Hebesphenomegacorona

Author

Eric W. Weisstein
May 13, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

To do

z-component of last expression in paper is wrong and needed to be computed based on the other coordinates

Simplify combinations of Root objects in RegionFunction

Graphic

PolyhedronData["Hebesphenomegacorona","StandardNames"]

PolyhedronData

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

PolyhedronData["Hebesphenomegacorona","VertexCoordinates"]

PolyhedronData (V12.1)

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

Net

PolyhedronData

PolyhedronData["Hebesphenomegacorona","Net"]

PolyhedronData["Hebesphenomegacorona","Net","Colored"]

Original (V12.1)

PolyhedronData["Hebesphenomegacorona","Net"]

v=PolyhedronData["Hebesphenomegacorona","Net","Coordinates"]

Graphics[{PolyhedronData["Hebesphenomegacorona","Net"][[1]],MapIndexed[Text[#2[[1]],#,Background->White]&,v]},Axes->True]

center=Mean[v[[{9,16}]]]//FullSimplify

v2=(#-center)&/@v;

Graphics[{Translate[PolyhedronData["Hebesphenomegacorona","Net"][[1]],-center],MapIndexed[Text[#2[[1]],#,Background->White]&,v2]},Axes->True]

ang=ArcTan@@Subtract@@v[[{13,9}]]//FullSimplify

v3=RotationMatrix[-ang].#&/@v2//FullSimplify

v3={{(1/2) (-3-(Sqrt[3])),1},{-2,(1/2) (1+(Sqrt[3]))},{(1/2) (-3-(Sqrt[3])),0},{(1/2) (-3-(Sqrt[3])),-1},{-(3/2),(1/2)},{-2,(1/2) (-1-(Sqrt[3]))},{-1,(1/2) (1+(Sqrt[3]))},{-(3/2),-(1/2)},{-(1/2),(1/2)},{-1,(1/2) (-1-(Sqrt[3]))},{0,(1/2) (1+(Sqrt[3]))},{-(1/2),-(1/2)},{(1/2),(1/2)},{0,(1/2) (-1-(Sqrt[3]))},{1,(1/2) (1+(Sqrt[3]))},{(1/2),-(1/2)},{(3/2),(1/2)},{1,(1/2) (-1-(Sqrt[3]))},{2,(1/2) (1+(Sqrt[3]))},{(3/2),-(1/2)},{(1/2) (3+(Sqrt[3])),1},{(1/2) (3+(Sqrt[3])),0},{2,(1/2) (-1-(Sqrt[3]))},{(1/2) (3+(Sqrt[3])),-1},{(3/2)+(Sqrt[3]),(1/2)},{(3/2)+(Sqrt[3]),-(1/2)}};

off=Min/@Transpose[v3]

v4=(#-off)&/@v3//FullSimplify

Graphics[{FaceForm[LightYellow],EdgeForm[Black],GraphicsComplex[v4,Polygon[PolyhedronData["Hebesphenomegacorona","NetFaceIndices"]]]},Axes->True]

Construction

Timofeenko, A.~V.  ``The Non-Platonic and Non-Archimedean Noncomposite Polyhedra.''  {\it J.\ Math.\ Sci.} {\bf 162}, 710--729, 2009.

Error in z-component of last coordinate provided

eqn=26880x^10+35328x^9-25600x^8-39680x^7+6112x^6+13696x^5+2128x^4-1808x^3-1119x^2+494x-47;

soln=Reduce[eqn==0,x,Reals][[All,2]]

N[soln]

soln[[3]]

(aroot=Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3])

v=With[{a=Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3]},
Flatten[Outer[Times,
{{1,1,1},{-1,1,1},{1,-1,1},{-1,-1,1}},
{{1,1,2(Sqrt[1-(a^2)])},{1+2a,1,0},{0,1+(Sqrt[2])(Sqrt[((2a-1)/(a-1))]),-((2(a^2)+a-1)/(Sqrt[1-(a^2)]))},{1,0,-(Sqrt[3-4(a^2)])},

(* the z-component of this one is wrong *)
{0,(((Sqrt[2(3-4(a^2))(1-2a)])+(Sqrt[1+a]))/(2(1-a)(Sqrt[1+a]))),(((2a-1)(Sqrt[3-4(a^2)]))/(4(1-a)))-((Sqrt[2(1-2a)])/(2(1-a)(Sqrt[1+a])))}},
1],1]//Union
];

Something is wrong with the z-component of this one; note the fact that triangles around v[[7]] are not equilateral:

ConvexHullMesh[v,MeshCellLabel->{0->"Index"}]

v[[7]]//N

v[[7]]/.aroot->a

v[[7]]//N

FindRoot[SquaredEuclideanDistance[v[[5]],{0,y,z}]==4&&SquaredEuclideanDistance[v[[9]],{0,y,z}]==4,{{y,1.6},{z,0}}]

So y component is correct but z is wrong:

With[{a=aroot},(((2a-1)(Sqrt[3-4(a^2)]))/(4(1-a)))-((Sqrt[2(1-2a)])/(2(1-a)(Sqrt[1+a])))]//N

ConvexHullMesh[ReplacePart[v,7->{0,1.6713181434447337`,-1.2222381073346047`}],MeshCellLabel->{0->"Index"}]

Correct z component:

With[{a=Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3]},FindRoot[SquaredEuclideanDistance[v[[5]],{0,(((Sqrt[1+a])+(Sqrt[2]) (Sqrt[(1-2 a) (3-4 (a^2))]))/(2 (1-a) (Sqrt[1+a]))),z}]==4,{z,0}]]

v[[5]]/.Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3]->a

With[{a=Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3]},Solve[SquaredEuclideanDistance[{0,1+(Sqrt[2]) (Sqrt[((-1+2 a)/(-1+a))]),-((-1+a+2 (a^2))/(Sqrt[1-(a^2)]))},{0,(((Sqrt[1+a])+(Sqrt[2]) (Sqrt[(1-2 a) (3-4 (a^2))]))/(2 (1-a) (Sqrt[1+a]))),z}]==4,z]]/.aroot->a

N[%/.a->aroot]

%382[[1,1,2]]

FullSimplify[%,a>0]

Together//@((((-1+2 a) (Sqrt[1-(a^2)]))/(-1+a))-(1/16) √(((32 a (8-8 a+((32 (-1+a) (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))-3 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])))/(((-1+a))^2))+32 (-32+((8 (Sqrt[2-4 a]))/(((1-a))^(3/2)))-((16 (Sqrt[2-4 a]))/(Sqrt[1-a]))-(4/(((-1+a))^2))-(35/(-1+a))+(3/(1+a))+((8 (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))+((8 (Sqrt[-(1+a) (-3+4 (a^2))]))/(((1-a))^(3/2)))+3 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])+(((Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(((-1+a))^2)))))//FullSimplify

N[%/.a->aroot]

FullSimplify[-((1+a)/(Sqrt[1-(a^2)]))+2 (Sqrt[1-(a^2)])-(1/4) √(-80+((16 (Sqrt[2-4 a]))/(((1-a))^(3/2)))-((32 (Sqrt[2-4 a]))/(Sqrt[1-a]))-(8/(((-1+a))^2))-(86/(-1+a))+(6/(1+a))-((32 (Sqrt[3-4 (a^2)]) (Sqrt[1-(a^2)]))/(((-1+a))^2))-((24 (Sqrt[3-4 (a^2)]) (Sqrt[1-(a^2)]))/(-1+a))+((24 (Sqrt[3-4 (a^2)]) (Sqrt[1-(a^2)]))/(1+a))+((16 (Sqrt[-(1+a) (-3+4 (a^2))]))/(((1-a))^(3/2)))+6 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])-((4 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(((-1+a))^2))-((6 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(-1+a))),0<a<1/4]

N[%%/.a->aroot]

Correct but messy and undersimplified z-component of v[[7]]:

aroot=Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3];

(basevorig=With[{a=aroot},
basev=
{{1,1,2(Sqrt[1-(a^2)])},{1+2a,1,0},{0,1+(Sqrt[2])(Sqrt[((2a-1)/(a-1))]),-((2(a^2)+a-1)/(Sqrt[1-(a^2)]))},{1,0,-(Sqrt[3-4(a^2)])},
{0,(((Sqrt[2(3-4(a^2))(1-2a)])+(Sqrt[1+a]))/(2(1-a)(Sqrt[1+a]))),(((-1+2 a) (Sqrt[1-(a^2)]))/(-1+a))-(1/16) √(((32 a (8-8 a+((32 (-1+a) (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))-3 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])))/(((-1+a))^2))+32 (-32+((8 (Sqrt[2-4 a]))/(((1-a))^(3/2)))-((16 (Sqrt[2-4 a]))/(Sqrt[1-a]))-(4/(((-1+a))^2))-(35/(-1+a))+(3/(1+a))+((8 (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))+((8 (Sqrt[-(1+a) (-3+4 (a^2))]))/(((1-a))^(3/2)))+3 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])+(((Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(((-1+a))^2))))}}])//N

RootReduce[2(Sqrt[1-(a^2)])/.a->aroot]

(basev=With[{a=aroot},{{1,1,},{1+2a,1,0},{0,1+(Sqrt[2])(Sqrt[((2a-1)/(a-1))]),-((2(a^2)+a-1)/(Sqrt[1-(a^2)]))},{1,0,-(Sqrt[3-4(a^2)])},
{0,(((Sqrt[2(3-4(a^2))(1-2a)])+(Sqrt[1+a]))/(2(1-a)(Sqrt[1+a]))),(((-1+2 a) (Sqrt[1-(a^2)]))/(-1+a))-(1/16) √(((32 a (8-8 a+((32 (-1+a) (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))-3 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])))/(((-1+a))^2))+32 (-32+((8 (Sqrt[2-4 a]))/(((1-a))^(3/2)))-((16 (Sqrt[2-4 a]))/(Sqrt[1-a]))-(4/(((-1+a))^2))-(35/(-1+a))+(3/(1+a))+((8 (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))+((8 (Sqrt[-(1+a) (-3+4 (a^2))]))/(((1-a))^(3/2)))+3 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])+(((Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(((-1+a))^2))))}}])==basevorig//N

RootReduce[1+2a/.a->aroot]

(basev=With[{a=aroot},{{1,1,},{,1,0},{0,1+(Sqrt[2])(Sqrt[((2a-1)/(a-1))]),-((2(a^2)+a-1)/(Sqrt[1-(a^2)]))},{1,0,-(Sqrt[3-4(a^2)])},
{0,(((Sqrt[2(3-4(a^2))(1-2a)])+(Sqrt[1+a]))/(2(1-a)(Sqrt[1+a]))),(((-1+2 a) (Sqrt[1-(a^2)]))/(-1+a))-(1/16) √(((32 a (8-8 a+((32 (-1+a) (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))-3 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])))/(((-1+a))^2))+32 (-32+((8 (Sqrt[2-4 a]))/(((1-a))^(3/2)))-((16 (Sqrt[2-4 a]))/(Sqrt[1-a]))-(4/(((-1+a))^2))-(35/(-1+a))+(3/(1+a))+((8 (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))+((8 (Sqrt[-(1+a) (-3+4 (a^2))]))/(((1-a))^(3/2)))+3 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])+(((Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(((-1+a))^2))))}}])==basevorig//N

RootReduce[1+(Sqrt[2])(Sqrt[((2a-1)/(a-1))])/.a->aroot]

(basev=With[{a=aroot},{{1,1,},{,1,0},{0,,-((2(a^2)+a-1)/(Sqrt[1-(a^2)]))},{1,0,-(Sqrt[3-4(a^2)])},
{0,(((Sqrt[2(3-4(a^2))(1-2a)])+(Sqrt[1+a]))/(2(1-a)(Sqrt[1+a]))),(((-1+2 a) (Sqrt[1-(a^2)]))/(-1+a))-(1/16) √(((32 a (8-8 a+((32 (-1+a) (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))-3 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])))/(((-1+a))^2))+32 (-32+((8 (Sqrt[2-4 a]))/(((1-a))^(3/2)))-((16 (Sqrt[2-4 a]))/(Sqrt[1-a]))-(4/(((-1+a))^2))-(35/(-1+a))+(3/(1+a))+((8 (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))+((8 (Sqrt[-(1+a) (-3+4 (a^2))]))/(((1-a))^(3/2)))+3 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])+(((Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(((-1+a))^2))))}}])==basevorig//N

With[{a=aroot},RootReduce[-((RootReduce[2(a^2)+a-1])/(RootReduce[(Sqrt[1-(a^2)])]))]]//Timing

(basev=With[{a=aroot},{{1,1,},{,1,0},{0,,},{1,0,-(Sqrt[3-4(a^2)])},
{0,(((Sqrt[2(3-4(a^2))(1-2a)])+(Sqrt[1+a]))/(2(1-a)(Sqrt[1+a]))),(((-1+2 a) (Sqrt[1-(a^2)]))/(-1+a))-(1/16) √(((32 a (8-8 a+((32 (-1+a) (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))-3 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])))/(((-1+a))^2))+32 (-32+((8 (Sqrt[2-4 a]))/(((1-a))^(3/2)))-((16 (Sqrt[2-4 a]))/(Sqrt[1-a]))-(4/(((-1+a))^2))-(35/(-1+a))+(3/(1+a))+((8 (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))+((8 (Sqrt[-(1+a) (-3+4 (a^2))]))/(((1-a))^(3/2)))+3 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])+(((Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(((-1+a))^2))))}}])==basevorig//N

With[{a=aroot},RootReduce[-(Sqrt[3-4(a^2)])]]//Timing

(basev=With[{a=aroot},{{1,1,},{,1,0},{0,,},{1,0,},
{0,(((Sqrt[2(3-4(a^2))(1-2a)])+(Sqrt[1+a]))/(2(1-a)(Sqrt[1+a]))),(((-1+2 a) (Sqrt[1-(a^2)]))/(-1+a))-(1/16) √(((32 a (8-8 a+((32 (-1+a) (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))-3 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])))/(((-1+a))^2))+32 (-32+((8 (Sqrt[2-4 a]))/(((1-a))^(3/2)))-((16 (Sqrt[2-4 a]))/(Sqrt[1-a]))-(4/(((-1+a))^2))-(35/(-1+a))+(3/(1+a))+((8 (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))+((8 (Sqrt[-(1+a) (-3+4 (a^2))]))/(((1-a))^(3/2)))+3 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])+(((Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(((-1+a))^2))))}}])==basevorig//N

With[{a=aroot},RootReduce[((Sqrt[2(3-4(a^2))(1-2a)])/(2(1-a)(Sqrt[1+a])))+(1/(2(1-a)))]]//Timing

(basev=With[{a=aroot},{{1,1,},{,1,0},{0,,},{1,0,},
{0,,(((-1+2 a) (Sqrt[1-(a^2)]))/(-1+a))-(1/16) √(((32 a (8-8 a+((32 (-1+a) (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))-3 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])))/(((-1+a))^2))+32 (-32+((8 (Sqrt[2-4 a]))/(((1-a))^(3/2)))-((16 (Sqrt[2-4 a]))/(Sqrt[1-a]))-(4/(((-1+a))^2))-(35/(-1+a))+(3/(1+a))+((8 (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))+((8 (Sqrt[-(1+a) (-3+4 (a^2))]))/(((1-a))^(3/2)))+3 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])+(((Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(((-1+a))^2))))}}])==basevorig//N

With[{a=aroot},part1=RootReduce[(((-1+2 a) (Sqrt[1-(a^2)]))/(-1+a))]]//Timing

With[{a=aroot},part2=RootReduce/@(Expand/@(((32 a (8-8 a+((32 (-1+a) (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))-3 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])))/(((-1+a))^2))+32 (-32+((8 (Sqrt[2-4 a]))/(((1-a))^(3/2)))-((16 (Sqrt[2-4 a]))/(Sqrt[1-a]))-(4/(((-1+a))^2))-(35/(-1+a))+(3/(1+a))+((8 (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))+((8 (Sqrt[-(1+a) (-3+4 (a^2))]))/(((1-a))^(3/2)))+3 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])+(((Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(((-1+a))^2)))))]//Timing

part1-(1/16)Sqrt[part2]//N

basevorig[[-1,-1]]//N

lastv=part1-RootReduce[(1/16)Sqrt[RootReduce[part2]]]

N[%]

RootReduce[lastv]//Timing

{-Root[2822400-26732544 ((#1)^2)+92214016 ((#1)^4)-149274624 ((#1)^6)+134025568 ((#1)^8)-73396864 ((#1)^10)+25664944 ((#1)^12)-5781824 ((#1)^14)+814945 ((#1)^16)-65568 ((#1)^18)+2304 ((#1)^20)&,8]+Root[1679616-22228992 ((#1)^2)+111570688 ((#1)^4)-326365696 ((#1)^6)+690379360 ((#1)^8)-1123657216 ((#1)^10)+1368768752 ((#1)^12)-1230375584 ((#1)^14)+781745665 ((#1)^16)-291156864 ((#1)^18)+45158400 ((#1)^20)&,6]}

N[Subtract@@{-(NumericalApproximation[Root, (RowDefault["1.93", FEExpressionsNumericalApproximationElider]), Root[2822400-26732544 ((#1)^2)+92214016 ((#1)^4)-149274624 ((#1)^6)+134025568 ((#1)^8)-73396864 ((#1)^10)+25664944 ((#1)^12)-5781824 ((#1)^14)+814945 ((#1)^16)-65568 ((#1)^18)+2304 ((#1)^20)&,8], ])+(NumericalApproximation[Root, (RowDefault["0.706", FEExpressionsNumericalApproximationElider]), Root[1679616-22228992 ((#1)^2)+111570688 ((#1)^4)-326365696 ((#1)^6)+690379360 ((#1)^8)-1123657216 ((#1)^10)+1368768752 ((#1)^12)-1230375584 ((#1)^14)+781745665 ((#1)^16)-291156864 ((#1)^18)+45158400 ((#1)^20)&,6], ]),(NumericalApproximation[Root, (RowDefault["-1.22", FEExpressionsNumericalApproximationElider]), Root[104857600-5077204992 ((#1)^2)+42444783616 ((#1)^4)-154255687680 ((#1)^6)+283845722112 ((#1)^8)-311955696640 ((#1)^10)+222271076224 ((#1)^12)-104429998080 ((#1)^14)+31258427913 ((#1)^16)-5397653376 ((#1)^18)+406425600 ((#1)^20)&,3], ])},100]

(basev=With[{a=aroot},{{1,1,},{,1,0},{0,,},{1,0,},
{0,,(NumericalApproximation[Root, (RowDefault["-1.22", FEExpressionsNumericalApproximationElider]), Root[104857600-5077204992 ((#1)^2)+42444783616 ((#1)^4)-154255687680 ((#1)^6)+283845722112 ((#1)^8)-311955696640 ((#1)^10)+222271076224 ((#1)^12)-104429998080 ((#1)^14)+31258427913 ((#1)^16)-5397653376 ((#1)^18)+406425600 ((#1)^20)&,3], ])}}])//N

v=With[{a=aroot},
Flatten[Outer[Times,
{{1,1,1},{-1,1,1},{1,-1,1},{-1,-1,1}},
basev,
1],1]//Union];

Length[v]

New

newv=RootReduce[v/2]

e=Select[Subsets[Range[14],{2}],N[EuclideanDistance@@newv[[#]],30]==1.&]

Length[e]

Old

v

e=Select[Subsets[Range[14],{2}],N[EuclideanDistance@@v[[#]],30]==2.&]

Length[e]

PolyhedronData["Hebesphenomegacorona","E"]

PolyhedronData["Hebesphenomegacorona","F"]

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

tri=Sort[SmallestRotation/@FindCycle[Graph[Range[14],UndirectedEdge@@@e],{3},All][[All,All,1]]]

Length[tri]

PolyhedronData["Hebesphenomegacorona","Faces","Tally"]

Graphics3D[{MapIndexed[Text[#2[[1]],#,Background->White]&,v],FaceForm[Green,Red],Polygon[v[[#]]]&/@tri}]

tri=Sort[SmallestRotation/@{{1,4,8},Reverse@{1,4,11},{2,7,9},{2,9,6},Reverse@{2,11,6},{2,11,12},{2,12,7},Reverse@{3,5,10},Reverse@{3,12,5},{4,6,13},Reverse@{4,8,13},{4,11,6},{5,7,12},{5,14,7},Reverse@{5,14,10},Reverse@{6,13,9},Reverse@{7,9,14},Reverse@{9,13,14}}]

Graphics3D[{MapIndexed[Text[#2[[1]],#,Background->White]&,v],FaceForm[Green,Red],Polygon[v[[#]]]&/@tri}]

squares={{1,3,12,11},{1,8,10,3},{8,13,14,10}};

Length[faces=Join[tri,squares]]

faces

PolyhedronData["Hebesphenomegacorona","F"]

Graphics3D[{MapIndexed[Text[#2[[1]],#,Background->White]&,v],FaceForm[Green,Red],Polygon[v[[#]]]&/@faces}]

p=Polyhedron[v/2,faces]

Graphics3D[p]

Properties

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

punsimplified=Polyhedron[]

AdjacentFaceIndices

AdjacentFaceIndices[p]

Length[%]

Centroid

PolyhedronData["Hebesphenomegacorona","Centroid"]

RegionCentroid[p//N]//Chop//N

RegionCentroid[punsimplified//N]//Chop//N

PolyhedronCentroid[p//N,PolyhedronData["Hebesphenomegacorona","Volume"]]//Chop

PolyhedronCentroid[punsimplified//N,PolyhedronData["Hebesphenomegacorona","Volume"]]//Chop

PolyhedronCentroid[punsimplified//N,PolyhedronData["Hebesphenomegacorona","Volume"]]//Chop

N[xbar1=RegionCentroid[p]]

N[xbar1u=RegionCentroid[punsimplified]]

N[xbar2=PolyhedronCentroid[p,PolyhedronData["Hebesphenomegacorona","Volume"],Identity,Identity]]

N[xbar2u=PolyhedronCentroid[punsimplified,PolyhedronData["Hebesphenomegacorona","Volume"],Identity,Identity]]

ByteCount/@{xbar1,xbar1u,xbar2,xbar2u}

N[expr1=xbar1[[3]]]

expr2=RootApproximant[N[expr1,1000],24]

N[expr1-expr2,3000]

Circumsphere

PolyhedronData[pname,"Circumsphere"]

Circumsphere[p]

(sphere=MyCircumsphere[p])//Timing

Convex

ConvexPolyhedronQ[p]

ConvexPolyhedronQ[punsimplified]

DihedralAngles

d=PolyhedronData["Hebesphenomegacorona","DihedralAngles"]

Union[%]

dih=DihedralAngles[p];

N[d-dih,2000]//Abs//Max

dih2=DihedralAngles[punsimplified];

dih2[[4]]//N

dih-dih0//N//Chop

d/.{d[[1]]->term1,d[[7]]->term7,d[[9]]->term9,d[[21]]->term21}

1

res=DihedralAngles[p][[1]]

res//N

roots=Cases[res,_Root,∞]//Union

nroots=Length[roots]

vars=Array[a,nroots]

FullSimplify[Cos[res]^2/.Thread[roots->vars],Element[vars,Reals]]

simp=FullSimplify[%/.Thread[vars->roots]]

N[%]

Cos[res]^2//N

term1=ArcCos[-Sqrt[simp]]//FullSimplify

N[term1-res,50]

4 (unsimplified)

res=DihedralAngles[punsimplified][[4]]

res//N

ByteCount[res]

roots=Cases[res,_Root,∞]//Union

rules={(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])->a}

Simplify[Cos[res]/.rules]

simp=FullSimplify[%/.Thread[vars->roots]]

N[%]

Cos[res]^2//N

term4=ArcCos[-Sqrt[simp]]//FullSimplify

N[term4-res,50]

4

<<MathWorld`PolyhedronOperations`

res=DihedralAngles[p][[4]]

res//N

d2=d/.ArcCos[r_]:>ArcCos[RootReduce[r]]

N[expr1=Cos[res]]

expr2=RootApproximant[N[expr1,200],24]

N[expr1-expr2,1000]

d2/. 2.74275`->ArcCos[(NumericalApproximation[Root, (RowDefault["-0.922", FEExpressionsNumericalApproximationElider]), Root[25+1786 #1-1499 ((#1)^2)-1896 ((#1)^3)-3942 ((#1)^4)-5076 ((#1)^5)+18954 ((#1)^6)+9720 ((#1)^7)-19683 ((#1)^8)-4374 ((#1)^9)+6561 ((#1)^10)&,1], ])]

Union[%]

roots=Cases[res,_Root,∞]//Union

nroots=Length[roots]

vars=Array[a,nroots]

FullSimplify[Cos[res]^2/.Thread[roots->vars],Element[vars,Reals]]

simp=FullSimplify[%/.Thread[vars->roots]]

N[%]

Cos[res]^2//N

term4=ArcCos[-Sqrt[simp]]//FullSimplify

N[term4-res,50]

7

res=DihedralAngles[p][[7]]

res//N

roots=Cases[res,_Root,∞]//Union

nroots=Length[roots]

vars=Array[a,nroots]

FullSimplify[Cos[res]^2/.Thread[roots->vars],Element[vars,Reals]]

simp=FullSimplify[%/.Thread[vars->roots]]

N[%]

Cos[res]^2//N

term7=ArcCos[-Sqrt[simp]]//FullSimplify

N[term7-res,50]

9

res=DihedralAngles[p][[9]]

N[res]

rules={(NumericalApproximation[Root, (RowDefault["0.836", FEExpressionsNumericalApproximationElider]), Root[67-26384 #1+99216 ((#1)^2)-9920 ((#1)^3)-426592 ((#1)^4)+643840 ((#1)^5)+70912 ((#1)^6)-1201152 ((#1)^7)+1514240 ((#1)^8)-860160 ((#1)^9)+196608 ((#1)^10)&,3], ])->a,(NumericalApproximation[Root, (RowDefault["-0.836", FEExpressionsNumericalApproximationElider]), Root[67+26384 #1+99216 ((#1)^2)+9920 ((#1)^3)-426592 ((#1)^4)-643840 ((#1)^5)+70912 ((#1)^6)+1201152 ((#1)^7)+1514240 ((#1)^8)+860160 ((#1)^9)+196608 ((#1)^10)&,2], ])->-a,(NumericalApproximation[Root, (RowDefault["-0.838", FEExpressionsNumericalApproximationElider]), Root[529-15032 ((#1)^2)+105616 ((#1)^4)-303984 ((#1)^6)+415496 ((#1)^8)-1085888 ((#1)^10)+5304000 ((#1)^12)-13361088 ((#1)^14)+17041168 ((#1)^16)-10916736 ((#1)^18)+2822400 ((#1)^20)&,2], ])->b,(NumericalApproximation[Root, (RowDefault["-0.611", FEExpressionsNumericalApproximationElider]), Root[1600-309888 ((#1)^2)+10362496 ((#1)^4)-150640320 ((#1)^6)+1108772352 ((#1)^8)-4874307760 ((#1)^10)+13891942264 ((#1)^12)-26107499520 ((#1)^14)+31258427913 ((#1)^16)-21590613504 ((#1)^18)+6502809600 ((#1)^20)&,3], ])->c};

FullSimplify[res/.rules,(a|b|c)∈Reals]

FullSimplify[ArcCos[1-((2 (a^2))/((a^2)+(((b-c))^2)))]/.Reverse/@rules]//RootReduce

term9=ArcCos[(NumericalApproximation[Root, (RowDefault["-0.862", FEExpressionsNumericalApproximationElider]), Root[1447696345-6491462808 #1+9879936732 ((#1)^2)+7918767576 ((#1)^3)-45958231818 ((#1)^4)-5635047528 ((#1)^5)+34091069724 ((#1)^6)+3069060840 ((#1)^7)+1154060217 ((#1)^8)-18895680 ((#1)^9)+544195584 ((#1)^10)&,2], ])]

N[ArcCos[(NumericalApproximation[Root, (RowDefault["-0.862", FEExpressionsNumericalApproximationElider]), Root[1447696345-6491462808 #1+9879936732 ((#1)^2)+7918767576 ((#1)^3)-45958231818 ((#1)^4)-5635047528 ((#1)^5)+34091069724 ((#1)^6)+3069060840 ((#1)^7)+1154060217 ((#1)^8)-18895680 ((#1)^9)+544195584 ((#1)^10)&,2], ])]-res,50]

21

res=DihedralAngles[p][[21]]

res//N

roots=Cases[res,_Root,∞]//Union

roots[[1]]+roots[[2]]//FullSimplify

nroots=Length[roots]-1

vars=Array[a,nroots]

rules=Thread[{(NumericalApproximation[Root, (RowDefault["0.717", FEExpressionsNumericalApproximationElider]), Root[-95+76 #1+756 ((#1)^2)-1984 ((#1)^3)+376 ((#1)^4)+8192 ((#1)^5)-13088 ((#1)^6)-5632 ((#1)^7)+29456 ((#1)^8)-24768 ((#1)^9)+6720 ((#1)^10)&,3], ]),(NumericalApproximation[Root, (RowDefault["-0.717", FEExpressionsNumericalApproximationElider]), Root[-95-76 #1+756 ((#1)^2)+1984 ((#1)^3)+376 ((#1)^4)-8192 ((#1)^5)-13088 ((#1)^6)+5632 ((#1)^7)+29456 ((#1)^8)+24768 ((#1)^9)+6720 ((#1)^10)&,2], ]),(NumericalApproximation[Root, (RowDefault["-1.10", FEExpressionsNumericalApproximationElider]), Root[345-4464 #1+12976 ((#1)^2)+10560 ((#1)^3)-47008 ((#1)^4)-68352 ((#1)^5)-3328 ((#1)^6)+207872 ((#1)^7)+387328 ((#1)^8)+270336 ((#1)^9)+65536 ((#1)^10)&,1], ]),(NumericalApproximation[Root, (RowDefault["-0.836", FEExpressionsNumericalApproximationElider]), Root[67+26384 #1+99216 ((#1)^2)+9920 ((#1)^3)-426592 ((#1)^4)-643840 ((#1)^5)+70912 ((#1)^6)+1201152 ((#1)^7)+1514240 ((#1)^8)+860160 ((#1)^9)+196608 ((#1)^10)&,2], ]),(NumericalApproximation[Root, (RowDefault["-0.611", FEExpressionsNumericalApproximationElider]), Root[1600-309888 ((#1)^2)+10362496 ((#1)^4)-150640320 ((#1)^6)+1108772352 ((#1)^8)-4874307760 ((#1)^10)+13891942264 ((#1)^12)-26107499520 ((#1)^14)+31258427913 ((#1)^16)-21590613504 ((#1)^18)+6502809600 ((#1)^20)&,3], ]),(NumericalApproximation[Root, (RowDefault["0.353", FEExpressionsNumericalApproximationElider]), Root[6561-347328 ((#1)^2)+6973168 ((#1)^4)-81591424 ((#1)^6)+690379360 ((#1)^8)-4494628864 ((#1)^10)+21900300032 ((#1)^12)-78744037376 ((#1)^14)+200126890240 ((#1)^16)-298144628736 ((#1)^18)+184968806400 ((#1)^20)&,6], ])}->{a[1],-a[1],a[2],a[3],a[4],a[5]}]

FullSimplify[Cos[res]/.rules]/.Abs[x_]:>x

FullSimplify[Cos[res]/.rules,Element[vars,Reals]]/.Abs[x_]:>x

((-4 ((a[1])^2) ((((a[2]-a[3]))^2)+(((a[4]-a[5]))^2))+((((1+2 a[2]) a[4]-(1+2 a[3]) a[5]))^2))/(4 ((a[1])^2) ((((a[2]-a[3]))^2)+(((a[4]-a[5]))^2))+((((1+2 a[2]) a[4]-(1+2 a[3]) a[5]))^2)))/.Reverse/@rules//N

((-4 ((a[1])^2)(( ((((a[2]-a[3]))^2)+(((a[4]-a[5]))^2)))/((((1+2 a[2]) a[4]-(1+2 a[3]) a[5]))^2))+1)/(4 ((a[1])^2)(( ((((a[2]-a[3]))^2)+(((a[4]-a[5]))^2)))/((((1+2 a[2]) a[4]-(1+2 a[3]) a[5]))^2))+1))

simp=FullSimplify[(( ((((a[2]-a[3]))^2)+(((a[4]-a[5]))^2)))/((((1+2 a[2]) a[4]-(1+2 a[3]) a[5]))^2))/.Reverse/@rules]

FullSimplify[((-4 ((a[1])^2)(NumericalApproximation[Root, (RowDefault["1.06", FEExpressionsNumericalApproximationElider]), Root[11025-118284 #1+340912 ((#1)^2)+544680 ((#1)^3)-4265082 ((#1)^4)+3515968 ((#1)^5)+17272772 ((#1)^6)-45521544 ((#1)^7)+42016081 ((#1)^8)-13986260 ((#1)^9)+206116 ((#1)^10)&,3], ])+1)/(4 ((a[1])^2)(NumericalApproximation[Root, (RowDefault["1.06", FEExpressionsNumericalApproximationElider]), Root[11025-118284 #1+340912 ((#1)^2)+544680 ((#1)^3)-4265082 ((#1)^4)+3515968 ((#1)^5)+17272772 ((#1)^6)-45521544 ((#1)^7)+42016081 ((#1)^8)-13986260 ((#1)^9)+206116 ((#1)^10)&,3], ])+1))/.Reverse/@rules]

N[%]

Cos[res]//N

term21=ArcCos[-1+(2/(1+(NumericalApproximation[Root, (RowDefault["2.18", FEExpressionsNumericalApproximationElider]), Root[2310400-5583872 #1-38783232 ((#1)^2)-61628928 ((#1)^3)+357824352 ((#1)^4)-620528256 ((#1)^5)-97678032 ((#1)^6)+310293984 ((#1)^7)+9790929 ((#1)^8)-39897620 ((#1)^9)+206116 ((#1)^10)&,3], ])))]//FullSimplify

RootReduce[-1+(2/(1+(NumericalApproximation[Root, (RowDefault["2.18", FEExpressionsNumericalApproximationElider]), Root[2310400-5583872 #1-38783232 ((#1)^2)-61628928 ((#1)^3)+357824352 ((#1)^4)-620528256 ((#1)^5)-97678032 ((#1)^6)+310293984 ((#1)^7)+9790929 ((#1)^8)-39897620 ((#1)^9)+206116 ((#1)^10)&,3], ])))]

term21=ArcCos[(NumericalApproximation[Root, (RowDefault["-0.370", FEExpressionsNumericalApproximationElider]), Root[-183674159-572579022 #1+1576540485 ((#1)^2)+842344920 ((#1)^3)-10142162766 ((#1)^4)+7356384684 ((#1)^5)+13051469250 ((#1)^6)-8402541480 ((#1)^7)-3664731843 ((#1)^8)+1853784306 ((#1)^9)+651015225 ((#1)^10)&,2], ])]

N[term21-res,50]

EdgeLengths

PolyhedronData["Hebesphenomegacorona","EdgeLengths"]

e=PolyhedronData["Hebesphenomegacorona","Lines"]/.Line[{l__}]:>SquaredEuclideanDistance[l]

FullSimplify/@e//Union

SimplifyNumericallyDistinct[e,Debug->True]

v=PolyhedronCoordinates[punsimplified];

e=SquaredEuclideanDistance@@v[[#]]&/@MeshCells[punsimplified,1][[All,1]]

SimplifyNumericallyDistinct[e,Debug->True]

FullSimplify/@e

GeneralizedDiameter

PolyhedronData["Hebesphenomegacorona","GeneralizedDiameter"]

N[%]

diam=GeneralizedDiameter[p]

N[%]

InertiaTensor

i=PolyhedronData["Hebesphenomegacorona","InertiaTensor"]

<<MathWorld`PolyhedronOperations`

(i1=MomentOfInertia[p,{0,0,0}]/PolyhedronData["Hebesphenomegacorona","Volume"])//N//Chop

(i2=PolyhedronInertiaTensor[p,PolyhedronData["Hebesphenomegacorona","Volume"],Identity,Identity])//N//Chop

N[i-i1,1000]

N[i-i2,1000]

<<MathWorld`PolyhedronOperations`

ix=RootApproximant[N[i[[1,1]],500],16]

N[i[[1,1]]-ix,2000]

iy=RootApproximant[N[i[[2,2]],500],16]

N[i[[2,2]]-iy,2000]

iz=RootApproximant[N[i[[3,3]],600],10]

N[i[[3,3]]-iz,2000]

DiagonalMatrix[{ix,iy,iz}]

N[i-%,2000]

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

sphere=MyInsphere[p]

MeanCylindricalRadius

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

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

N[i]

MeanSquareCylindricalRadius

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

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

N[i]

(r=RootApproximant[N[i,1000],32])//Timing

N[i-r,10^4]

MeanSphericalRadius

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

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

N[i]

MeanSquareSphericalRadius

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

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

N[i]

(r=RootApproximant[N[i,1000],32])//Timing

N[i-r,10^4]

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p])//Timing

RegionFunction

reg=PolyhedronData["Hebesphenomegacorona","RegionFunction"]

bounds=RegionBounds[ImplicitRegion[reg[x,y,z],{x,y,z}]]

ranges=Flatten/@Transpose[{{x,y,z},{-1,1}.1+#&/@bounds}]

RegionPlot3D[Evaluate[reg[x,y,z]],Evaluate[Sequence@@ranges],BoxRatios->Automatic,PlotPoints->40]

PolyhedronInequalities[p,{#1,#2,#3}]/.Times[r__Root]:>RootReduce[Times[r]]

SkeletonCoordinates

g=Graph[Range[Length[v]],UndirectedEdge@@@e]

r=RecognizeGraph[g]

GraphData[r,"Graph","All"]

ToCommonEdges[GraphData[r],g]

SurfaceArea

PolyhedronData["Hebesphenomegacorona","SurfaceArea"]

(s1=SurfaceArea[p])//Timing

(s2=Total[Area/@PolyhedronData[pname,"Polygons"]])//Timing

VertexCoordinates

PolyhedronData["Hebesphenomegacorona","VertexCoordinates"]/.r_Root?Negative:>-RootReduce[-r]

N[PolyhedronData["Hebesphenomegacorona","VertexCoordinates"]-%,1000]

Not RootReduce'd

z-component of last expression is wrong.  Here is the correct one:

orig=(((-1+2 a) (Sqrt[1-(a^2)]))/(-1+a))-(1/16) √(((32 a (8-8 a+((32 (-1+a) (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))-3 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])))/(((-1+a))^2))+32 (-32+((8 (Sqrt[2-4 a]))/(((1-a))^(3/2)))-((16 (Sqrt[2-4 a]))/(Sqrt[1-a]))-(4/(((-1+a))^2))-(35/(-1+a))+(3/(1+a))+((8 (Sqrt[3-4 (a^2)]))/(Sqrt[1-(a^2)]))+((8 (Sqrt[-(1+a) (-3+4 (a^2))]))/(((1-a))^(3/2)))+3 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])+(((Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(((-1+a))^2))));

Simplify by hand:

-(1/(4 (Sqrt[1-(a^2)])))(-4+4 a+8 (a^2)+(Sqrt[2]) √(10+(8/(-1+a))+40 a+40 (a^2)+3 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])(1- (a^2) )+(5 (Sqrt[2])+3(Sqrt[2])a) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])+((4 (Sqrt[2]) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))]))/(-1+a))- (Sqrt[2-6 a+4 (a^2)])(24)-((16 (Sqrt[2-6 a+4 (a^2)]))/(-1+a))-16 a (Sqrt[2-6 a+4 (a^2)])+32 (Sqrt[3-7 (a^2)+4 (a^4)])+((16 (Sqrt[3-7 (a^2)+4 (a^4)]))/(-1+a))))/.a->(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])//N

-(1/(4 (Sqrt[1-(a^2)])))(-4+4 a+8 (a^2)+(Sqrt[2]) √(10+(8/(-1+a))+40 a+40 (a^2)+3 (Sqrt[2]) (Sqrt[(((-1+2 a) (-3+4 (a^2)))/(1+a))])(1- (a^2) )+(5 (Sqrt[2])+3(Sqrt[2])a+((4 (Sqrt[2]))/(a-1))) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])- (Sqrt[2-6 a+4 (a^2)])(24+(16/(a-1))+16a)+32 (Sqrt[3-7 (a^2)+4 (a^4)])+((16 (Sqrt[3-7 (a^2)+4 (a^4)]))/(-1+a))))/.a->(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])//N

-(1/(4 (Sqrt[1-(a^2)])))(-4+4 a+8 (a^2)+(Sqrt[2]) √(10+(8/(-1+a))+40 a+40 (a^2)+(5 (Sqrt[2])+3(Sqrt[2])a+((4 (Sqrt[2]))/(a-1))+((3 (Sqrt[2]))/(1+a))(1- (a^2) )) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])- (Sqrt[2-6 a+4 (a^2)])(24+(16/(a-1))+16a)+32 (Sqrt[3-7 (a^2)+4 (a^4)])+((16 (Sqrt[3-7 (a^2)+4 (a^4)]))/(-1+a))))/.a->(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])//N

-(1/(4 (Sqrt[1-(a^2)])))(-4+4 a+8 (a^2)+(Sqrt[2]) √(10+(8/(-1+a))+40 a+40 (a^2)+(5 (Sqrt[2])+3(Sqrt[2])a+((4 (Sqrt[2]))/(a-1))+((3 (Sqrt[2]))/(1+a))(1- (a^2) )) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])- (Sqrt[2-6 a+4 (a^2)])(24+(16/(a-1))+16a)+(Sqrt[3-7 (a^2)+4 (a^4)])(32+(16/(-1+a)))))/.a->(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])//N

FullSimplify[-(1/(4 (Sqrt[1-(a^2)])))(-4+4 a+8 (a^2)+(Sqrt[2]) √(10+(8/(-1+a))+40 a+40 (a^2)+(5 (Sqrt[2])+3(Sqrt[2])a+((4 (Sqrt[2]))/(a-1))+((3 (Sqrt[2]))/(1+a))(1- (a^2) )) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])- (Sqrt[2-6 a+4 (a^2)])(24+(16/(a-1))+16a)+(Sqrt[3-7 (a^2)+4 (a^4)])(32+(16/(-1+a))))),0<a<1/4]

-(1/(4 (Sqrt[1-(a^2)])))(-4+4 a+8 (a^2)+(Sqrt[2])((-1)/(a-1)) √((10+(8/(-1+a))+40 a+40 (a^2))(((a-1))^2)+4 (Sqrt[2]) (-1+2 a)(a-1) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])-8 (-1+a+2 (a^2)) (a-1)(Sqrt[2-6 a+4 (a^2)])+16 (2+(1/(-1+a)))(((a-1))^2) (Sqrt[3-7 (a^2)+4 (a^4)])))/.a->(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])//N

FullSimplify[-(1/(4 (Sqrt[1-(a^2)])))(-4+4 a+8 (a^2)+(Sqrt[2])((-1)/(a-1)) √((10+(8/(-1+a))+40 a+40 (a^2))(((a-1))^2)+4 (Sqrt[2]) (-1+2 a)(a-1) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])-8 (-1+a+2 (a^2)) (a-1)(Sqrt[2-6 a+4 (a^2)])+16 (2+(1/(-1+a)))(((a-1))^2) (Sqrt[3-7 (a^2)+4 (a^4)]))),a>0]

FullSimplify[((-2+4 a+2 (a^2)-4 (a^3))/(2 (-1+a) (Sqrt[1-(a^2)])))]

-(((1+a) (2 a-1))/(Sqrt[1-(a^2)]))+((-(Sqrt[a-1]))/(2 (-1+a) (Sqrt[1-(a^2)])))√(-1-15 a+20 (a^3)+2 (Sqrt[2]) (-1+2 a) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])-4 (-1+a+2 (a^2)) (Sqrt[2-6 a+4 (a^2)])+8 (-1+2 a) (Sqrt[3-7 (a^2)+4 (a^4)]))/.a->(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])//N

(new=-(((1+a) (2 a-1))/(Sqrt[1-(a^2)]))+(1/(2 (-1+a) (Sqrt[-1-a])))√(-1-15 a+20 (a^3)+2 (Sqrt[2])(-1+2 a) (Sqrt[(1+a) (-1+2 a) (-3+4 (a^2))])-4 (1+a) (-1+2 a) (Sqrt[2 (-1+a) (-1+2 a)])+8 (-1+2 a) (Sqrt[(-1+a) (1+a) (-3+4 (a^2))])))/.a->(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])//N

(new=-(((1+a) (2 a-1))/(Sqrt[1-(a^2)]))+(1/(2 (-1+a) (Sqrt[-1-a])))√(-1-15 a+20 (a^3)+2(-1+2 a) ((Sqrt[2(1+a) (-1+2 a) (-3+4 (a^2))])-2 (1+a) (Sqrt[2 (-1+a) (-1+2 a)])+4 (Sqrt[(-1+a) (1+a) (-3+4 (a^2))]))))/.a->(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])//N

N[new-orig/.a->(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ]),20]

v=
Union[Flatten[Outer[Times,
{{1,1,1},{-1,1,1},{1,-1,1},{-1,-1,1}},
{{1,1,2(Sqrt[1-(a^2)])},{1+2a,1,0},{0,1+(Sqrt[2])(Sqrt[((2a-1)/(a-1))]),-((2(a^2)+a-1)/(Sqrt[1-(a^2)]))},{1,0,-(Sqrt[3-4(a^2)])},
{0,(((Sqrt[2(3-4(a^2))(1-2a)])+(Sqrt[1+a]))/(2(1-a)(Sqrt[1+a]))),new}},
1],1]/.a->(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])]/2;

N[v-PolyhedronCoordinates[p],100]//Chop

faces={{1,4,8},{1,11,4},{2,6,11},{2,7,9},{2,9,6},{2,11,12},{2,12,7},{3,5,12},{3,10,5},{4,6,13},{4,11,6},{4,13,8},{5,7,12},{5,10,14},{5,14,7},{6,9,13},{7,14,9},{9,14,13},{1,3,12,11},{1,8,10,3},{8,13,14,10}};

Graphics3D[punsimplified=Polyhedron[v,faces]]

punsimplified

Volume

PolyhedronData["Hebesphenomegacorona","Volume"]

N[PolyhedronData["Hebesphenomegacorona","Volume"]-Volume[p],2000]

expr1=PolyhedronData["Hebesphenomegacorona","Volume"]

N[expr1]

expr2=RootApproximant[N[expr1,500],20]

N[expr1-expr2,2000]

Volume[p]//FullSimplify

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

Simplify by replacing roots with variables

ByteCount[v=Volume[punsimplified]]

N[v,10]

v2=PolyhedronVolume[punsimplified,Method->"GaussLaw"];//Timing

N[v2]

ByteCount[v2]

v3=PolyhedronVolume[punsimplified];//Timing

v3//N

ByteCount[v3]

roots=Cases[v,_Root,∞]//Union

expr=Simplify[v/.(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])->a,0<a<1/4]

expr=(53188547/522208038)-(53188547/(522208038 (-1+a)))+((53188547 a)/522208038)+((53188547 (a^2))/(522208038 (-1+a)))+(1/24) (Sqrt[3-4 (a^2)])+((Sqrt[3-4 (a^2)])/(24 (-1+a)))+((a (Sqrt[3-4 (a^2)]))/(24-24 a))+(((a^2) (Sqrt[3-4 (a^2)]))/(12-12 a))+(1/6) (1+2 a) (Sqrt[3-4 (a^2)])-(((1+a) (-1+2 a))/(12 (Sqrt[2]) (Sqrt[1-a-2 (a^2)])))+((Sqrt[1-a-2 (a^2)])/(6 (Sqrt[2])))+((Sqrt[1-a-2 (a^2)])/((Sqrt[2]) (12-12 a)))+((a (Sqrt[1-a-2 (a^2)]))/(12 (Sqrt[2]) (-1+a)))+(((1+2 a) (Sqrt[1-a-2 (a^2)]))/(3 (Sqrt[2])))+((Sqrt[1-(a^2)])/2)+(1/2) (1+2 a) (Sqrt[1-(a^2)])+(((Sqrt[((1-2 a)/(2+2 a))]) (-3+4 (a^2)))/(12 (-1+a)))+((a (Sqrt[((1-2 a)/(2+2 a))]) (-3+4 (a^2)))/(12 (-1+a)))+(((Sqrt[((-1+2 a)/(-1+a))]) (1+2 a) ((53188547/87034673)-((-1+a+2 (a^2))/(2 (Sqrt[1-(a^2)])))))/(6 (Sqrt[2])))+(((Sqrt[((3-4 (a^2))/(1+a))]) ((Sqrt[1+a])+(Sqrt[2]) (Sqrt[(-1+2 a) (-3+4 (a^2))])))/(48 (1-a)))+(((-1+4 (a^2)) (-87034673+87034673 a+174069346 (a^2)-106377094 (Sqrt[1-(a^2)])) (-3 (Sqrt[2])+4 (Sqrt[2]) (a^2)+(Sqrt[3-3 a-10 (a^2)+4 (a^3)+8 (a^4)])))/(4177664304 (-1+(a^2)) (Sqrt[3-9 a+2 (a^2)+12 (a^3)-8 (a^4)])))-(1/48) (Sqrt[3-4 (a^2)]) (2+((1+(Sqrt[2]) (Sqrt[((3-6 a-4 (a^2)+8 (a^3))/(1+a))]))/(-1+a)))-(1/24) a (Sqrt[3-4 (a^2)]) (2+((1+(Sqrt[2]) (Sqrt[((3-6 a-4 (a^2)+8 (a^3))/(1+a))]))/(-1+a)))+((53188547 (1+2 a) (2+2 (Sqrt[((-2+4 a)/(-1+a))])+((1+(Sqrt[2]) (Sqrt[((3-6 a-4 (a^2)+8 (a^3))/(1+a))]))/(-1+a))))/2088832152)+(((-1-a+4 (a^2)+4 (a^3)) (2+2 (Sqrt[((-2+4 a)/(-1+a))])+((1+(Sqrt[2]) (Sqrt[((3-6 a-4 (a^2)+8 (a^3))/(1+a))]))/(-1+a))))/(48 (Sqrt[1-(a^2)])));

Monitor[Replace[TimeConstrained[RootReduce[(f=#)/.a->(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])],5],$Aborted->f]&/@expr,f]

N[%/.a->(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])]

RootReduce[(NumericalApproximation[Root, (RowDefault["0.130", FEExpressionsNumericalApproximationElider]), Root[6342336919091022052148778142753084033828482833671760764536203631604571450626715-704533870134529417303674931983088810977542788374567376502591285761341723987945816 #1+34160714826833974653063791863412312129410334812056157071745729926979051195823689476 ((#1)^2)-949695439796058475895732818682012551132347948406002323606183215721611548403189528056 ((#1)^3)+16714722361274053282026475199438524386022036029021889045527988767328795783438281882282 ((#1)^4)-193911390580735491695585727964090789431413102396547970699741179145507388484566809150632 ((#1)^5)+1495752012818430141412722777602460999426931902740665773097553588766785185978492432733748 ((#1)^6)-7545712391070589560745265107469321223408591856171227434299100820142337969886313921636872 ((#1)^7)+23757124953407514272287214727416932385461877696585632023697478096945498199086067327045939 ((#1)^8)-42139432060209654167826887939543175457290459558009100645395718797925171876171558212266496 ((#1)^9)+32173198814598293101832844830830072522158995144745021925221375566036751739245175719002112 ((#1)^10)&,3], ])+(NumericalApproximation[Root, (RowDefault["0.0388", FEExpressionsNumericalApproximationElider]), Root[-17214914494675631284403826387472656663248739119966207789455409857212408223129655+811281426215518722955746891374465903549897756310107888093892995725181379137634576 #1-25502630124686008007571059887803975988889938039339691149522123245298831470548450640 ((#1)^2)+1371979206350773696592884346497821057435523431152955863874579978168824243765517888 ((#1)^3)+3512345843997297777144616529839579448479129692113444673701298356946987085260482373984 ((#1)^4)+226016655306145878484108134277030891646540251291833888169725110403800284062347480278272 ((#1)^5)+2661164550558735187063992457924493504563097682410858998293118936745186685791512560163584 ((#1)^6)-66813372277717853311132067791715191592467898370750565334896892358133588339471786234856448 ((#1)^7)-888638983509213083615199781806580673256913888961317806422390349515917902523163117312161536 ((#1)^8)+1614713401519771146824530665931268020924801862237393679165491358458527606667006390986678272 ((#1)^9)+30403672879795386981232038365134418533440250411784045719334199909904730393586691054456995840 ((#1)^10)&,2], ])+(NumericalApproximation[Root, (RowDefault["0.0221", FEExpressionsNumericalApproximationElider]), Root[-8516852434207943898599787791696998559712534090930650169520044876726138805127303+878888211733478616535392465655671395512389236002616878768384078702279827399104124 #1-19546193910632632164706860283871677487375233332891174244051545144855748229824463196 ((#1)^2)-310067300635274855429991862308507558980428295440568025235655075066154279091007042688 ((#1)^3)+3583064216695229947154239681447087759522333645645930136930183558764711657446935039232 ((#1)^4)+226413407773037884122172689223099406962479175079941261876284019316153356532907481706496 ((#1)^5)+992012547756464348825025732058703604530962005055816674650374660977468957279685579880448 ((#1)^6)-63231125071070511815152653357547672256943479473177409666301962620939247884712852456570880 ((#1)^7)-400520396095632932631720882660759260468969445895345619542829855237019988108821912413798400 ((#1)^8)+5426616728395212621291573379568097093153672011902839031350692419294412413273592254640160768 ((#1)^9)+40538230506393849308309384486845891377920333882378727625778933213206307191448921405942661120 ((#1)^10)&,3], ])+(NumericalApproximation[Root, (RowDefault["-6.12", FEExpressionsNumericalApproximationElider, " × ", (Superscript["10", "-3"])]), Root[400292064407773363234190026209758932306489102273740557967442109206128523840983241+205723890079282589852673080139061932805442494205373673938756655442311783404480178272 #1+36390366691999901859088273915740189606415256573022500443899475083842465197651529999808 ((#1)^2)+2980674217048484483701118586602990168836742333023318120678624487450984747253363170323968 ((#1)^3)+165508848416921149008723870128716938733904364747140780134249912176781985471590966987316736 ((#1)^4)+«1»+240999299825504656838466158146696372098581232218583545578652676596059591158835085424504127488 ((#1)^6)+4704305957884319398320910096358071586962423727220630898333339654921901935303057920655709306880 ((#1)^7)+51151104623210995795418610995226648976901442976102480578262264768386754283558501637645179879424 ((#1)^8)+282342621460963299041140421707587560202538738862078008231499817338342743105084182197597846372352 ((#1)^9)+664178368616756827067340955432483084335846750328893073420762041765172137024699128314964559790080 ((#1)^10)&,3], ])+(NumericalApproximation[Root, (RowDefault["0.488", FEExpressionsNumericalApproximationElider]), Root[331776-15094656 ((#1)^2)+354890305 ((#1)^4)-6204273024 ((#1)^6)+82738662400 ((#1)^8)-779173789696 ((#1)^10)+4925123067904 ((#1)^12)-20263227883520 ((#1)^14)+51948367642624 ((#1)^16)-75383118495744 ((#1)^18)+47352014438400 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["0.700", FEExpressionsNumericalApproximationElider]), Root[2994278400-129615067008 ((#1)^2)+1773310978609 ((#1)^4)-17259185925248 ((#1)^6)+27142734408448 ((#1)^8)+95952869883904 ((#1)^10)+141206521339904 ((#1)^12)-2402132177715200 ((#1)^14)+5835588946100224 ((#1)^16)-5706859344297984 ((#1)^18)+2039281090560000 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["0.281", FEExpressionsNumericalApproximationElider]), Root[81225-11715438 ((#1)^2)+552938617 ((#1)^4)-16258273200 ((#1)^6)+149065482600 ((#1)^8)+991789663392 ((#1)^10)+12326914032984 ((#1)^12)-501184409553216 ((#1)^14)+210191471356176 ((#1)^16)+17258309290755936 ((#1)^18)+83717693468010000 ((#1)^20)&,8], ])+(NumericalApproximation[Root, (RowDefault["0.401", FEExpressionsNumericalApproximationElider]), Root[76387600-29921738688 ((#1)^2)+30174332544 ((#1)^4)-7933759767936 ((#1)^6)+134570970441288 ((#1)^8)+348775927637040 ((#1)^10)-7372137605206320 ((#1)^12)-76296336756447168 ((#1)^14)+1154231116392942513 ((#1)^16)-4804087726182096504 ((#1)^18)+6781133170908810000 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["0.0978", FEExpressionsNumericalApproximationElider]), Root[9-4152 ((#1)^2)+776080 ((#1)^4)-108334080 ((#1)^6)+11094589440 ((#1)^8)+200084815872 ((#1)^10)-108716572016640 ((#1)^12)-6579091452985344 ((#1)^14)+2010331000269176832 ((#1)^16)-114659054558071750656 ((#1)^18)+2038348954317776486400 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["0.144", FEExpressionsNumericalApproximationElider]), Root[1146228736-1672782225408 ((#1)^2)+21388194312192 ((#1)^4)+22257731542265856 ((#1)^6)-358666473605010048 ((#1)^8)-83659011675514227072 ((#1)^10)+1968493422356724484224 ((#1)^12)+71396636025218034718656 ((#1)^14)-1590644556161758897713351 ((#1)^16)-17480521001922851059531776 ((#1)^18)+380404835250600718997913600 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["0.0625", FEExpressionsNumericalApproximationElider]), Root[9-23232 ((#1)^2)+74795008 ((#1)^4)-240261414912 ((#1)^6)+252136518156288 ((#1)^8)-129097931437375488 ((#1)^10)+35673014856132329472 ((#1)^12)-4930275958466965143552 ((#1)^14)+221306526515667639730176 ((#1)^16)+4728829883688298532044800 ((#1)^18)+794071845499378503449051136 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["0.0489", FEExpressionsNumericalApproximationElider]), Root[9-16608 ((#1)^2)+12417280 ((#1)^4)-6933381120 ((#1)^6)+2840214896640 ((#1)^8)+204886851452928 ((#1)^10)-445303078980157440 ((#1)^12)-107791834365711876096 ((#1)^14)+131749052433640772861952 ((#1)^16)-30057183198071161003966464 ((#1)^18)+2137363793122716797003366400 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["-0.0892", FEExpressionsNumericalApproximationElider]), Root[529-4188672 ((#1)^2)-7775419392 ((#1)^4)-4549929698304 ((#1)^6)+6225128276871168 ((#1)^8)-3050161153475346432 ((#1)^10)+767553755445403582464 ((#1)^12)-104061276675697303093248 ((#1)^14)+7591331583900400858693632 ((#1)^16)-278251744411842779384119296 ((#1)^18)+4019988717840603673710821376 ((#1)^20)&,2], ])+(NumericalApproximation[Root, (RowDefault["0.0699", FEExpressionsNumericalApproximationElider]), Root[529-2164608 ((#1)^2)+2190053376 ((#1)^4)-907691360256 ((#1)^6)+178655674761216 ((#1)^8)-67235322802470912 ((#1)^10)+47290996777549824000 ((#1)^12)-17154547589662252204032 ((#1)^14)+3150643716385809931173888 ((#1)^16)-290639900867438053995577344 ((#1)^18)+10820404202683753784829542400 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["0.0469", FEExpressionsNumericalApproximationElider]), Root[529-2203530048 ((#1)^2)+1100821155840 ((#1)^4)+323454853545984 ((#1)^6)-122787695284420608 ((#1)^8)-33629197658785579008 ((#1)^10)+3112245204444896034816 ((#1)^12)+1276907340175458869182464 ((#1)^14)+40768119464019660493553664 ((#1)^16)-7952703167373138058757013504 ((#1)^18)+173126467242940060557272678400 ((#1)^20)&,6], ])+(NumericalApproximation[Root, (RowDefault["0.0313", FEExpressionsNumericalApproximationElider]), Root[618168769-2915863001064 ((#1)^2)+4559214219633072 ((#1)^4)-2414341895296345056 ((#1)^6)-139209907767204667752 ((#1)^8)+234870719307287933648160 ((#1)^10)+51335645884580495253691008 ((#1)^12)+3960066273290755697228137344 ((#1)^14)+252671544346903133235628833936 ((#1)^16)+6827727801467759921517922615296 ((#1)^18)+67103412938205966831231959040000 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["0.0203", FEExpressionsNumericalApproximationElider]), Root[1168561-1747134622656 ((#1)^2)+10004583206670336 ((#1)^4)-14642830517378973696 ((#1)^6)+1065123761771658903552 ((#1)^8)+1163034652061800508424192 ((#1)^10)+172231920498655721164898304 ((#1)^12)+11321466262105704354268839936 ((#1)^14)+411531120017413604256132366336 ((#1)^16)+9483469994538080344793054969856 ((#1)^18)+119294956334588385477745704960000 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["-0.0135", FEExpressionsNumericalApproximationElider]), Root[19881-393026304 ((#1)^2)+3027694354432 ((#1)^4)-12831056927391744 ((#1)^6)+38047019613441490944 ((#1)^8)-83802148319517248323584 ((#1)^10)+118173567784064252601434112 ((#1)^12)-103578046322106564059653472256 ((#1)^14)+64166295811112643970403464642560 ((#1)^16)+1933153482229710394603902332829696 ((#1)^18)+573744224444786147362458093119078400 ((#1)^20)&,3], ])+(NumericalApproximation[Root, (RowDefault["0.0193", FEExpressionsNumericalApproximationElider]), Root[1168561-23966977536 ((#1)^2)+124631120904192 ((#1)^4)-218748633234997248 ((#1)^6)+184227648586847354880 ((#1)^8)-404185059532726578708480 ((#1)^10)+648444160211456830646255616 ((#1)^12)-741694474580964151475237289984 ((#1)^14)+400317900477393246751003294826496 ((#1)^16)-58361985323701727413265987989929984 ((#1)^18)+2904580136251729871022444096415334400 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["8.39", FEExpressionsNumericalApproximationElider, " × ", (Superscript["10", "-3"])]), Root[2581351249-123448200966144 ((#1)^2)+1544089905429331968 ((#1)^4)-4071279168260313513984 ((#1)^6)-5404006463572569503563776 ((#1)^8)+5350048027199952098118598656 ((#1)^10)+6458624702247070414319212560384 ((#1)^12)+1703797397454061999888811635507200 ((#1)^14)+530396961400285052484762907126530048 ((#1)^16)-51455138967960308059232553443707060224 ((#1)^18)+2001437250135957614251402885186191360000 ((#1)^20)&,6], ])]

expr=FullSimplify[v/.(NumericalApproximation[Root, (RowDefault["0.217", FEExpressionsNumericalApproximationElider]), Root[-47+494 #1-1119 ((#1)^2)-1808 ((#1)^3)+2128 ((#1)^4)+13696 ((#1)^5)+6112 ((#1)^6)-39680 ((#1)^7)-25600 ((#1)^8)+35328 ((#1)^9)+26880 ((#1)^10)&,3], ])->a,0<a<1/4]

RootReduce'd coordinates

vol=PolyhedronData["Hebesphenomegacorona","Volume"]

(v1=Volume[p])//N

(v2=PolyhedronVolume[p])//N

(v3=PolyhedronVolume[p,Method->"GaussLaw"])//N

ByteCount/@{v1,v2,v3}

{v1simp,v2simp,v3simp}=Simplify[{v1,v2,v2}];

ByteCount/@{v1simp,v2simp,v3simp}

v1simp===v2simp===v3simp

v1simp

v3simp

vol=(1/6) (-(NumericalApproximation[Root, (RowDefault["0.717", FEExpressionsNumericalApproximationElider]), Root[-95+76 #1+756 ((#1)^2)-1984 ((#1)^3)+376 ((#1)^4)+8192 ((#1)^5)-13088 ((#1)^6)-5632 ((#1)^7)+29456 ((#1)^8)-24768 ((#1)^9)+6720 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["-0.838", FEExpressionsNumericalApproximationElider]), Root[529-15032 ((#1)^2)+105616 ((#1)^4)-303984 ((#1)^6)+415496 ((#1)^8)-1085888 ((#1)^10)+5304000 ((#1)^12)-13361088 ((#1)^14)+17041168 ((#1)^16)-10916736 ((#1)^18)+2822400 ((#1)^20)&,2], ])+(NumericalApproximation[Root, (RowDefault["-0.717", FEExpressionsNumericalApproximationElider]), Root[-95-76 #1+756 ((#1)^2)+1984 ((#1)^3)+376 ((#1)^4)-8192 ((#1)^5)-13088 ((#1)^6)+5632 ((#1)^7)+29456 ((#1)^8)+24768 ((#1)^9)+6720 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["-0.838", FEExpressionsNumericalApproximationElider]), Root[529-15032 ((#1)^2)+105616 ((#1)^4)-303984 ((#1)^6)+415496 ((#1)^8)-1085888 ((#1)^10)+5304000 ((#1)^12)-13361088 ((#1)^14)+17041168 ((#1)^16)-10916736 ((#1)^18)+2822400 ((#1)^20)&,2], ])-(NumericalApproximation[Root, (RowDefault["0.836", FEExpressionsNumericalApproximationElider]), Root[67-26384 #1+99216 ((#1)^2)-9920 ((#1)^3)-426592 ((#1)^4)+643840 ((#1)^5)+70912 ((#1)^6)-1201152 ((#1)^7)+1514240 ((#1)^8)-860160 ((#1)^9)+196608 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["-0.838", FEExpressionsNumericalApproximationElider]), Root[529-15032 ((#1)^2)+105616 ((#1)^4)-303984 ((#1)^6)+415496 ((#1)^8)-1085888 ((#1)^10)+5304000 ((#1)^12)-13361088 ((#1)^14)+17041168 ((#1)^16)-10916736 ((#1)^18)+2822400 ((#1)^20)&,2], ])-(NumericalApproximation[Root, (RowDefault["0.717", FEExpressionsNumericalApproximationElider]), Root[-95+76 #1+756 ((#1)^2)-1984 ((#1)^3)+376 ((#1)^4)+8192 ((#1)^5)-13088 ((#1)^6)-5632 ((#1)^7)+29456 ((#1)^8)-24768 ((#1)^9)+6720 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["0.836", FEExpressionsNumericalApproximationElider]), Root[67-26384 #1+99216 ((#1)^2)-9920 ((#1)^3)-426592 ((#1)^4)+643840 ((#1)^5)+70912 ((#1)^6)-1201152 ((#1)^7)+1514240 ((#1)^8)-860160 ((#1)^9)+196608 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["-0.838", FEExpressionsNumericalApproximationElider]), Root[529-15032 ((#1)^2)+105616 ((#1)^4)-303984 ((#1)^6)+415496 ((#1)^8)-1085888 ((#1)^10)+5304000 ((#1)^12)-13361088 ((#1)^14)+17041168 ((#1)^16)-10916736 ((#1)^18)+2822400 ((#1)^20)&,2], ])+(NumericalApproximation[Root, (RowDefault["-0.717", FEExpressionsNumericalApproximationElider]), Root[-95-76 #1+756 ((#1)^2)+1984 ((#1)^3)+376 ((#1)^4)-8192 ((#1)^5)-13088 ((#1)^6)+5632 ((#1)^7)+29456 ((#1)^8)+24768 ((#1)^9)+6720 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["0.836", FEExpressionsNumericalApproximationElider]), Root[67-26384 #1+99216 ((#1)^2)-9920 ((#1)^3)-426592 ((#1)^4)+643840 ((#1)^5)+70912 ((#1)^6)-1201152 ((#1)^7)+1514240 ((#1)^8)-860160 ((#1)^9)+196608 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["-0.838", FEExpressionsNumericalApproximationElider]), Root[529-15032 ((#1)^2)+105616 ((#1)^4)-303984 ((#1)^6)+415496 ((#1)^8)-1085888 ((#1)^10)+5304000 ((#1)^12)-13361088 ((#1)^14)+17041168 ((#1)^16)-10916736 ((#1)^18)+2822400 ((#1)^20)&,2], ])+(NumericalApproximation[Root, (RowDefault["-0.836", FEExpressionsNumericalApproximationElider]), Root[67+26384 #1+99216 ((#1)^2)+9920 ((#1)^3)-426592 ((#1)^4)-643840 ((#1)^5)+70912 ((#1)^6)+1201152 ((#1)^7)+1514240 ((#1)^8)+860160 ((#1)^9)+196608 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["-0.838", FEExpressionsNumericalApproximationElider]), Root[529-15032 ((#1)^2)+105616 ((#1)^4)-303984 ((#1)^6)+415496 ((#1)^8)-1085888 ((#1)^10)+5304000 ((#1)^12)-13361088 ((#1)^14)+17041168 ((#1)^16)-10916736 ((#1)^18)+2822400 ((#1)^20)&,2], ])+(NumericalApproximation[Root, (RowDefault["0.717", FEExpressionsNumericalApproximationElider]), Root[-95+76 #1+756 ((#1)^2)-1984 ((#1)^3)+376 ((#1)^4)+8192 ((#1)^5)-13088 ((#1)^6)-5632 ((#1)^7)+29456 ((#1)^8)-24768 ((#1)^9)+6720 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["-0.836", FEExpressionsNumericalApproximationElider]), Root[67+26384 #1+99216 ((#1)^2)+9920 ((#1)^3)-426592 ((#1)^4)-643840 ((#1)^5)+70912 ((#1)^6)+1201152 ((#1)^7)+1514240 ((#1)^8)+860160 ((#1)^9)+196608 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["-0.838", FEExpressionsNumericalApproximationElider]), Root[529-15032 ((#1)^2)+105616 ((#1)^4)-303984 ((#1)^6)+415496 ((#1)^8)-1085888 ((#1)^10)+5304000 ((#1)^12)-13361088 ((#1)^14)+17041168 ((#1)^16)-10916736 ((#1)^18)+2822400 ((#1)^20)&,2], ])-(NumericalApproximation[Root, (RowDefault["-0.717", FEExpressionsNumericalApproximationElider]), Root[-95-76 #1+756 ((#1)^2)+1984 ((#1)^3)+376 ((#1)^4)-8192 ((#1)^5)-13088 ((#1)^6)+5632 ((#1)^7)+29456 ((#1)^8)+24768 ((#1)^9)+6720 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["-0.836", FEExpressionsNumericalApproximationElider]), Root[67+26384 #1+99216 ((#1)^2)+9920 ((#1)^3)-426592 ((#1)^4)-643840 ((#1)^5)+70912 ((#1)^6)+1201152 ((#1)^7)+1514240 ((#1)^8)+860160 ((#1)^9)+196608 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["-0.838", FEExpressionsNumericalApproximationElider]), Root[529-15032 ((#1)^2)+105616 ((#1)^4)-303984 ((#1)^6)+415496 ((#1)^8)-1085888 ((#1)^10)+5304000 ((#1)^12)-13361088 ((#1)^14)+17041168 ((#1)^16)-10916736 ((#1)^18)+2822400 ((#1)^20)&,2], ])+2 (NumericalApproximation[Root, (RowDefault["0.976", FEExpressionsNumericalApproximationElider]), Root[5308416-60378624 ((#1)^2)+354890305 ((#1)^4)-1551068256 ((#1)^6)+5171166400 ((#1)^8)-12174590464 ((#1)^10)+19238761984 ((#1)^12)-19788308480 ((#1)^14)+12682706944 ((#1)^16)-4601020416 ((#1)^18)+722534400 ((#1)^20)&,7], ])+2 (NumericalApproximation[Root, (RowDefault["0.717", FEExpressionsNumericalApproximationElider]), Root[-95+76 #1+756 ((#1)^2)-1984 ((#1)^3)+376 ((#1)^4)+8192 ((#1)^5)-13088 ((#1)^6)-5632 ((#1)^7)+29456 ((#1)^8)-24768 ((#1)^9)+6720 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["0.976", FEExpressionsNumericalApproximationElider]), Root[5308416-60378624 ((#1)^2)+354890305 ((#1)^4)-1551068256 ((#1)^6)+5171166400 ((#1)^8)-12174590464 ((#1)^10)+19238761984 ((#1)^12)-19788308480 ((#1)^14)+12682706944 ((#1)^16)-4601020416 ((#1)^18)+722534400 ((#1)^20)&,7], ])-2 (NumericalApproximation[Root, (RowDefault["-0.717", FEExpressionsNumericalApproximationElider]), Root[-95-76 #1+756 ((#1)^2)+1984 ((#1)^3)+376 ((#1)^4)-8192 ((#1)^5)-13088 ((#1)^6)+5632 ((#1)^7)+29456 ((#1)^8)+24768 ((#1)^9)+6720 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["0.976", FEExpressionsNumericalApproximationElider]), Root[5308416-60378624 ((#1)^2)+354890305 ((#1)^4)-1551068256 ((#1)^6)+5171166400 ((#1)^8)-12174590464 ((#1)^10)+19238761984 ((#1)^12)-19788308480 ((#1)^14)+12682706944 ((#1)^16)-4601020416 ((#1)^18)+722534400 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["1.10", FEExpressionsNumericalApproximationElider]), Root[345+4464 #1+12976 ((#1)^2)-10560 ((#1)^3)-47008 ((#1)^4)+68352 ((#1)^5)-3328 ((#1)^6)-207872 ((#1)^7)+387328 ((#1)^8)-270336 ((#1)^9)+65536 ((#1)^10)&,4], ]) (NumericalApproximation[Root, (RowDefault["0.976", FEExpressionsNumericalApproximationElider]), Root[5308416-60378624 ((#1)^2)+354890305 ((#1)^4)-1551068256 ((#1)^6)+5171166400 ((#1)^8)-12174590464 ((#1)^10)+19238761984 ((#1)^12)-19788308480 ((#1)^14)+12682706944 ((#1)^16)-4601020416 ((#1)^18)+722534400 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["0.717", FEExpressionsNumericalApproximationElider]), Root[-95+76 #1+756 ((#1)^2)-1984 ((#1)^3)+376 ((#1)^4)+8192 ((#1)^5)-13088 ((#1)^6)-5632 ((#1)^7)+29456 ((#1)^8)-24768 ((#1)^9)+6720 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["1.10", FEExpressionsNumericalApproximationElider]), Root[345+4464 #1+12976 ((#1)^2)-10560 ((#1)^3)-47008 ((#1)^4)+68352 ((#1)^5)-3328 ((#1)^6)-207872 ((#1)^7)+387328 ((#1)^8)-270336 ((#1)^9)+65536 ((#1)^10)&,4], ]) (NumericalApproximation[Root, (RowDefault["0.976", FEExpressionsNumericalApproximationElider]), Root[5308416-60378624 ((#1)^2)+354890305 ((#1)^4)-1551068256 ((#1)^6)+5171166400 ((#1)^8)-12174590464 ((#1)^10)+19238761984 ((#1)^12)-19788308480 ((#1)^14)+12682706944 ((#1)^16)-4601020416 ((#1)^18)+722534400 ((#1)^20)&,7], ])-(NumericalApproximation[Root, (RowDefault["-0.717", FEExpressionsNumericalApproximationElider]), Root[-95-76 #1+756 ((#1)^2)+1984 ((#1)^3)+376 ((#1)^4)-8192 ((#1)^5)-13088 ((#1)^6)+5632 ((#1)^7)+29456 ((#1)^8)+24768 ((#1)^9)+6720 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["1.10", FEExpressionsNumericalApproximationElider]), Root[345+4464 #1+12976 ((#1)^2)-10560 ((#1)^3)-47008 ((#1)^4)+68352 ((#1)^5)-3328 ((#1)^6)-207872 ((#1)^7)+387328 ((#1)^8)-270336 ((#1)^9)+65536 ((#1)^10)&,4], ]) (NumericalApproximation[Root, (RowDefault["0.976", FEExpressionsNumericalApproximationElider]), Root[5308416-60378624 ((#1)^2)+354890305 ((#1)^4)-1551068256 ((#1)^6)+5171166400 ((#1)^8)-12174590464 ((#1)^10)+19238761984 ((#1)^12)-19788308480 ((#1)^14)+12682706944 ((#1)^16)-4601020416 ((#1)^18)+722534400 ((#1)^20)&,7], ])-(NumericalApproximation[Root, (RowDefault["-1.10", FEExpressionsNumericalApproximationElider]), Root[345-4464 #1+12976 ((#1)^2)+10560 ((#1)^3)-47008 ((#1)^4)-68352 ((#1)^5)-3328 ((#1)^6)+207872 ((#1)^7)+387328 ((#1)^8)+270336 ((#1)^9)+65536 ((#1)^10)&,1], ]) (NumericalApproximation[Root, (RowDefault["0.976", FEExpressionsNumericalApproximationElider]), Root[5308416-60378624 ((#1)^2)+354890305 ((#1)^4)-1551068256 ((#1)^6)+5171166400 ((#1)^8)-12174590464 ((#1)^10)+19238761984 ((#1)^12)-19788308480 ((#1)^14)+12682706944 ((#1)^16)-4601020416 ((#1)^18)+722534400 ((#1)^20)&,7], ])-(NumericalApproximation[Root, (RowDefault["0.717", FEExpressionsNumericalApproximationElider]), Root[-95+76 #1+756 ((#1)^2)-1984 ((#1)^3)+376 ((#1)^4)+8192 ((#1)^5)-13088 ((#1)^6)-5632 ((#1)^7)+29456 ((#1)^8)-24768 ((#1)^9)+6720 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["-1.10", FEExpressionsNumericalApproximationElider]), Root[345-4464 #1+12976 ((#1)^2)+10560 ((#1)^3)-47008 ((#1)^4)-68352 ((#1)^5)-3328 ((#1)^6)+207872 ((#1)^7)+387328 ((#1)^8)+270336 ((#1)^9)+65536 ((#1)^10)&,1], ]) (NumericalApproximation[Root, (RowDefault["0.976", FEExpressionsNumericalApproximationElider]), Root[5308416-60378624 ((#1)^2)+354890305 ((#1)^4)-1551068256 ((#1)^6)+5171166400 ((#1)^8)-12174590464 ((#1)^10)+19238761984 ((#1)^12)-19788308480 ((#1)^14)+12682706944 ((#1)^16)-4601020416 ((#1)^18)+722534400 ((#1)^20)&,7], ])+(NumericalApproximation[Root, (RowDefault["-0.717", FEExpressionsNumericalApproximationElider]), Root[-95-76 #1+756 ((#1)^2)+1984 ((#1)^3)+376 ((#1)^4)-8192 ((#1)^5)-13088 ((#1)^6)+5632 ((#1)^7)+29456 ((#1)^8)+24768 ((#1)^9)+6720 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["-1.10", FEExpressionsNumericalApproximationElider]), Root[345-4464 #1+12976 ((#1)^2)+10560 ((#1)^3)-47008 ((#1)^4)-68352 ((#1)^5)-3328 ((#1)^6)+207872 ((#1)^7)+387328 ((#1)^8)+270336 ((#1)^9)+65536 ((#1)^10)&,1], ]) (NumericalApproximation[Root, (RowDefault["0.976", FEExpressionsNumericalApproximationElider]), Root[5308416-60378624 ((#1)^2)+354890305 ((#1)^4)-1551068256 ((#1)^6)+5171166400 ((#1)^8)-12174590464 ((#1)^10)+19238761984 ((#1)^12)-19788308480 ((#1)^14)+12682706944 ((#1)^16)-4601020416 ((#1)^18)+722534400 ((#1)^20)&,7], ])-(NumericalApproximation[Root, (RowDefault["-0.611", FEExpressionsNumericalApproximationElider]), Root[1600-309888 ((#1)^2)+10362496 ((#1)^4)-150640320 ((#1)^6)+1108772352 ((#1)^8)-4874307760 ((#1)^10)+13891942264 ((#1)^12)-26107499520 ((#1)^14)+31258427913 ((#1)^16)-21590613504 ((#1)^18)+6502809600 ((#1)^20)&,3], ])-(NumericalApproximation[Root, (RowDefault["0.717", FEExpressionsNumericalApproximationElider]), Root[-95+76 #1+756 ((#1)^2)-1984 ((#1)^3)+376 ((#1)^4)+8192 ((#1)^5)-13088 ((#1)^6)-5632 ((#1)^7)+29456 ((#1)^8)-24768 ((#1)^9)+6720 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["1.10", FEExpressionsNumericalApproximationElider]), Root[345+4464 #1+12976 ((#1)^2)-10560 ((#1)^3)-47008 ((#1)^4)+68352 ((#1)^5)-3328 ((#1)^6)-207872 ((#1)^7)+387328 ((#1)^8)-270336 ((#1)^9)+65536 ((#1)^10)&,4], ]) (NumericalApproximation[Root, (RowDefault["-0.611", FEExpressionsNumericalApproximationElider]), Root[1600-309888 ((#1)^2)+10362496 ((#1)^4)-150640320 ((#1)^6)+1108772352 ((#1)^8)-4874307760 ((#1)^10)+13891942264 ((#1)^12)-26107499520 ((#1)^14)+31258427913 ((#1)^16)-21590613504 ((#1)^18)+6502809600 ((#1)^20)&,3], ])+(NumericalApproximation[Root, (RowDefault["-0.717", FEExpressionsNumericalApproximationElider]), Root[-95-76 #1+756 ((#1)^2)+1984 ((#1)^3)+376 ((#1)^4)-8192 ((#1)^5)-13088 ((#1)^6)+5632 ((#1)^7)+29456 ((#1)^8)+24768 ((#1)^9)+6720 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["1.10", FEExpressionsNumericalApproximationElider]), Root[345+4464 #1+12976 ((#1)^2)-10560 ((#1)^3)-47008 ((#1)^4)+68352 ((#1)^5)-3328 ((#1)^6)-207872 ((#1)^7)+387328 ((#1)^8)-270336 ((#1)^9)+65536 ((#1)^10)&,4], ]) (NumericalApproximation[Root, (RowDefault["-0.611", FEExpressionsNumericalApproximationElider]), Root[1600-309888 ((#1)^2)+10362496 ((#1)^4)-150640320 ((#1)^6)+1108772352 ((#1)^8)-4874307760 ((#1)^10)+13891942264 ((#1)^12)-26107499520 ((#1)^14)+31258427913 ((#1)^16)-21590613504 ((#1)^18)+6502809600 ((#1)^20)&,3], ])+(NumericalApproximation[Root, (RowDefault["0.717", FEExpressionsNumericalApproximationElider]), Root[-95+76 #1+756 ((#1)^2)-1984 ((#1)^3)+376 ((#1)^4)+8192 ((#1)^5)-13088 ((#1)^6)-5632 ((#1)^7)+29456 ((#1)^8)-24768 ((#1)^9)+6720 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["-1.10", FEExpressionsNumericalApproximationElider]), Root[345-4464 #1+12976 ((#1)^2)+10560 ((#1)^3)-47008 ((#1)^4)-68352 ((#1)^5)-3328 ((#1)^6)+207872 ((#1)^7)+387328 ((#1)^8)+270336 ((#1)^9)+65536 ((#1)^10)&,1], ]) (NumericalApproximation[Root, (RowDefault["-0.611", FEExpressionsNumericalApproximationElider]), Root[1600-309888 ((#1)^2)+10362496 ((#1)^4)-150640320 ((#1)^6)+1108772352 ((#1)^8)-4874307760 ((#1)^10)+13891942264 ((#1)^12)-26107499520 ((#1)^14)+31258427913 ((#1)^16)-21590613504 ((#1)^18)+6502809600 ((#1)^20)&,3], ])-(NumericalApproximation[Root, (RowDefault["-0.717", FEExpressionsNumericalApproximationElider]), Root[-95-76 #1+756 ((#1)^2)+1984 ((#1)^3)+376 ((#1)^4)-8192 ((#1)^5)-13088 ((#1)^6)+5632 ((#1)^7)+29456 ((#1)^8)+24768 ((#1)^9)+6720 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["-1.10", FEExpressionsNumericalApproximationElider]), Root[345-4464 #1+12976 ((#1)^2)+10560 ((#1)^3)-47008 ((#1)^4)-68352 ((#1)^5)-3328 ((#1)^6)+207872 ((#1)^7)+387328 ((#1)^8)+270336 ((#1)^9)+65536 ((#1)^10)&,1], ]) (NumericalApproximation[Root, (RowDefault["-0.611", FEExpressionsNumericalApproximationElider]), Root[1600-309888 ((#1)^2)+10362496 ((#1)^4)-150640320 ((#1)^6)+1108772352 ((#1)^8)-4874307760 ((#1)^10)+13891942264 ((#1)^12)-26107499520 ((#1)^14)+31258427913 ((#1)^16)-21590613504 ((#1)^18)+6502809600 ((#1)^20)&,3], ])-(NumericalApproximation[Root, (RowDefault["0.717", FEExpressionsNumericalApproximationElider]), Root[-95+76 #1+756 ((#1)^2)-1984 ((#1)^3)+376 ((#1)^4)+8192 ((#1)^5)-13088 ((#1)^6)-5632 ((#1)^7)+29456 ((#1)^8)-24768 ((#1)^9)+6720 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["0.353", FEExpressionsNumericalApproximationElider]), Root[6561-347328 ((#1)^2)+6973168 ((#1)^4)-81591424 ((#1)^6)+690379360 ((#1)^8)-4494628864 ((#1)^10)+21900300032 ((#1)^12)-78744037376 ((#1)^14)+200126890240 ((#1)^16)-298144628736 ((#1)^18)+184968806400 ((#1)^20)&,6], ])+(NumericalApproximation[Root, (RowDefault["-0.717", FEExpressionsNumericalApproximationElider]), Root[-95-76 #1+756 ((#1)^2)+1984 ((#1)^3)+376 ((#1)^4)-8192 ((#1)^5)-13088 ((#1)^6)+5632 ((#1)^7)+29456 ((#1)^8)+24768 ((#1)^9)+6720 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["0.353", FEExpressionsNumericalApproximationElider]), Root[6561-347328 ((#1)^2)+6973168 ((#1)^4)-81591424 ((#1)^6)+690379360 ((#1)^8)-4494628864 ((#1)^10)+21900300032 ((#1)^12)-78744037376 ((#1)^14)+200126890240 ((#1)^16)-298144628736 ((#1)^18)+184968806400 ((#1)^20)&,6], ])+(NumericalApproximation[Root, (RowDefault["0.717", FEExpressionsNumericalApproximationElider]), Root[-95+76 #1+756 ((#1)^2)-1984 ((#1)^3)+376 ((#1)^4)+8192 ((#1)^5)-13088 ((#1)^6)-5632 ((#1)^7)+29456 ((#1)^8)-24768 ((#1)^9)+6720 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["0.836", FEExpressionsNumericalApproximationElider]), Root[67-26384 #1+99216 ((#1)^2)-9920 ((#1)^3)-426592 ((#1)^4)+643840 ((#1)^5)+70912 ((#1)^6)-1201152 ((#1)^7)+1514240 ((#1)^8)-860160 ((#1)^9)+196608 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["0.353", FEExpressionsNumericalApproximationElider]), Root[6561-347328 ((#1)^2)+6973168 ((#1)^4)-81591424 ((#1)^6)+690379360 ((#1)^8)-4494628864 ((#1)^10)+21900300032 ((#1)^12)-78744037376 ((#1)^14)+200126890240 ((#1)^16)-298144628736 ((#1)^18)+184968806400 ((#1)^20)&,6], ])-(NumericalApproximation[Root, (RowDefault["-0.717", FEExpressionsNumericalApproximationElider]), Root[-95-76 #1+756 ((#1)^2)+1984 ((#1)^3)+376 ((#1)^4)-8192 ((#1)^5)-13088 ((#1)^6)+5632 ((#1)^7)+29456 ((#1)^8)+24768 ((#1)^9)+6720 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["0.836", FEExpressionsNumericalApproximationElider]), Root[67-26384 #1+99216 ((#1)^2)-9920 ((#1)^3)-426592 ((#1)^4)+643840 ((#1)^5)+70912 ((#1)^6)-1201152 ((#1)^7)+1514240 ((#1)^8)-860160 ((#1)^9)+196608 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["0.353", FEExpressionsNumericalApproximationElider]), Root[6561-347328 ((#1)^2)+6973168 ((#1)^4)-81591424 ((#1)^6)+690379360 ((#1)^8)-4494628864 ((#1)^10)+21900300032 ((#1)^12)-78744037376 ((#1)^14)+200126890240 ((#1)^16)-298144628736 ((#1)^18)+184968806400 ((#1)^20)&,6], ])-(NumericalApproximation[Root, (RowDefault["0.717", FEExpressionsNumericalApproximationElider]), Root[-95+76 #1+756 ((#1)^2)-1984 ((#1)^3)+376 ((#1)^4)+8192 ((#1)^5)-13088 ((#1)^6)-5632 ((#1)^7)+29456 ((#1)^8)-24768 ((#1)^9)+6720 ((#1)^10)&,3], ]) (NumericalApproximation[Root, (RowDefault["-0.836", FEExpressionsNumericalApproximationElider]), Root[67+26384 #1+99216 ((#1)^2)+9920 ((#1)^3)-426592 ((#1)^4)-643840 ((#1)^5)+70912 ((#1)^6)+1201152 ((#1)^7)+1514240 ((#1)^8)+860160 ((#1)^9)+196608 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["0.353", FEExpressionsNumericalApproximationElider]), Root[6561-347328 ((#1)^2)+6973168 ((#1)^4)-81591424 ((#1)^6)+690379360 ((#1)^8)-4494628864 ((#1)^10)+21900300032 ((#1)^12)-78744037376 ((#1)^14)+200126890240 ((#1)^16)-298144628736 ((#1)^18)+184968806400 ((#1)^20)&,6], ])+(NumericalApproximation[Root, (RowDefault["-0.717", FEExpressionsNumericalApproximationElider]), Root[-95-76 #1+756 ((#1)^2)+1984 ((#1)^3)+376 ((#1)^4)-8192 ((#1)^5)-13088 ((#1)^6)+5632 ((#1)^7)+29456 ((#1)^8)+24768 ((#1)^9)+6720 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["-0.836", FEExpressionsNumericalApproximationElider]), Root[67+26384 #1+99216 ((#1)^2)+9920 ((#1)^3)-426592 ((#1)^4)-643840 ((#1)^5)+70912 ((#1)^6)+1201152 ((#1)^7)+1514240 ((#1)^8)+860160 ((#1)^9)+196608 ((#1)^10)&,2], ]) (NumericalApproximation[Root, (RowDefault["0.353", FEExpressionsNumericalApproximationElider]), Root[6561-347328 ((#1)^2)+6973168 ((#1)^4)-81591424 ((#1)^6)+690379360 ((#1)^8)-4494628864 ((#1)^10)+21900300032 ((#1)^12)-78744037376 ((#1)^14)+200126890240 ((#1)^16)-298144628736 ((#1)^18)+184968806400 ((#1)^20)&,6], ]));

N[vol]

vol/.r:Times[_Root..]:>N[Times[r]]

volrr=vol/.r:HoldPattern[Times[_Root..]]:>RootReduce[Print[Times[r]];Times[r]];//Timing

volrr//N

ByteCount[volrr]

volrr

Simplify[volrr]

N[%]

(volrr2=(1/6) (2 (NumericalApproximation[Root, (RowDefault["0.976", FEExpressionsNumericalApproximationElider]), Root[5308416-60378624 ((#1)^2)+354890305 ((#1)^4)-1551068256 ((#1)^6)+5171166400 ((#1)^8)-12174590464 ((#1)^10)+19238761984 ((#1)^12)-19788308480 ((#1)^14)+12682706944 ((#1)^16)-4601020416 ((#1)^18)+722534400 ((#1)^20)&,7], ])-(NumericalApproximation[Root, (RowDefault["-0.611", FEExpressionsNumericalApproximationElider]), Root[1600-309888 ((#1)^2)+10362496 ((#1)^4)-150640320 ((#1)^6)+1108772352 ((#1)^8)-4874307760 ((#1)^10)+13891942264 ((#1)^12)-26107499520 ((#1)^14)+31258427913 ((#1)^16)-21590613504 ((#1)^18)+6502809600 ((#1)^20)&,3], ])+2(NumericalApproximation[Root, (RowDefault["1.08", FEExpressionsNumericalApproximationElider]), Root[1071225-117616302 ((#1)^2)+3457487449 ((#1)^4)-31626185600 ((#1)^6)+68472163328 ((#1)^8)+91517714432 ((#1)^10)+1067343806464 ((#1)^12)-1603708911616 ((#1)^14)+4017606361088 ((#1)^16)-9120363053056 ((#1)^18)+5261334937600 ((#1)^20)&,8], ])+2(NumericalApproximation[Root, (RowDefault["0.601", FEExpressionsNumericalApproximationElider]), Root[4774225-831159408 ((#1)^2)+372522624 ((#1)^4)-43532289536 ((#1)^6)+328171852928 ((#1)^8)+378019261440 ((#1)^10)-3551226245120 ((#1)^12)-16334508720128 ((#1)^14)+109827892645888 ((#1)^16)-203164715188224 ((#1)^18)+127455068160000 ((#1)^20)&,7], ])+4 (NumericalApproximation[Root, (RowDefault["0.700", FEExpressionsNumericalApproximationElider]), Root[2994278400-129615067008 ((#1)^2)+1773310978609 ((#1)^4)-17259185925248 ((#1)^6)+27142734408448 ((#1)^8)+95952869883904 ((#1)^10)+141206521339904 ((#1)^12)-2402132177715200 ((#1)^14)+5835588946100224 ((#1)^16)-5706859344297984 ((#1)^18)+2039281090560000 ((#1)^20)&,7], ])+2(NumericalApproximation[Root, (RowDefault["0.701", FEExpressionsNumericalApproximationElider]), Root[2374681-584975639040 ((#1)^2)+52935062348800 ((#1)^4)-382739457868800 ((#1)^6)+1006163582060544 ((#1)^8)-1379131333279744 ((#1)^10)-2922108687220736 ((#1)^12)+26337817805193216 ((#1)^14)-29055700071088128 ((#1)^16)-70688768633339904 ((#1)^18)+109099041266073600 ((#1)^20)&,7], ])-2(NumericalApproximation[Root, (RowDefault["0.253", FEExpressionsNumericalApproximationElider]), Root[59213025-7910755488 ((#1)^2)+408012789184 ((#1)^4)-11300253115904 ((#1)^6)+190374822512128 ((#1)^8)-1774123262959616 ((#1)^10)+8910611782418432 ((#1)^12)-93641547151179776 ((#1)^14)+952543005873602560 ((#1)^16)-1615013035087233024 ((#1)^18)+8352895346933760000 ((#1)^20)&,7], ])+4 (NumericalApproximation[Root, (RowDefault["0.771", FEExpressionsNumericalApproximationElider]), Root[9667805625-3471756842232 ((#1)^2)+255824816217232 ((#1)^4)+4373850348576768 ((#1)^6)+371655957447114752 ((#1)^8)-2149301237441363968 ((#1)^10)+16509712745814294528 ((#1)^12)+9086343295709216768 ((#1)^14)-59367521006164901888 ((#1)^16)-133581981030971080704 ((#1)^18)+237593467646115840000 ((#1)^20)&,7], ])+4 (NumericalApproximation[Root, (RowDefault["0.482", FEExpressionsNumericalApproximationElider]), Root[2983890625+53042349552 ((#1)^2)-11246205017600 ((#1)^4)-372191357151408640 ((#1)^6)+103796924883287400448 ((#1)^8)-6406692234329811771392 ((#1)^10)+39868062879286838198272 ((#1)^12)-149042711391905761984512 ((#1)^14)+498106854223225907576832 ((#1)^16)-1027062110642280358477824 ((#1)^18)+2189661397826603581440000 ((#1)^20)&,8], ])+4 (NumericalApproximation[Root, (RowDefault["0.502", FEExpressionsNumericalApproximationElider]), Root[21431496025-7088325527373248 ((#1)^2)+5048991838827214848 ((#1)^4)+99941012309993521152 ((#1)^6)+1232143059367874035712 ((#1)^8)+716118844637881303040 ((#1)^10)-32189837930183568392192 ((#1)^12)-178756289998022579847168 ((#1)^14)+601551084851681562722304 ((#1)^16)-661238816140073197633536 ((#1)^18)+4926738145109858058240000 ((#1)^20)&,8], ])+4 (NumericalApproximation[Root, (RowDefault["0.211", FEExpressionsNumericalApproximationElider]), Root[29534141025-151390756763620992 ((#1)^2)+38840459182863600640 ((#1)^4)-890872427456268566528 ((#1)^6)+27118287364852387545088 ((#1)^8)-917664148876685238861824 ((#1)^10)+7937913366253232847847424 ((#1)^12)-123623844097768812311478272 ((#1)^14)+1918547904943753861280038912 ((#1)^16)+18621778253898807209619357696 ((#1)^18)+35875412341991073078312960000 ((#1)^20)&,7], ])))//N

N[vol-volrr2,100]

l=List@@(2 (NumericalApproximation[Root, (RowDefault["0.976", FEExpressionsNumericalApproximationElider]), Root[5308416-60378624 ((#1)^2)+354890305 ((#1)^4)-1551068256 ((#1)^6)+5171166400 ((#1)^8)-12174590464 ((#1)^10)+19238761984 ((#1)^12)-19788308480 ((#1)^14)+12682706944 ((#1)^16)-4601020416 ((#1)^18)+722534400 ((#1)^20)&,7], ])-(NumericalApproximation[Root, (RowDefault["-0.611", FEExpressionsNumericalApproximationElider]), Root[1600-309888 ((#1)^2)+10362496 ((#1)^4)-150640320 ((#1)^6)+1108772352 ((#1)^8)-4874307760 ((#1)^10)+13891942264 ((#1)^12)-26107499520 ((#1)^14)+31258427913 ((#1)^16)-21590613504 ((#1)^18)+6502809600 ((#1)^20)&,3], ])+2(NumericalApproximation[Root, (RowDefault["1.08", FEExpressionsNumericalApproximationElider]), Root[1071225-117616302 ((#1)^2)+3457487449 ((#1)^4)-31626185600 ((#1)^6)+68472163328 ((#1)^8)+91517714432 ((#1)^10)+1067343806464 ((#1)^12)-1603708911616 ((#1)^14)+4017606361088 ((#1)^16)-9120363053056 ((#1)^18)+5261334937600 ((#1)^20)&,8], ])+2(NumericalApproximation[Root, (RowDefault["0.601", FEExpressionsNumericalApproximationElider]), Root[4774225-831159408 ((#1)^2)+372522624 ((#1)^4)-43532289536 ((#1)^6)+328171852928 ((#1)^8)+378019261440 ((#1)^10)-3551226245120 ((#1)^12)-16334508720128 ((#1)^14)+109827892645888 ((#1)^16)-203164715188224 ((#1)^18)+127455068160000 ((#1)^20)&,7], ])+4 (NumericalApproximation[Root, (RowDefault["0.700", FEExpressionsNumericalApproximationElider]), Root[2994278400-129615067008 ((#1)^2)+1773310978609 ((#1)^4)-17259185925248 ((#1)^6)+27142734408448 ((#1)^8)+95952869883904 ((#1)^10)+141206521339904 ((#1)^12)-2402132177715200 ((#1)^14)+5835588946100224 ((#1)^16)-5706859344297984 ((#1)^18)+2039281090560000 ((#1)^20)&,7], ])+2(NumericalApproximation[Root, (RowDefault["0.701", FEExpressionsNumericalApproximationElider]), Root[2374681-584975639040 ((#1)^2)+52935062348800 ((#1)^4)-382739457868800 ((#1)^6)+1006163582060544 ((#1)^8)-1379131333279744 ((#1)^10)-2922108687220736 ((#1)^12)+26337817805193216 ((#1)^14)-29055700071088128 ((#1)^16)-70688768633339904 ((#1)^18)+109099041266073600 ((#1)^20)&,7], ])-2(NumericalApproximation[Root, (RowDefault["0.253", FEExpressionsNumericalApproximationElider]), Root[59213025-7910755488 ((#1)^2)+408012789184 ((#1)^4)-11300253115904 ((#1)^6)+190374822512128 ((#1)^8)-1774123262959616 ((#1)^10)+8910611782418432 ((#1)^12)-93641547151179776 ((#1)^14)+952543005873602560 ((#1)^16)-1615013035087233024 ((#1)^18)+8352895346933760000 ((#1)^20)&,7], ])+4 (NumericalApproximation[Root, (RowDefault["0.771", FEExpressionsNumericalApproximationElider]), Root[9667805625-3471756842232 ((#1)^2)+255824816217232 ((#1)^4)+4373850348576768 ((#1)^6)+371655957447114752 ((#1)^8)-2149301237441363968 ((#1)^10)+16509712745814294528 ((#1)^12)+9086343295709216768 ((#1)^14)-59367521006164901888 ((#1)^16)-133581981030971080704 ((#1)^18)+237593467646115840000 ((#1)^20)&,7], ])+4 (NumericalApproximation[Root, (RowDefault["0.482", FEExpressionsNumericalApproximationElider]), Root[2983890625+53042349552 ((#1)^2)-11246205017600 ((#1)^4)-372191357151408640 ((#1)^6)+103796924883287400448 ((#1)^8)-6406692234329811771392 ((#1)^10)+39868062879286838198272 ((#1)^12)-149042711391905761984512 ((#1)^14)+498106854223225907576832 ((#1)^16)-1027062110642280358477824 ((#1)^18)+2189661397826603581440000 ((#1)^20)&,8], ])+4 (NumericalApproximation[Root, (RowDefault["0.502", FEExpressionsNumericalApproximationElider]), Root[21431496025-7088325527373248 ((#1)^2)+5048991838827214848 ((#1)^4)+99941012309993521152 ((#1)^6)+1232143059367874035712 ((#1)^8)+716118844637881303040 ((#1)^10)-32189837930183568392192 ((#1)^12)-178756289998022579847168 ((#1)^14)+601551084851681562722304 ((#1)^16)-661238816140073197633536 ((#1)^18)+4926738145109858058240000 ((#1)^20)&,8], ])+4 (NumericalApproximation[Root, (RowDefault["0.211", FEExpressionsNumericalApproximationElider]), Root[29534141025-151390756763620992 ((#1)^2)+38840459182863600640 ((#1)^4)-890872427456268566528 ((#1)^6)+27118287364852387545088 ((#1)^8)-917664148876685238861824 ((#1)^10)+7937913366253232847847424 ((#1)^12)-123623844097768812311478272 ((#1)^14)+1918547904943753861280038912 ((#1)^16)+18621778253898807209619357696 ((#1)^18)+35875412341991073078312960000 ((#1)^20)&,7], ]))

Total/@N[Subsets[l,{2}]]

Total/@N[Subsets[l,{3}]]

Total/@N[Subsets[l,{4}]]

Unsimplified

ByteCount[v1]

N[v2=PolyhedronVolume[punsimplified]]

ByteCount/@{v1,v2}

FullSimplify[v1]//Timing