowDefault["0.754", FEExpressionsNumericalApproximationElider]), Root[-1-8 ((#1)^2)+80 ((#1)^4)-256 ((#1)^6)+256 ((#1)^8)&,2], ])-Sqrt[1+Sqrt[GoldenRatio]]/2]

Ran overnight:

Midsphere[p,Debug->True,"Simplification"->RootReduce]//Timing

Region

reg=BoundaryMeshRegion[PolyhedronData[pname,"Vertices"],Polygon[PolyhedronData[pname,"Faces","List"]],WorkingPrecision->∞]

PolyhedronData[p,"Region"]

RegionFunction

ByteCount[ineq=PolyhedronInequalities[PolyhedronData[p,"Faces","Polygon"][[1]],{x,y,z}]]

ineq//N

And@@ineq

RegionPlot3D[And@@ineq,{x,-3.2,3.2},{y,-3.2,3.2},{z,-3.2,3.2},PlotPoints->50,Boxed->False,Axes->False,Method->{"ShrinkWrap"->True}]

RegionPlot3D[Evaluate[PolyhedronData[p,"RegionFunction"][x,y,z]],{x,-3.2,3.2},{y,-3.2,3.2},{z,-3.2,3.2},PlotPoints->50,Boxed->False,Axes->False,Method->{"ShrinkWrap"->True}]

InertiaTensor

PolyhedronData[p,"InertiaTensor"]

N[%]

(i=InertiaTensor[PolyhedronData[p,"GraphicsComplex"]]//RootReduce)//Timing

ToRadicals[i]//FullSimplify

SkeletonGraph

PolyhedronData[p,"SkeletonGraph"]

SurfaceArea

PolyhedronData[p,"SurfaceArea"]

(area=PolyhedronSurfaceArea[PolyhedronData[p,"GraphicsComplex"]])//RootReduce//Timing

Vertices

PolyhedronData[p,"VertexCount"]

PolyhedronData[p,"Vertices","Count"]

PolyhedronData[p,"Vertices"]//Length

Volume

PolyhedronData[p,"Volume"]

N[%]

(vol=PolyhedronVolume[PolyhedronData[p,"GraphicsComplex"]])//Timing

FullSimplify[%]

N[vol2=Volume[reg]]

N[vol-vol2,1000]

vol2//ByteCount

Volume for unit midradius

((PolyhedronData["TetragonalAntiwedge","Volume"])/((PolyhedronData["TetragonalAntiwedge","Midradius"])^3))

FullSimplify[%]

N[%,20]

N[Pi-%%%,20]

Construction

Number fields

First, an aside. Number fields are incredibly powerful for polyhedra. Here are vertex types for the snub cube and snub dodecahedron

Tribonacci space

RootReduce[t/.Solve[t^3-t^2-t==1][[1]]]

With[{snubroot=RootReduce[t/.Solve[t^3-t^2-t==1][[1]]],
snubvert=Union[RootReduce[Abs[Flatten[PolyhedronData["SnubCube","Vertices"]]]]]},
12Last[ToNumberField[((#)^2),snubroot]]&/@snubvert]

Phi-tribonacci space

RootReduce[t/.Solve[t^3-t^2-t==GoldenRatio][[1]]]

With[{snubroot=RootReduce[t/.Solve[t^3-t^2-t==GoldenRatio][[1]]],
snubvert=Union[RootReduce[Abs[Flatten[PolyhedronData["SnubDodecahedron","Vertices"]]]]]},
12Last[ToNumberField[((#)^2),snubroot]]&/@snubvert]

General to/from routines.

ToSqrtSpace[root_,coord_] := Module[{ρ, dim, order,algebraic, converted},
dim=Dimensions[coord];
order={1,2}.NumberFieldSignature[root];
algebraic= Map[ToNumberField[Sign[#]RootReduce[#^2], root]&,coord,{Length[dim]-1} ];
Map[If[Head[#]===AlgebraicNumber,Last[#],PadRight[{#},order]]&,algebraic,{Length[dim]} ]];

FromSqrtSpace[root_,coord_]:= Module[{ dim, order, vector},
dim=Dimensions[coord];
order={1,2}.NumberFieldSignature[root];
vector =(root^Range[0,order-1]);
Map[With[{k=(#).vector},RootReduce[Sign[k](Sqrt[Abs[k]])]]&,coord,{Length[dim]-1}]];

Coming On Pi Day

https://community.wolfram.com/groups/-/m/t/1301599

The canonical form of the least symmetric hexahedron, the self-dual tetragonal antiwedge, happens to live in the space of (Sqrt[ϕ]) and has a volume close to Pi in canonical form, where all edges are tangent to a unit sphere.

Module[{name,root,vals,points, faces},
name= "Canonical Self-dual hexehedron  and Dual (in (Sqrt[ϕ])-space)";
root=Root[-1-((#1)^2)+((#1)^4)&,2];
vals={{{-3,3,2,-2},{1,0,0,0},{-3,-3,2,2}},{{3,-3,-2,2},{1,0,0,0},{3,3,-2,-2}},{{11,-9,18,-14},{-9,8,-16,12},{1,-1,2,-2}},{{-11,9,-18,14},{-9,8,-16,12},{-1,1,-2,2}},{{-1,-1,0,0},{-1,0,0,0},{-1,1,0,0}},{{1,1,0,0},{-1,0,0,0},{1,-1,0,0}}};
points={FromSqrtSpace[root,#]&/@vals, {-1,-1,1}FromSqrtSpace[root,#]&/@vals};
faces ={{1,2,6,4},{1,5,3,2},{1,4,5},{2,3,6},{3,5,6},{4,6,5}};
Text@Column[{Style[name,20],
Style[Row[{"volume = ",Volume[ConvexHullMesh[points[[1]]]]}],24],
 Graphics3D[{Opacity[.3],Sphere[{0,0,0},1],Opacity[.6],
EdgeForm[{Black, Thick}],Polygon[points[[1]][[#]]]&/@faces,
EdgeForm[{Red, Thick}],Polygon[points[[2]][[#]]]&/@faces},
 ImageSize-> {600,400}, ViewAngle-> Pi/12,ViewPoint-> {0,0,3},Boxed-> False, SphericalRegion-> True]}, Alignment-> Center]]

root=Root[-1-((#1)^2)+((#1)^4)&,2];
vals={{{-3,3,2,-2},{1,0,0,0},{-3,-3,2,2}},{{3,-3,-2,2},{1,0,0,0},{3,3,-2,-2}},{{11,-9,18,-14},{-9,8,-16,12},{1,-1,2,-2}},{{-11,9,-18,14},{-9,8,-16,12},{-1,1,-2,2}},{{-1,-1,0,0},{-1,0,0,0},{-1,1,0,0}},{{1,1,0,0},{-1,0,0,0},{1,-1,0,0}}};
points={FromSqrtSpace[root,#]&/@vals, {-1,-1,1}FromSqrtSpace[root,#]&/@vals}

p1=ConvexHullRegion[points[[1]]]

<<MathWorld`PolyhedronOperations`

PolyhedronEdgeLengths[p1]//RootReduce//Tally

Volume[p1]//RootReduce

N[%25,20]

p2=ConvexHullRegion[points[[2]]]

Volume[p2]//RootReduce

N[%24,20]

Self-Dual Tetragonal Antiwedge Hexahedron in Canonical Form, with poly = {-1,-1,1} dual

p2=ConvexHullRegion[{{Root[-1+14 ((#1)^2)+25 ((#1)^4)-8 ((#1)^6)+((#1)^8)&,2],-1,Root[-1-14 ((#1)^2)+25 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,2]},{Root[-1+14 ((#1)^2)+25 ((#1)^4)-8 ((#1)^6)+((#1)^8)&,1],-1,Root[-1-14 ((#1)^2)+25 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,1]},{Root[-1+10 ((#1)^2)-31 ((#1)^4)+80 ((#1)^6)+((#1)^8)&,2],Root[1-8 #1-2 ((#1)^2)-8 ((#1)^3)+((#1)^4)&,1],Root[-1+6 ((#1)^2)-15 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,1]},{Root[-1+10 ((#1)^2)-31 ((#1)^4)+80 ((#1)^6)+((#1)^8)&,1],Root[1-8 #1-2 ((#1)^2)-8 ((#1)^3)+((#1)^4)&,1],Root[-1+6 ((#1)^2)-15 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,2]},{Root[-1-2 ((#1)^2)+5 ((#1)^4)-4 ((#1)^6)+((#1)^8)&,2],1,Root[-1+2 ((#1)^2)+5 ((#1)^4)+4 ((#1)^6)+((#1)^8)&,2]},{Root[-1-2 ((#1)^2)+5 ((#1)^4)-4 ((#1)^6)+((#1)^8)&,1],1,Root[-1+2 ((#1)^2)+5 ((#1)^4)+4 ((#1)^6)+((#1)^8)&,1]}}]

Volume[p]//FullSimplify

antiwedge={"AlternateNames"->{"Self-dual hexahedron"},"AlternateStandardNames"->{},"Amphichiral"->True,"Antiprism"->False,"Archimedean"->False,"ArchimedeanDual"->False,"BoundaryMeshRegion"->,"Centroid"->{0,Root[16+64 #1-8 ((#1)^2)+16 ((#1)^3)+((#1)^4)&,2],0},"Chiral"->True,"Circumcenter"->Missing["NotApplicable"],"Circumradius"->Missing["NotApplicable"],"Circumsphere"->Missing["NotApplicable"],"Classes"->{"Chiral","Convex","Rigid","SelfDual"},"Compound"->False,"Concave"->False,"Convex"->True,"DefaultOrientation"->Missing["NotAvailable"],"Deltahedron"->False,"DihedralAngles"->Missing["NotAvailable"],"Dipyramid"->False,"Dual"->,"DualCompound"->,"EdgeCount"->10,"EdgeLengths"->{Root[-256+256 ((#1)^2)-80 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,2],Root[-256+256 ((#1)^2)-80 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,2],Root[-4096+704 ((#1)^4)+((#1)^8)&,2],Root[-256+384 ((#1)^2)-240 ((#1)^4)+32 ((#1)^6)+((#1)^8)&,2],Root[-256+384 ((#1)^2)-240 ((#1)^4)+32 ((#1)^6)+((#1)^8)&,2],Root[-256+384 ((#1)^2)-240 ((#1)^4)+32 ((#1)^6)+((#1)^8)&,2],Root[-256+384 ((#1)^2)-240 ((#1)^4)+32 ((#1)^6)+((#1)^8)&,2],Root[-256-384 ((#1)^2)+8 ((#1)^6)+((#1)^8)&,2],Root[-256-384 ((#1)^2)+8 ((#1)^6)+((#1)^8)&,2],Root[-4096-64 ((#1)^4)+((#1)^8)&,2]},"Edges"->{{1,4},{2,3},{1,2},{3,5},{3,6},{4,5},{4,6},{1,5},{2,6},{5,6}},"Entity"->Missing["NotApplicable"],"Equilateral"->False,

"FaceAreas"->Missing["NotAvailable"],"FaceCount"->6,"Faces"->{{1,2,6,4},{1,5,3,2},{1,4,5},{2,3,6},{3,5,6},{4,6,5}},
"GeneralizedDiameter"->Missing["NotApplicable"],"Graphics3D"->,"GraphicsComplex"->GraphicsComplex[{{Root[-1+14 ((#1)^2)+25 ((#1)^4)-8 ((#1)^6)+((#1)^8)&,2],-1,Root[-1-14 ((#1)^2)+25 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,2]},{Root[-1+14 ((#1)^2)+25 ((#1)^4)-8 ((#1)^6)+((#1)^8)&,1],-1,Root[-1-14 ((#1)^2)+25 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,1]},{Root[-1+10 ((#1)^2)-31 ((#1)^4)+80 ((#1)^6)+((#1)^8)&,2],Root[1-8 #1-2 ((#1)^2)-8 ((#1)^3)+((#1)^4)&,1],Root[-1+6 ((#1)^2)-15 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,1]},{Root[-1+10 ((#1)^2)-31 ((#1)^4)+80 ((#1)^6)+((#1)^8)&,1],Root[1-8 #1-2 ((#1)^2)-8 ((#1)^3)+((#1)^4)&,1],Root[-1+6 ((#1)^2)-15 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,2]},{Root[-1-2 ((#1)^2)+5 ((#1)^4)-4 ((#1)^6)+((#1)^8)&,2],1,Root[-1+2 ((#1)^2)+5 ((#1)^4)+4 ((#1)^6)+((#1)^8)&,2]},{Root[-1-2 ((#1)^2)+5 ((#1)^4)-4 ((#1)^6)+((#1)^8)&,1],1,Root[-1+2 ((#1)^2)+5 ((#1)^4)+4 ((#1)^6)+((#1)^8)&,1]}}, Polygon[{{1,2,6,4},{1,5,3,2},{1,4,5},{2,3,6},{3,5,6},{4,6,5}}]],
"Image"->,"ImplicitRegion"->Missing["NotAvailable"],"Incenter"->Missing["NotAvailable"],"InertiaTensor"->Missing["NotAvailable"],"Information"->Missing["NotAvailable"],"Inradius"->Missing["NotAvailable"],"Insphere"->Missing["NotAvailable"],"Isohedron"->False,"Johnson"->False,"KeplerPoinsot"->False,"MeshRegion"->,"Midcenter"->Missing["NotApplicable"],"Midradius"->Missing["NotAvailable"],"Midsphere"->Sphere[Missing["NotApplicable"],Missing["NotAvailable"]],"MultipieceNetCoordinates"->Missing["NotAvailable"],"MultipieceNetFaceIndices"->Missing["NotAvailable"],"MultipieceNetImage"->Missing["NotAvailable"],"Name"->"Tetragonal antiwedge","Net"->Missing["NotApplicable"],"NetCount"->0,"NotationRules"->{},"Orientations"->Missing["NotAvailable"],"Platonic"->False,"PlatonicDual"->False,"Polyhedron"->,"Prism"->False,"Pyramid"->False,"Quasiregular"->False,"Region"->,"RegionFunction"->Missing["NotAvailable"],"Rhombohedron"->False,"Rigid"->True,"SchlaefliSymbol"->Missing["NotApplicable"],"SelfDual"->True,"SelfDual"->True,"Shaky"->False,"Skeleton"->,"SpaceFilling"->False,"StandardName"->"TetragonalAntiwedge","StandardNames"->{"TetragonalAntiwedge"},"Stellation"->False,"StellationCount"->Missing["NotAvailable"],"SurfaceArea"->Missing["NotAvailable"],"SymmetryGroup"->Missing["NotAvailable"],"SymmetryGroupString"->Missing["NotApplicable"],"Uniform"->False,"UniformDual"->False,"VertexCount"->6,"VertexSubsetHulls"->Missing["NotAvailable"],"Vertices"->{{-(Sqrt[(1/2) (4-2 (Sqrt[5])+(Sqrt[2 (-11+5 (Sqrt[5]))]))]),1,(Sqrt[-2+(Sqrt[5])+(Sqrt[(1/2) (-11+5 (Sqrt[5]))])])},{(Sqrt[(1/2) (4-2 (Sqrt[5])+(Sqrt[2 (-11+5 (Sqrt[5]))]))]),1,-(Sqrt[-2+(Sqrt[5])+(Sqrt[(1/2) (-11+5 (Sqrt[5]))])])},{-(Sqrt[(1/2) (-40-18 (Sqrt[5])+(Sqrt[2 (1621+725 (Sqrt[5]))]))]),-2-(Sqrt[5])+2 (Sqrt[2+(Sqrt[5])]),-(Sqrt[(1/2) (-4-2 (Sqrt[5])+(Sqrt[2 (29+13 (Sqrt[5]))]))])},{(Sqrt[(1/2) (-40-18 (Sqrt[5])+(Sqrt[2 (1621+725 (Sqrt[5]))]))]),-2-(Sqrt[5])+2 (Sqrt[2+(Sqrt[5])]),(Sqrt[(1/2) (-4-2 (Sqrt[5])+(Sqrt[2 (29+13 (Sqrt[5]))]))])},{-(Sqrt[1+(Sqrt[(1/2) (1+(Sqrt[5]))])]),-1,(Sqrt[-1+(Sqrt[(1/2) (1+(Sqrt[5]))])])},{(Sqrt[1+(Sqrt[(1/2) (1+(Sqrt[5]))])]),-1,-(Sqrt[-1+(Sqrt[(1/2) (1+(Sqrt[5]))])])}},"Volume"->Root[45056+6144 #1-4608 ((#1)^2)-864 ((#1)^3)+81 ((#1)^4)&,1],"WythoffSymbol"->Missing["NotApplicable"],"Zonohedron"->False};

From polygons [not working yet]

poly=Polyhedron[{{0,0,0},{1,0,0},{1,1,0},{0,1,-1/2},{1/2,1/4,1/2},{1/2,3/4,1/2}},{{1,2,3,4},{1,2,5},{3,4,6},Reverse@{1,2,4},Reverse@{2,3,4},{2,3,6,5},{1,5,6,4}}]

Graphics3D[{Opacity[.5],poly},Boxed->False]

poly=ConvexHullRegion[{{0,0,0},{1,0,0},{1,1,0},{0,1,-1/2},{1/2,1/4,1/2},{1/2,3/4,1/2}}]

Graphics3D[{FaceForm[Green,Red],poly},Boxed->False]

SimplePolyhedronQ[poly]

<<MathWorld`PolyhedronOperations`

Length/@Through[{PolyhedronFaces,PolyhedronEdges,PolyhedronCoordinates}[poly]]

Skeleton construction

v=PolyhedronData[p="TetragonalAntiwedge","Vertices"]

Graphics3D[{PolyhedronData[p,"GraphicsComplex"],MapIndexed[Text[#2[[1]],#,Background->White]&,v]}]

PolyhedronData[p,"Faces"]

EuclideanDistance@@v[[{4,2}]]//FullSimplify

s34=EuclideanDistance@@v[[{4,3}]]//FullSimplify

ang243=VectorAngle@@(Subtract@@@(v[[{#,4}]]&/@{2,3}))//FullSimplify

{x3,y3}=s34 Through[{Cos,Sin}[ang243]]//FullSimplify

ang421=VectorAngle@@(Subtract@@@(v[[{#,2}]]&/@{4,1}))//FullSimplify

s12=EuclideanDistance@@v[[{1,2}]]//FullSimplify

s13=EuclideanDistance@@v[[{1,3}]]//FullSimplify

{x1,y1}={1,0}+s12 Through[{Cos,Sin}[π-ang421]]//FullSimplify

Graphics3D[{Polygon[v[[{4,2,1,3}]]],Text[#,v[[#]]]&/@{4,2,1,3}}]

square1={{0,0},{1,0},{x1,y1},{x3,y3}}

square1={{0,0},{1,0},{(1/2) (3-(Sqrt[5])+(Sqrt[2 (1+(Sqrt[5]))])),(Sqrt[(1/2) (-1+(Sqrt[5]))+(Sqrt[2 (-1+(Sqrt[5]))])])},{(1/2) (3-(Sqrt[5])-2 (Sqrt[-2+(Sqrt[5])])),(Sqrt[(1/2) (3-(Sqrt[5])+4 (Sqrt[-2+(Sqrt[5])]))])}};

RootReduce[EuclideanDistance[v[[6]],#]&/@v[[{4,2}]]]//FullSimplify

Solve[x5^2+y5^2==(((1+Root[-1+((#1)^2)+((#1)^4)&,2]))^2)&&(x5-1)^2+y5^2==((((1/2) (1+(Sqrt[5]))))^2),{x5,y5}]//RootReduce

N[%]

triangle1={{1,0},{0,0},{Root[-1+((#1)^2)+((#1)^4)&,2],Root[-11-12 ((#1)^2)+10 ((#1)^4)-4 ((#1)^6)+((#1)^8)&,1]}};

{s1,s2}=RootReduce[EuclideanDistance[v[[6]],#]&/@v[[{2,1}]]]//FullSimplify

{x,y}/.Solve[SquaredEuclideanDistance[{x,y},z[[2]]]==s1^2&&SquaredEuclideanDistance[{x,y},z[[3]]]==s2^2,{x,y}][[2]]//RootReduce

N[%]

triangle2={z[[3]],z[[2]],{Root[-211-352 #1+310 ((#1)^2)+572 ((#1)^3)-97 ((#1)^4)-492 ((#1)^5)+238 ((#1)^6)-28 ((#1)^7)+((#1)^8)&,2],Root[841-9848 ((#1)^2)+42628 ((#1)^4)-83096 ((#1)^6)+72606 ((#1)^8)-34280 ((#1)^10)+18324 ((#1)^12)+248 ((#1)^14)+((#1)^16)&,1]}}

(triangle2={{(1/2) (3-(Sqrt[5])+(Sqrt[2 (1+(Sqrt[5]))])),(Sqrt[(1/2) (-1+(Sqrt[5]))+(Sqrt[2 (-1+(Sqrt[5]))])])},{1,0},{Root[-211-352 #1+310 ((#1)^2)+572 ((#1)^3)-97 ((#1)^4)-492 ((#1)^5)+238 ((#1)^6)-28 ((#1)^7)+((#1)^8)&,2],Root[841-9848 ((#1)^2)+42628 ((#1)^4)-83096 ((#1)^6)+72606 ((#1)^8)-34280 ((#1)^10)+18324 ((#1)^12)+248 ((#1)^14)+((#1)^16)&,1]}})//N

{s1,s2}=RootReduce[EuclideanDistance[v[[5]],#]&/@v[[{1,3}]]]//FullSimplify

{x,y}/.Solve[SquaredEuclideanDistance[{x,y},z[[3]]]==s1^2&&SquaredEuclideanDistance[{x,y},z[[4]]]==s2^2/.Abs[e_]^2:>e^2,{x,y}][[1]]//RootReduce

triangle3={z[[4]],z[[3]],{Root[841-7888 #1+20606 ((#1)^2)-21272 ((#1)^3)+11875 ((#1)^4)-4008 ((#1)^5)+786 ((#1)^6)-72 ((#1)^7)+((#1)^8)&,1],Root[2247001-17671052 ((#1)^2)+753970 ((#1)^4)+1110928 ((#1)^6)-11861 ((#1)^8)+56032 ((#1)^10)-32710 ((#1)^12)+3572 ((#1)^14)+((#1)^16)&,4]}}

{s1,s2}=RootReduce[EuclideanDistance[v[[6]],#]&/@v[[{1,5}]]]//FullSimplify

N[%299]

RootReduce[{x,y}/.Solve[SquaredEuclideanDistance[{x,y},z[[3]]]==s2^2&&SquaredEuclideanDistance[{x,y},z[[7]]]==s1^2/.Abs[e_]^2:>e^2,{x,y}][[2]]]

triangle4={z[[7]],z[[3]],{Root[68105441-27865960 #1-192918132 ((#1)^2)+141486200 ((#1)^3)-8965326 ((#1)^4)-92512912 ((#1)^5)+135191512 ((#1)^6)-137167784 ((#1)^7)+113956299 ((#1)^8)-63914032 ((#1)^9)+25294888 ((#1)^10)-6298024 ((#1)^11)+969458 ((#1)^12)-66728 ((#1)^13)+2564 ((#1)^14)-64 ((#1)^15)+((#1)^16)&,3],Root[37504923377810641-192350660296392208 ((#1)^2)+384882026327606792 ((#1)^4)-402081594309036304 ((#1)^6)+246488589316912556 ((#1)^8)-90668925030830928 ((#1)^10)+17899512538367640 ((#1)^12)-651858377110352 ((#1)^14)-434635889298506 ((#1)^16)+40936106053584 ((#1)^18)+16513222741240 ((#1)^20)-3749497314096 ((#1)^22)+180292851276 ((#1)^24)+223075728 ((#1)^26)+92648 ((#1)^28)-1136 ((#1)^30)+((#1)^32)&,8]}};

{s1,s2}=RootReduce[EuclideanDistance[v[[5]],#]&/@v[[{4,3}]]]//FullSimplify

{x5,y5}=RootReduce[{x,y}/.Solve[SquaredEuclideanDistance[{x,y},z[[1]]]==s1^2&&SquaredEuclideanDistance[{x,y},z[[4]]]==s2^2/.Abs[e_]^2:>e^2,{x,y}][[1]]]

N[%]

{s1,s2}=RootReduce[EuclideanDistance[v[[6]],#]&/@v[[{4,3}]]]//FullSimplify

{x6,y6}=RootReduce[{x,y}/.Solve[SquaredEuclideanDistance[{x,y},z[[1]]]==s1^2&&SquaredEuclideanDistance[{x,y},z[[4]]]==s2^2/.Abs[e_]^2:>e^2,{x,y}][[2]]]

N[%]

square2={z[[1]],z[[4]],{x5,y5},{x6,y6}}

square2={{0,0},{(1/2) (3-(Sqrt[5])-2 (Sqrt[-2+(Sqrt[5])])),(Sqrt[(1/2) (3-(Sqrt[5])+4 (Sqrt[-2+(Sqrt[5])]))])},{Root[1+8 #1+3 ((#1)^2)-2 ((#1)^3)+((#1)^4)&,1],Root[-19-14 ((#1)^2)+27 ((#1)^4)-6 ((#1)^6)+((#1)^8)&,2]},{Root[-25+5 ((#1)^2)+((#1)^4)&,1],Root[-19+188 ((#1)^2)+18 ((#1)^4)-12 ((#1)^6)+((#1)^8)&,1]}};

polys=net=Polygon/@{square1,square2,triangle1,triangle2,triangle3,triangle4};
z=DeleteDuplicates[Flatten[polys[[All,1]],1]];
Graphics[{EdgeForm[Black],FaceForm[None],polys,MapIndexed[Text[#2[[1]],#,Background->White]&,z],
MapIndexed[Text[Subscript["S",#2[[1]]],Mean[#]]&,{square1}],
MapIndexed[Text[Subscript["T",#2[[1]]],Mean[#]]&,{triangle1,triangle2,triangle3,triangle4}]
}]

(Subscript[S, 1])

square1

EuclideanDistance@@@Partition[z[[{1,2,3,4}]],2,1,1]//RootReduce

EuclideanDistance@@@Partition[v[[{4,2,1,3}]],2,1,1]//RootReduce

(Subscript[T, 1])

triangle1

EuclideanDistance@@@Partition[z[[{2,1,5}]],2,1,1]//RootReduce

EuclideanDistance@@@Partition[v[[{2,4,6}]],2,1,1]//RootReduce

(Subscript[T, 2])

triangle2

EuclideanDistance@@@Partition[z[[{3,2,6}]],2,1,1]//RootReduce

EuclideanDistance@@@Partition[v[[{1,2,6}]],2,1,1]//RootReduce

(Subscript[T, 3])

triangle3

EuclideanDistance@@@Partition[z[[{4,3,7}]],2,1,1]//RootReduce

EuclideanDistance@@@Partition[v[[{3,1,5}]],2,1,1]//RootReduce

(Subscript[T, 4])

triangle4

EuclideanDistance@@@Partition[z[[{7,3,8}]],2,1,1]//N[#,100]&

EuclideanDistance@@@Partition[z[[{7,3,8}]],2,1,1]

EuclideanDistance@@@Partition[v[[{5,1,6}]],2,1,1]//RootReduce

N[%,100]

(Subscript[S, 2])

EuclideanDistance@@@Partition[z[[{1,4,5,6}]],2,1,1]//RootReduce

EuclideanDistance@@@Partition[v[[{4,3,5,6}]],2,1,1]//RootReduce

Generate

?NetFromPolygons

newnet=NetFromPolygons[net]//RootReduce

Graphics[{EdgeForm[Black],FaceForm[None],%%}]

PolyhedronData

PolyhedronDataString

file=OpenWrite["~/TetragonalAntiwedge.m"]

gc0=GraphicsComplex[{{Root[-1+14 ((#1)^2)+25 ((#1)^4)-8 ((#1)^6)+((#1)^8)&,2],-1,Root[-1-14 ((#1)^2)+25 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,2]},{Root[-1+14 ((#1)^2)+25 ((#1)^4)-8 ((#1)^6)+((#1)^8)&,1],-1,Root[-1-14 ((#1)^2)+25 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,1]},{Root[-1+10 ((#1)^2)-31 ((#1)^4)+80 ((#1)^6)+((#1)^8)&,2],Root[1-8 #1-2 ((#1)^2)-8 ((#1)^3)+((#1)^4)&,1],Root[-1+6 ((#1)^2)-15 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,1]},{Root[-1+10 ((#1)^2)-31 ((#1)^4)+80 ((#1)^6)+((#1)^8)&,1],Root[1-8 #1-2 ((#1)^2)-8 ((#1)^3)+((#1)^4)&,1],Root[-1+6 ((#1)^2)-15 ((#1)^4)+8 ((#1)^6)+((#1)^8)&,2]},{Root[-1-2 ((#1)^2)+5 ((#1)^4)-4 ((#1)^6)+((#1)^8)&,2],1,Root[-1+2 ((#1)^2)+5 ((#1)^4)+4 ((#1)^6)+((#1)^8)&,2]},{Root[-1-2 ((#1)^2)+5 ((#1)^4)-4 ((#1)^6)+((#1)^8)&,1],1,Root[-1+2 ((#1)^2)+5 ((#1)^4)+4 ((#1)^6)+((#1)^8)&,1]}}, Polygon[{{1,2,6,4},{1,5,3,2},{1,4,5},{2,3,6},{3,5,6},{4,6,5}}]];

Graphics3D[{FaceForm[Blue,Red],PolyhedronData["Cube","GraphicsComplex"]}]

Graphics3D[{FaceForm[Blue,Red],gc0}]

s=Min[PolyhedronEdgeLengths[gc0]//RootReduce]

Graphics3D[{FaceForm[Blue,Red],gc=GraphicsComplex[RootReduce[gc0[[1]]/s],Polygon/@Reverse/@gc0[[2,1]]]},Boxed->False,Method->{"ShrinkWrap"->True}]

PolyhedronEdgeLengths[gc]//RootReduce

N[%]

WriteString[file,PolyhedronDataString[gc,
"Name"->"tetragonal antiwedge",
"StandardName"->"TetragonalAntiwedge",
"Classes"->{"Convex"},
Debug->True
]]

Close[file]