Endododecahedron

Author

Eric W. Weisstein
April 7, 2022

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

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

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

Source

https://twitter.com/johncarlosbaez/status/1173068560433729537

When you fold the dodecahedron into the cube, the shape of the empty space inside is called the 'concave pyrohedral dodecahedron' or 'endo-docahedron'.  It looks like this.

Import["https://pbs.twimg.com/media/EEeTodVU8AEAp60?format=jpg&name=small"]

Unfolding animation

https://twitter.com/johncarlosbaez/status/1173068560433729537

Unfolding net

Rotate[Import["https://pbs.twimg.com/media/EEiK109X4AAXXSE?format=jpg&name=4096x4096"],Pi/2]

Length[v={-#2,#1}&@@@{{794.4935557839772`,4067.4177962809526`},{1076.7746604281429`,4010.7883498750343`},{1513.3028626826706`,3392.704563034371`},{1511.672505251281`,3099.4694943980576`},{1060.1399197609971`,3251.092735517277`},{806.1862754872232`,3628.1638401958207`},{354.959382015325`,3492.156366349128`},{791.6659046139112`,2870.455223957569`},{325.7657942595074`,2242.1562288358564`},{756.94`,2085.95`},{48.88524860681946`,1863.5821384002506`},{44.8857780329422`,1107.1981875750034`},{317.2064177447128`,1479.454642963975`},{1508.36`,2614.82`},{1064.0120186605473`,2486.68436920941`},{1763.92`,2223.36`},{1499.3429271763985`,1872.9821679656056`},{1050.99`,2003.05`},{759.62`,1626.06`},{330.88613556746486`,1013.7838016233604`},{764.54`,854.34`},{1042.6135773735605`,1252.6821139916474`},{1474.7856683660948`,1386.1421543517936`},{1220.9084470969174`,1016.3821837796377`},{1488.3380145145195`,629.6817805219498`},{1489.1022445604833`,167.34807704871127`},{1922.6499496357524`,25.81267253621354`},{2199.76`,391.06`},{1943.2586865419098`,778.7575881546236`},{1928.8402130080592`,2004.9392225686902`},{1928.79`,1231.69`},{2207.5549107710603`,861.1161127746568`},{2659.3167652751335`,1014.1659166463423`},{2215.5793262536804`,1630.593871720751`},{2212.6497777441527`,2092.3416654920843`},{2663.061492500356`,1482.1039737899828`},{2659.5969829586534`,2241.977908491798`},{2931.05`,1851.74`},{2934.3122101477775`,1097.492465991263`}}]

Graphics[{Point[v],MapIndexed[Text[#2[[1]],#,{1,1},Background->White]&,v]}]

EuclideanDistance@@v[[#]]&/@{{1,7},{6,7}}

723 .01

EuclideanDistance@@v[[{1,6}]]

ee=Select[Subsets[Range[39],{2}],MatchQ[Round[.01EuclideanDistance@@v[[#]]],5|7]&]

g=Graph[Range[39],UndirectedEdge@@@ee,VertexCoordinates->v]

Graphics

Graphics3D

Show[PolyhedronData["Endododecahedron"],Boxed->False]

GraphicsComplex

Graphics3D[{FaceForm[Green,Red],PolyhedronData["Endododecahedron","GraphicsComplex"]}]

Polyhedron

Rendering issues

Graphics3D@PolyhedronData["Endododecahedron","Polyhedron"]

Properties

bug(421961): Graphics3D[PolyhedronData["Endododecahedron", "Polyhedron"]] displays incorrectly starting in V13

Graphics3D[p=PolyhedronData[pname="Endododecahedron","Polyhedron"]]

AdjacentFaceIndices

AdjacentFaceIndices[p]

Centroid

RegionCentroid[p]//FullSimplify

Circumsphere

Circumsphere[p]

MyCircumsphere[p]

Classes

PolyhedronData["Endododecahedron","Classes"]

Convex

ConvexPolyhedronQ[p]

ConvexHull

Graphics3D[p]

hull=ConvexHullRegion[p]

InputForm[%]

hull=Polyhedron[{{(-1 + Sqrt[5])/4, -1/2, 0}, {(1 + Sqrt[5])/4, (-1 - Sqrt[5])/4, 
   (1 + Sqrt[5])/4}, {1/2, 0, (-1 + Sqrt[5])/4}, {(1 + Sqrt[5])/4, (1 + Sqrt[5])/4, 
   (1 + Sqrt[5])/4}, {(-1 + Sqrt[5])/4, 1/2, 0}, {(1 + Sqrt[5])/4, (-1 - Sqrt[5])/4, 
   (-1 - Sqrt[5])/4}, {1/2, 0, (1 - Sqrt[5])/4}, {(1 + Sqrt[5])/4, (1 + Sqrt[5])/4, 
   (-1 - Sqrt[5])/4}, {(1 - Sqrt[5])/4, -1/2, 0}, {(-1 - Sqrt[5])/4, (-1 - Sqrt[5])/4, 
   (1 + Sqrt[5])/4}, {-1/2, 0, (-1 + Sqrt[5])/4}, {(-1 - Sqrt[5])/4, (1 + Sqrt[5])/4, 
   (1 + Sqrt[5])/4}, {(1 - Sqrt[5])/4, 1/2, 0}, {(-1 - Sqrt[5])/4, (-1 - Sqrt[5])/4, 
   (-1 - Sqrt[5])/4}, {-1/2, 0, (1 - Sqrt[5])/4}, {(-1 - Sqrt[5])/4, (1 + Sqrt[5])/4, 
   (-1 - Sqrt[5])/4}, {0, (-1 + Sqrt[5])/4, -1/2}, {0, (-1 + Sqrt[5])/4, 1/2}, 
  {0, (1 - Sqrt[5])/4, -1/2}, {0, (1 - Sqrt[5])/4, 1/2}}, {{2, 4, 12, 10}, {2, 10, 14, 
 6}, {10, 12, 16, 14}, {4, 2, 6, 8}, {12, 4, 8, 16}, {6, 14, 16, 8}}]

Graphics3D[{Opacity[.5],p,hull}]

EuclideanDistance@@hull[[1]][[{2,4}]]//FullSimplify

N[%]

DihedralAngles

DihedralAngles[p]//FullSimplify

GeneralizedDiameter

GeneralizedDiameter[p]//FullSimplify

EdgeCount

PolyhedronData["Endododecahedron","EdgeCount"]

PolyhedronData["Endododecahedron","Lines"]//Length

EdgeLengths

PolyhedronData["Endododecahedron","EdgeLengths"]

e=PolyhedronData["Endododecahedron","Lines"]/.Line[{l__}]:>EuclideanDistance[l]//RootReduce//Union//Quiet

Edges

Graphics3D[{PolyhedronData["Endododecahedron","Lines"]},Boxed-> False]

FaceCount

PolyhedronData["Endododecahedron","FaceCount"]

PolyhedronData["Endododecahedron","Polygons"]//Length

InertiaTensor

MomentOfInertia[p,{0,0,0}]/Volume[p]//FullSimplify

PolyhedronInertiaTensor[p,Method->"GaussLaw"]//FullSimplify

PolyhedronInertiaTensor[p,(1/4) (1+(Sqrt[5]))]//FullSimplify

Insphere

PolyhedronData["Endododecahedron","Insphere"]

Graphics3D[{{Opacity[.5],p},Yellow,PolyhedronData["Endododecahedron","Insphere"]}]

Insphere[p]

MyInsphere[p]

MeanCylindricalRadius

PolyhedronData["Endododecahedron","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

N[i]

MeanSquareCylindricalRadius

PolyhedronData["Endododecahedron","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["Endododecahedron","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["Endododecahedron","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]

Midsphere

Midsphere[p]

Polygons

Graphics3D/@PolyhedronData["Endododecahedron","Polygons"]

PolyhedralEmbeddings

g=Graph3D[Range[20],UndirectedEdge@@@PolyhedronData["Endododecahedron","EdgeIndices"],VertexCoordinates->PolyhedronData["Endododecahedron","Vertices"]]

r=RecognizeGraph[g]

ToCommonEdges[g,GraphData[r]]

RegionFunction

PolyhedronData["Endododecahedron","RegionFunction"]

Simple

SimplePolyhedronQ[p]

Skeleton

PolyhedronData["Endododecahedron","Skeleton"]

e=Union[Sort/@Join@@(Partition[#,2,1,1]&/@faces)]

r=RecognizeGraph[skel=Graph[UndirectedEdge@@@e]]

ToCommonEdges[GraphData[r],Graph[Range[12],UndirectedEdge@@@e]]

SurfaceArea

SurfaceArea[p]//FullSimplify

PolyhedronSurfaceArea[p]//FullSimplify

VertexCount

PolyhedronData["Endododecahedron","VertexCount"]

PolyhedronData["Endododecahedron","Points"]//Length

VertexSubsetHulls

(hulls=SortHulls@VertexSubsetHulls[PolyhedronData["Endododecahedron","Polyhedron"],"Platonic",Debug->True])//Timing

hulls/.{i__Integer}:>Graphics3D[ConvexHullRegion[PolyhedronData["Endododecahedron","VertexCoordinates"][[{i}]]]]

Volume

PolyhedronData["Endododecahedron","Volume"]

Volume[p]//FullSimplify

GoldenRatio/2//FunctionExpand

PolyhedronVolume[p]//FullSimplify

(GoldenRatio/2)//N

From removing 6 oblique wedges

(1/6)b h(2a+c)/.b->a

(a^3)-6((1/6)b h(2a+c))/.b->a/.{h->(1/2),a->GoldenRatio,c->1}//FullSimplify

Construction

Cube with 6 dimples

v=PolyhedronData["Endododecahedron","Vertices"];

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

v[[4]]

RegionDistance[Polygon[v[[{4,12,10,2}]]],Mean[v[[{3,11}]]]]//FullSimplify

top=Polyhedron[v,{{4,12,10,2},{3,4,2},{10,12,11},{11,3,2,10},{12,4,3,11}}];

bottom=Polyhedron[{#,#2,-#3}&@@@top[[1]],top[[2]]];

Graphics3D/@{top,bottom}

Region[RegionDifference[Cube[GoldenRatio],RegionUnion[top,bottom]]]

From Harry Calkins

Original construction by
Harry Calkins
Oct 23, 2019

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

endoDodecEdgeIndx={{1,2},{1,5},{1,6},{2,3},{2,20},{3,4},{3,11},{4,5},{4,18},{5,8},{6,7},{6,19},{7,8},{7,15},{8,17},{9,10},{9,13},{9,14},{10,11},{10,20},{11,12},{12,13},{12,18},{13,16},{14,15},{14,19},{15,16},{16,17},{17,18},{19,20}};

polyhedronData["Endododecahedron",opts:OptionsPattern[Graphics3D]]:=Graphics3D[GraphicsComplex[endododecVrts,Polygon[endoDodecIndx]],{opts}]

polyhedronData["Endododecahedron","Faces"]:=GraphicsComplex[endododecVrts,Polygon[endoDodecIndx]]

polyhedronData["Endododecahedron","FaceCount"]:=12

polyhedronData["Endododecahedron","FaceIndices"]:=endoDodecIndx

polyhedronData["Endododecahedron","EdgeIndices"]:=endoDodecEdgeIndx

polyhedronData["Endododecahedron","Edges"]:= GraphicsComplex[endododecVrts,Line[endoDodecEdgeIndx]]

polyhedronData["Geododecahedron","EdgeLength"]:={1}

polyhedronData["Endododecahedron","EdgeCount"]:=30

polyhedronData["Endododecahedron","Equalateral"]:=True

polyhedronData["Endododecahedron","Circumcenter"]:={0,0,0}

polyhedronData["Endododecahedron","CircumRadius"]:=1.4012585384440737`

polyhedronData["Endododecahedron","CircumSphere"]:= Sphere[{0,0,0},1.4012585384440737`]

polyhedronData["Endododecahedron","Name"]="Endododecahedron";

polyhedronData["Endododecahedron","Vertices"]:= endododecVrts

From specified coordinates

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

Length[endododecVrts]

v=Take[endododecVrts,20]

v2=v/.{(1/4) (-1+(Sqrt[5]))->1/(2GoldenRatio),(1/4) (-1-(Sqrt[5]))->-GoldenRatio/2,(1/4) (1+(Sqrt[5]))->GoldenRatio/2,(1/4) (1-(Sqrt[5]))->-1/(2GoldenRatio)}

N[v-v2]//Chop

(1/4) (-1+(Sqrt[5]))//N

1/(2GoldenRatio)//N

SquaredEuclideanDistance@@v[[{1,2}]]//FullSimplify

Fix orientation of faces in original list:

endoDodecIndx={
{3,4,18,12,11},Reverse@{3,2,20,10,11},
Reverse@{7,8,17,16,15},{7,6,19,14,15},Reverse@{17,8,5,4,18},{17,16,13,12,18},{19,6,1,2,20},Reverse@{19,14,9,10,20},Reverse@{1,2,3,4,5},{1,6,7,8,5},{9,10,11,12,13},Reverse@{9,14,15,16,13}}

SmallestRotation[l_]:=Sort[NestList[RotateRight,l,Length[l]-1]][[1]]

faces=Sort[SmallestRotation/@endoDodecIndx]

faces={{1,2,20,19,6},{1,5,4,3,2},{1,6,7,8,5},{2,3,11,10,20},{3,4,18,12,11},{4,5,8,17,18},{6,19,14,15,7},{7,15,16,17,8},{9,10,11,12,13},{9,13,16,15,14},{9,14,19,20,10},{12,18,17,16,13}};

Length[faces]

Union[Sort/@Join@@(Partition[#,2,1,1]&/@faces)]

Length[%]

Graphics3D[{FaceForm[Green,Red],p=Polyhedron[v,faces],MapIndexed[Text[#2[[1]],#,Background->White]&,v]}]

Net

Construction

poly=Normal[PolyhedronData["Endododecahedron","Faces"]][[1,1]];
v=poly[[1]];

Graphics3D[{poly,MapIndexed[Text[#2[[1]],#,Background->White]&,v]}]

EuclideanDistance@@v[[#]]&/@{{3,4},{2,3},{1,2}}//FullSimplify

y=RegionDistance[Line[v[[{3,4}]]],v[[1]]]//FullSimplify

eqn=Thread[SquaredEuclideanDistance@@@{{{0,y},{a,b}},{{a,b},{1/2,0}}}==1/.Abs->Identity]

Solve[eqn,{a,b},PositiveReals]//FullSimplify

N[%]

p1

p1=With[{a=(1/4) (1+(Sqrt[5])),b=(1/2) (Sqrt[(1/2) (5+(Sqrt[5]))]),y=(1/2) (Sqrt[5-2 (Sqrt[5])])},{{1/2,0},{a,-b},{0,-y},{-a,-b},{-1/2,0}}];

Graphics[{Style[Text["6×",{-1/2,0},{3,0}],24],FaceForm[LightYellow],EdgeForm[Black],Polygon@p1,Polygon[{#,-#2}&@@@p1]}]

p=Polyhedron[v,faces]

Graphics3D[{Polygon[v[[faces[[1]]]]],MapIndexed[Text[#2[[1]],#,Background->White]&,v]}]

EuclideanDistance[Mean[v[[{19,20}]]],v[[1]]]//FullSimplify

Solve[{
(x-1/2)^2+y^2==1,
x^2+(h-y)^2==1,
h==(1/2) (Sqrt[5-2 (Sqrt[5])])},{x,y,h}]//FullSimplify

N[%[[2]],20]

N[5-2 (Sqrt[5]),10]

v=With[{x=(1/4) (1+(Sqrt[5])),h=(1/2) (Sqrt[5-2 (Sqrt[5])]),y=(1/2) (Sqrt[(1/2) (5+(Sqrt[5]))])},
{
{{-1/2,0},{1/2,0},{x,y},{0,h},{-x,y}},
{{1/2,0},{-1/2,0},{-x,-y},{0,-h},{x,-y}}
}]

Graphics[{FaceForm[LightYellow],EdgeForm[Black],Polygon/@v}]

Need 6 of those.  However, can't be done in a flat net; needs a multi-piece net.