Tritetrahedron

Author

Eric W. Weisstein
September 22, 2023

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

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

©2023 Wolfram Research, Inc. except for portions noted otherwise

Polyhedron

PolyhedronData["Tritetrahedron","Orientations"]

PolyhedronData["Tritetrahedron"]

Show[PolyhedronData[{"Tritetrahedron",#}],Boxed->False]&/@PolyhedronData["Tritetrahedron","Orientations"]

Polyhedron

Graphics3D[{FaceForm[Green,Red],PolyhedronData["Tritetrahedron","Polyhedron"]}]

Region

PolyhedronData["Tritetrahedron","Region"]

Solid and Dual

PolyhedronData["Tritetrahedron",#]&/@{"Graphics3D","Dual","DualCompound"}

Net

PolyhedronData["Tritetrahedron","Net"]

PolyhedronData["Tritetrahedron","Net","Colored"]

NetPrintout[PolyhedronData["Tritetrahedron","Polyhedron"],"Tritetrahedron.pdf"]

Properties

Initialization

<<MathWorld`Polyhedra`

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

vol=PolyhedronData[pname,"Volume"]

Centroid

PolyhedronData[pname,"Centroid"]

Graphics3D[{AbsolutePointSize[10],
Point[{0,0,0}],{Blue,Point[PolyhedronData[pname,"Centroid"]]},Opacity[.2],PolyhedronData[pname,"GraphicsComplex"]}]

RegionCentroid[p]

PolyhedronCentroid[p]//FullSimplify

Circumsphere

sphere=PolyhedronData[pname,"Circumsphere"]

sphere=Circumsphere[p]

(sphere=MyCircumsphere[p])//Timing

Graphics3D[{{Opacity[.2],Yellow,sphere},p},Boxed->False]

Classes

PolyhedronData[pname,"Classes"]

Compound

PolyhedronData[pname,"Compound"]

Convex

PolyhedronData[pname,"Convex"]

ConvexPolyhedronQ[p]

ConvexHull

PolyhedronData[pname,"ConvexHull"]

convexhull=ConvexHullRegion[PolyhedronData[pname,"Vertices"]]

v=PolyhedronCoordinates[convexhull]

Sort[(convexhull[[2]]/.i_Integer:>v[[i]])/.Thread[PolyhedronData[pname,"Vertices"]->Range[PolyhedronData[pname,"V"]]]]

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

{Graphics3D[{Red,Opacity[.5],p}],Graphics3D[{Blue,Opacity[.5],PolyhedronData[pname,"ConvexHull"][[1]]}]}

Show[%]

SimplifyNumericallyDistinct[PolyhedronEdgeLengths[convexhull]]//Tally

DihedralAngles

PolyhedronData[pname,"DihedralAngles"]//Tally

SimplifyNumericallyDistinct[DihedralAngles[p]]//Tally

EdgeCount

PolyhedronData[pname,"EdgeCount"]

EdgeLengths

PolyhedronData[pname,"EdgeLengths"]

PolyhedronData[pname,"Lines"]/.Line[l_]:>EuclideanDistance@@l//FullSimplify//Union//Quiet

lens=SimplifyNumericallyDistinct[PolyhedronEdgeLengths[p]];//Quiet

(distinctlens=Mean/@Split[Sort[lens],Abs[#1-#2]<1.*^-10&])//N//Quiet

v=PolyhedronCoordinates[p];

colors=With[{n=Length[distinctlens]},Table[Hue[i/n],{i,0,n-1}]]

Graphics3D[{Thick,Sequence@@Nearest[Thread[distinctlens->colors],EuclideanDistance@@v[[#]]],Line[v[[#]]]}&/@PolyhedronEdges[p],Boxed->False]//Quiet

FaceCount

PolyhedronData[pname,"FaceCount"]

GeneralizedDiameter

PolyhedronData[pname,"GeneralizedDiameter"]

GeneralizedDiameter[p]//FullSimplify//Quiet

Hull

PolyhedronData[pname,"Hull","Name"]

PolyhedronData[pname,"Hull","Scale"]

PolyhedronData[pname,"Hull"]

Max[Norm/@PolyhedronData[pname,"Hull","Polyhedron"][[1]]]

Max[Norm/@p[[1]]]

{Graphics3D[{Opacity[.5],Red,p}],Graphics3D[{Opacity[.5],Blue,PolyhedronData[pname,"Hull"][[1]]}]}

Show[%]

InertiaTensor

PolyhedronData[pname,"InertiaTensor"]

(i=((MomentOfInertia[p,{0,0,0}])/(Volume[p])))//Timing

PolyhedronInertiaTensor[p]//Timing

Insphere

PolyhedronData[pname,"Insphere"]

Insphere[p]

sphere=MyInsphere[p]

Graphics3D[{{Opacity[.5],p},{Red,sphere}},Boxed->False]

MeanCylindricalRadius

PolyhedronData[pname,"MeanCylindricalRadius"]

ResourceFunction["LogSimplify"][%]

N[%]

Integrate over polyhedron

((NIntegrate[(Sqrt[(x^2)+(y^2)]),{x,y,z}∈p])/vol)

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

Divergence theorem

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

$Version

NIntegrate[(1/(3 (Sqrt[2])))((1/2) (Sqrt[((((5/6)-((5 s)/6)-((5 t)/6)))^2)+((((s/2)-(t/2)))^2)]) ((5/6)-((5 s)/6)-((5 t)/6))-(1/2) ((((s/2)-(t/2)))^2) Log[-(5/6)+((5 s)/6)+(Sqrt[((((5/6)-((5 s)/6)-((5 t)/6)))^2)+((((s/2)-(t/2)))^2)])+((5 t)/6)]),{t,0,1},{s,0,1-t}]

bug (428636): Integrate[1/4 Sqrt[1/4 + (1/2 - s)^2] - 1/2 (1/2 - s)^2 Log[-(1/2) + Sqrt[1/4 + (1/2 - s)^2]], {t, 0, 1}, {s, 0, 1 - t}] returns result with incorrect imaginary part

(int1=Integrate[(1/(3 (Sqrt[2])))((1/2) (Sqrt[((((5/6)-((5 s)/6)-((5 t)/6)))^2)+((((s/2)-(t/2)))^2)]) ((5/6)-((5 s)/6)-((5 t)/6))-(1/2) ((((s/2)-(t/2)))^2) Log[-(5/6)+((5 s)/6)+(Sqrt[((((5/6)-((5 s)/6)-((5 t)/6)))^2)+((((s/2)-(t/2)))^2)])+((5 t)/6)]),{t,0,1},{s,0,1-t}])//Timing

int1//N

(int2=Integrate[(1/(3 (Sqrt[2])))((1/2) (Sqrt[((((5/6)-((5 s)/6)-((5 t)/6)))^2)+((((s/2)-(t/2)))^2)]) ((5/6)-((5 s)/6)-((5 t)/6))-(1/2) ((((s/2)-(t/2)))^2) Log[-(5/6)+((5 s)/6)+(Sqrt[((((5/6)-((5 s)/6)-((5 t)/6)))^2)+((((s/2)-(t/2)))^2)])+((5 t)/6)]),{t,0,1},{s,0,1-t},GenerateConditions->True])//Timing

(int3=Integrate[(1/(3 (Sqrt[2])))((1/2) (Sqrt[((((5/6)-((5 s)/6)-((5 t)/6)))^2)+((((s/2)-(t/2)))^2)]) ((5/6)-((5 s)/6)-((5 t)/6))-(1/2) ((((s/2)-(t/2)))^2) Log[-(5/6)+((5 s)/6)+(Sqrt[((((5/6)-((5 s)/6)-((5 t)/6)))^2)+((((s/2)-(t/2)))^2)])+((5 t)/6)]),{s,0,1-t},Assumptions->0<t<1])//Timing

(ints=Integrate[(1/(3 (Sqrt[2])))((1/2) (Sqrt[((((5/6)-((5 s)/6)-((5 t)/6)))^2)+((((s/2)-(t/2)))^2)]) ((5/6)-((5 s)/6)-((5 t)/6))-(1/2) ((((s/2)-(t/2)))^2) Log[-(5/6)+((5 s)/6)+(Sqrt[((((5/6)-((5 s)/6)-((5 t)/6)))^2)+((((s/2)-(t/2)))^2)])+((5 t)/6)]),s])//Timing

intss=FullSimplify[Limit[ints,s->1-t]-Limit[ints,s->0]]

NIntegrate[intss,{t,0,1}]

(int4=Integrate[intss,{t,0,1}])//Timing

N[int4]

int=FullSimplify[int4]

int=(1/(117768619008 (Sqrt[2])))(1254151336-2601 (-157216+5125 (Sqrt[34])) Log[2]-1125 (Sqrt[34]) (63002 Log[15]-949 Log[3+(Sqrt[34])]-16555 Log[5+(Sqrt[34])]-19652 Log[43+6 (Sqrt[34])]-10900 Log[49+8 (Sqrt[34])]-11849 Log[342+36 (Sqrt[34])]));

N[%]

rxy=(1/((19/(9 (Sqrt[2])))+((4 (Sqrt[2]))/9)))6 ((2525/(88128 (Sqrt[2])))-((ArcSinh[1])/192)-int-((Log[2])/(144 (Sqrt[2])))+(1/13855131648)(254611312 (Sqrt[2])+8696250 (Sqrt[17]) ArcCoth[4 (Sqrt[(2/17)])]+21661875 (Sqrt[17]) ArcSinh[(3/5)]-72162144 ArcSinh[1]-62730000 (Sqrt[17]) ArcTanh[(3/5)-((Sqrt[34])/5)]-120270240 (Sqrt[2]) Log[3]+9703125 (Sqrt[17]) Log[3]+120270240 (Sqrt[2]) Log[6]-24054048 (Sqrt[2]) Log[8]-9703125 (Sqrt[17]) Log[5+(Sqrt[34])])+(1/2996352)(55063 (Sqrt[2])-15606 ArcSinh[1]-45 (Sqrt[17 (54929+8500 (Sqrt[34]))]) Log[3]+5202 (Sqrt[2]) Log[972]+5625 (Sqrt[17]) Log[(1/5) (49+8 (Sqrt[34]))])-((625 (12850 Log[15]-4113 Log[49-8 (Sqrt[34])]-8737 Log[49+8 (Sqrt[34])]))/(90556416 (Sqrt[17])))+(1/(117768619008 (Sqrt[2])))(-777079384+453625875 (Sqrt[34]) ArcSinh[(8/15)]+70929585 (Sqrt[34]) ArcSinh[(5/3)]+44217000 (Sqrt[34]) ArcTanh[(5/(Sqrt[34]))]+3271350528 ArcTanh[(1/25) (13-4 (Sqrt[34]))]-481080960 (Sqrt[34]) Log[(5/3)]-569055960 (Sqrt[34]) Log[(9/5)]+4293647568 Log[2]+1635675264 Log[3]+118903500 (Sqrt[34]) Log[3]-1226756448 Log[4]+204459408 Log[(9/2)]-684509625 (Sqrt[34]) Log[5]+102229704 Log[16]+1912500 (Sqrt[34]) Log[25]-1912500 (Sqrt[34]) Log[34]+17556750 (Sqrt[34]) Log[5625]+17556750 (Sqrt[34]) Log[3073-516 (Sqrt[34])]-11054250 (Sqrt[34]) Log[34-5 (Sqrt[34])]+36987750 (Sqrt[34]) Log[34-3 (Sqrt[34])]-2657972304 Log[-5+(Sqrt[34])]+569055960 (Sqrt[34]) Log[-5+(Sqrt[34])]-50677875 (Sqrt[34]) Log[-3+(Sqrt[34])]+152123625 (Sqrt[34]) Log[3+(Sqrt[34])]-1022297040 Log[5+(Sqrt[34])]+533163375 (Sqrt[34]) Log[5+(Sqrt[34])]-33162750 (Sqrt[34]) Log[34+3 (Sqrt[34])]+11054250 (Sqrt[34]) Log[34+5 (Sqrt[34])]-19469250 (Sqrt[34]) Log[59+10 (Sqrt[34])]-26660250 (Sqrt[34]) Log[3073+516 (Sqrt[34])]-26660250 (Sqrt[34]) Log[6881+1180 (Sqrt[34])])+(1/203751936)(2918900 (Sqrt[2])+4140000 (Sqrt[17]) ArcCoth[4]-1061208 ArcSinh[1]-1158750 (Sqrt[17]) ArcTanh[((8 (Sqrt[34]))/49)]+1845000 (Sqrt[17]) ArcTanh[(4/33) (-1+(Sqrt[34]))]-707472 (Sqrt[2]) Log[2]+1944000 (Sqrt[17]) Log[3]-2070000 (Sqrt[17]) Log[5]+63000 (Sqrt[17]) Log[(3125/27)]-5625 (Sqrt[17]) Log[34-5 (Sqrt[34])]-5625 (Sqrt[17]) Log[34-3 (Sqrt[34])]-243000 (Sqrt[17]) Log[-5+(Sqrt[34])]+5625 (Sqrt[17]) Log[34+3 (Sqrt[34])]+5625 (Sqrt[17]) Log[34+5 (Sqrt[34])]+121500 (Sqrt[17]) Log[(1/729) (34119931+5851502 (Sqrt[34]))]));

N[rxy,20]

FullSimplify[rxy]//Timing

N[rxy=-((ArcSinh[1])/(12 (Sqrt[2])))+(1/13085402112)(16184000 (Sqrt[34]) ArcTanh[(5/(Sqrt[34]))]+2537250 (Sqrt[34]) Log[2]-514027385 (Sqrt[34]) Log[3]-422649125 (Sqrt[34]) Log[5]-425000 (Sqrt[34]) Log[17]+3901500 (Sqrt[34]) Log[3073-516 (Sqrt[34])]+118118125 (Sqrt[34]) Log[49-8 (Sqrt[34])]+7497000 (Sqrt[34]) Log[34-3 (Sqrt[34])]-67458125 (Sqrt[34]) Log[8-(Sqrt[34])]+95244880 (Sqrt[34]) Log[-5+(Sqrt[34])]-11261750 (Sqrt[34]) Log[-3+(Sqrt[34])]+59245000 (Sqrt[34]) Log[2+(Sqrt[34])]+74484875 (Sqrt[34]) Log[3+(Sqrt[34])]+111776005 (Sqrt[34]) Log[5+(Sqrt[34])]+8213125 (Sqrt[34]) Log[8+(Sqrt[34])]-6647000 (Sqrt[34]) Log[34+3 (Sqrt[34])]-4913000 (Sqrt[34]) Log[43+6 (Sqrt[34])]+183308125 (Sqrt[34]) Log[49+8 (Sqrt[34])]-4326500 (Sqrt[34]) Log[59+10 (Sqrt[34])]-578 (-5141888+5125 (Sqrt[34]) Log[342+36 (Sqrt[34])]-27000 (Sqrt[34]) Log[34119931+5851502 (Sqrt[34])]+10250 (Sqrt[34]) Log[41847233+7176736 (Sqrt[34])])),10]

Table[(((6+(Sqrt[34])))^n),{n,5}]//Expand

Log[8+(Sqrt[34]),#]&/@{3073-516 (Sqrt[34]),342+36 (Sqrt[34]),34119931+5851502 (Sqrt[34]),41847233+7176736 (Sqrt[34]),59+10 (Sqrt[34]),49+8 (Sqrt[34]),43+6 (Sqrt[34])}//N

Log[3+(Sqrt[34]),#]&/@{3073-516 (Sqrt[34]),342+36 (Sqrt[34]),34119931+5851502 (Sqrt[34]),41847233+7176736 (Sqrt[34]),59+10 (Sqrt[34]),49+8 (Sqrt[34]),43+6 (Sqrt[34])}//N

N[Subtract@@{rxy,-((ArcSinh[1])/(12 (Sqrt[2])))+(1/13085402112)(16184000 (Sqrt[34]) ArcTanh[(5/(Sqrt[34]))]+2537250 (Sqrt[34]) Log[2]-514027385 (Sqrt[34]) Log[3]-422649125 (Sqrt[34]) Log[5]-425000 (Sqrt[34]) Log[17]+3901500 (Sqrt[34]) Log[3073-516 (Sqrt[34])]+118118125 (Sqrt[34]) Log[49-8 (Sqrt[34])]+7497000 (Sqrt[34]) Log[34-3 (Sqrt[34])]-67458125 (Sqrt[34]) Log[8-(Sqrt[34])]+95244880 (Sqrt[34]) Log[-5+(Sqrt[34])]-11261750 (Sqrt[34]) Log[-3+(Sqrt[34])]+59245000 (Sqrt[34]) Log[2+(Sqrt[34])]+74484875 (Sqrt[34]) Log[3+(Sqrt[34])]+111776005 (Sqrt[34]) Log[5+(Sqrt[34])]+8213125 (Sqrt[34]) Log[8+(Sqrt[34])]-6647000 (Sqrt[34]) Log[34+3 (Sqrt[34])]-4913000 2(Sqrt[34]) Log[3+(Sqrt[34])]+183308125 (Sqrt[34]) Log[49+8 (Sqrt[34])]-4326500 (Sqrt[34]) Log[59+10 (Sqrt[34])]-578 (-5141888+5125 (Sqrt[34]) Log[342+36 (Sqrt[34])]-27000 (Sqrt[34]) Log[34119931+5851502 (Sqrt[34])]+10250 (Sqrt[34]) Log[41847233+7176736 (Sqrt[34])]))},10]//Quiet

Log[5+(Sqrt[34]),#]&/@{3073-516 (Sqrt[34]),342+36 (Sqrt[34]),34119931+5851502 (Sqrt[34]),41847233+7176736 (Sqrt[34]),59+10 (Sqrt[34]),49+8 (Sqrt[34])}//N

N[Subtract@@{rxy,-((ArcSinh[1])/(12 (Sqrt[2])))+(1/13085402112)(16184000 (Sqrt[34]) ArcTanh[(5/(Sqrt[34]))]+2537250 (Sqrt[34]) Log[2]-514027385 (Sqrt[34]) Log[3]-422649125 (Sqrt[34]) Log[5]-425000 (Sqrt[34]) Log[17]+3901500 (Sqrt[34]) Log[3073-516 (Sqrt[34])]+118118125 (Sqrt[34]) Log[49-8 (Sqrt[34])]+7497000 (Sqrt[34]) Log[34-3 (Sqrt[34])]-67458125 (Sqrt[34]) Log[8-(Sqrt[34])]+95244880 (Sqrt[34]) Log[-5+(Sqrt[34])]-11261750 (Sqrt[34]) Log[-3+(Sqrt[34])]+59245000 (Sqrt[34]) Log[2+(Sqrt[34])]+74484875 (Sqrt[34]) Log[3+(Sqrt[34])]+111776005 (Sqrt[34]) Log[5+(Sqrt[34])]+8213125 (Sqrt[34]) Log[8+(Sqrt[34])]-6647000 (Sqrt[34]) Log[34+3 (Sqrt[34])]-4913000 2(Sqrt[34]) Log[3+(Sqrt[34])]+183308125 (Sqrt[34]) Log[49+8 (Sqrt[34])]-4326500  2(Sqrt[34]) Log[5+(Sqrt[34])]-578 (-5141888+5125 (Sqrt[34]) Log[342+36 (Sqrt[34])]-27000 (Sqrt[34]) Log[34119931+5851502 (Sqrt[34])]+10250 (Sqrt[34]) Log[41847233+7176736 (Sqrt[34])]))},10]//Quiet

N[Subtract@@{rxy,-((ArcSinh[1])/(12 (Sqrt[2])))+(1/13085402112)(16184000 (Sqrt[34]) ArcTanh[(5/(Sqrt[34]))]+2537250 (Sqrt[34]) Log[2]-514027385 (Sqrt[34]) Log[3]-422649125 (Sqrt[34]) Log[5]-425000 (Sqrt[34]) Log[17]+3901500 (Sqrt[34]) Log[3073-516 (Sqrt[34])]+118118125 (Sqrt[34]) Log[49-8 (Sqrt[34])]+7497000 (Sqrt[34]) Log[34-3 (Sqrt[34])]-67458125 (Sqrt[34]) Log[8-(Sqrt[34])]+95244880 (Sqrt[34]) Log[-5+(Sqrt[34])]-11261750 (Sqrt[34]) Log[-3+(Sqrt[34])]+59245000 (Sqrt[34]) Log[2+(Sqrt[34])]+74484875 (Sqrt[34]) Log[3+(Sqrt[34])]+111776005 (Sqrt[34]) Log[5+(Sqrt[34])]+8213125 (Sqrt[34]) Log[8+(Sqrt[34])]-6647000 (Sqrt[34]) Log[34+3 (Sqrt[34])]-4913000 2(Sqrt[34]) Log[3+(Sqrt[34])]+183308125 (Sqrt[34]) Log[49+8 (Sqrt[34])]-4326500  2(Sqrt[34]) Log[5+(Sqrt[34])]-578 (-5141888+5125 (Sqrt[34]) Log[342+36 (Sqrt[34])]-27000 (Sqrt[34]) Log[34119931+5851502 (Sqrt[34])]+10250 (Sqrt[34]) Log[41847233+7176736 (Sqrt[34])]))},10]//Quiet

TrigToExp[ArcTanh[(5/(Sqrt[34]))]]

N[Subtract@@{rxy,-((ArcSinh[1])/(12 (Sqrt[2])))+(1/13085402112)(16184000 (Sqrt[34]) (-(1/2) (Log[(Sqrt[34])-5]-(1/2)Log[34])+(1/2) (Log[(Sqrt[34])+5]-(1/2)Log[34]))+2537250 (Sqrt[34]) Log[2]-514027385 (Sqrt[34]) Log[3]-422649125 (Sqrt[34]) Log[5]-425000 (Sqrt[34]) Log[17]+3901500 (Sqrt[34]) 4Log[-3+(Sqrt[34])]+118118125 (Sqrt[34]) Log[49-8 (Sqrt[34])]+7497000 (Sqrt[34]) Log[34-3 (Sqrt[34])]-67458125 (Sqrt[34]) Log[8-(Sqrt[34])]+95244880 (Sqrt[34]) Log[-5+(Sqrt[34])]-11261750 (Sqrt[34]) Log[-3+(Sqrt[34])]+59245000 (Sqrt[34]) Log[2+(Sqrt[34])]+74484875 (Sqrt[34]) Log[3+(Sqrt[34])]+111776005 (Sqrt[34]) Log[5+(Sqrt[34])]+8213125 (Sqrt[34]) Log[8+(Sqrt[34])]-6647000 (Sqrt[34]) Log[34+3 (Sqrt[34])]-4913000 2(Sqrt[34]) Log[3+(Sqrt[34])]+183308125 (Sqrt[34]) Log[49+8 (Sqrt[34])]-4326500  2(Sqrt[34]) Log[5+(Sqrt[34])]-578 (-5141888+5125 2 (Sqrt[34]) (Log[3]+Log[2+(Sqrt[34])])-27000 (Sqrt[34]) Log[34119931+5851502 (Sqrt[34])]+10250 (Sqrt[34]) Log[41847233+7176736 (Sqrt[34])]))},10]//Quiet

Expand[-((ArcSinh[1])/(12 (Sqrt[2])))+(1/13085402112)(16184000 (Sqrt[34]) (-(1/2) (Log[(Sqrt[34])-5]-(1/2)Log[34])+(1/2) (Log[(Sqrt[34])+5]-(1/2)Log[34]))+2537250 (Sqrt[34]) Log[2]-514027385 (Sqrt[34]) Log[3]-422649125 (Sqrt[34]) Log[5]-425000 (Sqrt[34]) Log[17]+3901500 (Sqrt[34]) 4Log[-3+(Sqrt[34])]+118118125 (Sqrt[34]) Log[49-8 (Sqrt[34])]+7497000 (Sqrt[34]) Log[34-3 (Sqrt[34])]-67458125 (Sqrt[34]) Log[8-(Sqrt[34])]+95244880 (Sqrt[34]) Log[-5+(Sqrt[34])]-11261750 (Sqrt[34]) Log[-3+(Sqrt[34])]+59245000 (Sqrt[34]) Log[2+(Sqrt[34])]+74484875 (Sqrt[34]) Log[3+(Sqrt[34])]+111776005 (Sqrt[34]) Log[5+(Sqrt[34])]+8213125 (Sqrt[34]) Log[8+(Sqrt[34])]-6647000 (Sqrt[34]) Log[34+3 (Sqrt[34])]-4913000 2(Sqrt[34]) Log[3+(Sqrt[34])]+183308125 (Sqrt[34]) Log[49+8 (Sqrt[34])]-4326500  2(Sqrt[34]) Log[5+(Sqrt[34])]-578 (-5141888+5125 2 (Sqrt[34]) (Log[3]+Log[2+(Sqrt[34])])-27000 (Sqrt[34]) Log[34119931+5851502 (Sqrt[34])]+10250 (Sqrt[34]) Log[41847233+7176736 (Sqrt[34])]))]

N[Subtract@@{rxy,(139/612)-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))+((24875 Log[2])/(3773184 (Sqrt[34])))-((10195135 Log[3])/(7546368 (Sqrt[34])))-((422649125 Log[5])/(384864768 (Sqrt[34])))-((3125 Log[17])/(2829888 (Sqrt[34])))+((6948125 Log[49-8 (Sqrt[34])])/(22639104 (Sqrt[34])))+((6125 Log[34-3 (Sqrt[34])])/(314432 (Sqrt[34])))-((3968125 Log[8-(Sqrt[34])])/(22639104 (Sqrt[34])))+((106805 Log[-5+(Sqrt[34])])/(471648 (Sqrt[34])))+((2172125 Log[-3+(Sqrt[34])])/(192432384 (Sqrt[34])))+((5125 Log[2+(Sqrt[34])])/(36992 (Sqrt[34])))+((64658875 Log[3+(Sqrt[34])])/(384864768 (Sqrt[34])))+((111215005 Log[5+(Sqrt[34])])/(384864768 (Sqrt[34])))+((483125 Log[8+(Sqrt[34])])/(22639104 (Sqrt[34])))-((2875 Log[34+3 (Sqrt[34])])/(166464 (Sqrt[34])))+((183308125 Log[49+8 (Sqrt[34])])/(384864768 (Sqrt[34])))+((375 Log[34119931+5851502 (Sqrt[34])])/(9248 (Sqrt[34])))-((5125 Log[41847233+7176736 (Sqrt[34])])/(332928 (Sqrt[34])))},10]//Quiet

Cases[Flatten[Table[{{a,b,n},N[(((a+b(Sqrt[34])))^n)- (34119931+5851502 (Sqrt[34])),50]//Chop},{n,30},{a,-30,30},{b,-30,30}],2],{_,0}]//Timing

Cases[Flatten[Table[{{a,b,n},N[(((a+b(Sqrt[34])))^n)- (41847233+7176736 (Sqrt[34])),50]//Chop},{n,30},{a,-30,30},{b,-30,30}],2],{_,0}]//Timing

NIntegrate[(1/(3 (Sqrt[2])))((1/2) (Sqrt[((((5/6)-((5 s)/6)-((5 t)/6)))^2)+((((s/2)-(t/2)))^2)]) ((5/6)-((5 s)/6)-((5 t)/6))-(1/2) ((((s/2)-(t/2)))^2) Log[-(5/6)+((5 s)/6)+(Sqrt[((((5/6)-((5 s)/6)-((5 t)/6)))^2)+((((s/2)-(t/2)))^2)])+((5 t)/6)]),{t,0,1},{s,0,1-t},WorkingPrecision->20,PrecisionGoal->10]

rxy=(139/612)-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))+((24875 Log[2])/(3773184 (Sqrt[34])))-((10195135 Log[3])/(7546368 (Sqrt[34])))-((422649125 Log[5])/(384864768 (Sqrt[34])))-((3125 Log[17])/(2829888 (Sqrt[34])))+((6948125 Log[49-8 (Sqrt[34])])/(22639104 (Sqrt[34])))+((6125 Log[34-3 (Sqrt[34])])/(314432 (Sqrt[34])))-((3968125 Log[8-(Sqrt[34])])/(22639104 (Sqrt[34])))+((106805 Log[-5+(Sqrt[34])])/(471648 (Sqrt[34])))+((2172125 Log[-3+(Sqrt[34])])/(192432384 (Sqrt[34])))+((5125 Log[2+(Sqrt[34])])/(36992 (Sqrt[34])))+((64658875 Log[3+(Sqrt[34])])/(384864768 (Sqrt[34])))+((111215005 Log[5+(Sqrt[34])])/(384864768 (Sqrt[34])))+((483125 Log[8+(Sqrt[34])])/(22639104 (Sqrt[34])))-((2875 Log[34+3 (Sqrt[34])])/(166464 (Sqrt[34])))+((183308125 Log[49+8 (Sqrt[34])])/(384864768 (Sqrt[34])))+((375 Log[34119931+5851502 (Sqrt[34])])/(9248 (Sqrt[34])))-((5125 Log[41847233+7176736 (Sqrt[34])])/(332928 (Sqrt[34])));

N[rxy,20]

Select[Flatten[Table[{u+v Sqrt[34],N[Log[u+v Sqrt[34],41847233+7176736 Sqrt[34]],20]},{u,-100,100},{v,-100,100}],1],Im[#[[2]]]==0&&#[[2]]-Round[#[[2]]]==0&]//Quiet

From mtrott:

Log[34119931+5851502 Sqrt[34]]
==5 Log[3+Sqrt[34]]+3 Log[5+Sqrt[34]]
==2 Log[3+Sqrt[34]]+3 Log[49+8 Sqrt[34]]

5 Log[3+Sqrt[34]]+3 Log[5+Sqrt[34]]==2 Log[3+Sqrt[34]]+3 Log[49+8 Sqrt[34]]

Log[49+8 Sqrt[34]]==((5 Log[3+Sqrt[34]]+3 Log[5+Sqrt[34]]-2 Log[3+Sqrt[34]])/3)//Expand

Log[(3+(Sqrt[34]))(5+(Sqrt[34]))//Expand]

N[%]

Log[49-8 (Sqrt[34])]//TrigToExp

N[rxy=(139/612)-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))+((24875 Log[2])/(3773184 (Sqrt[34])))-((10195135 Log[3])/(7546368 (Sqrt[34])))-((422649125 Log[5])/(384864768 (Sqrt[34])))-((3125 Log[17])/(2829888 (Sqrt[34])))+((6948125 Log[49-8 (Sqrt[34])])/(22639104 (Sqrt[34])))+((6125 Log[34-3 (Sqrt[34])])/(314432 (Sqrt[34])))-((3968125 Log[8-(Sqrt[34])])/(22639104 (Sqrt[34])))+((106805 Log[-5+(Sqrt[34])])/(471648 (Sqrt[34])))+((2172125 Log[-3+(Sqrt[34])])/(192432384 (Sqrt[34])))+((5125 Log[2+(Sqrt[34])])/(36992 (Sqrt[34])))+((64658875 Log[3+(Sqrt[34])])/(384864768 (Sqrt[34])))+((111215005 Log[5+(Sqrt[34])])/(384864768 (Sqrt[34])))+((483125 Log[8+(Sqrt[34])])/(22639104 (Sqrt[34])))-((2875 Log[34+3 (Sqrt[34])])/(166464 (Sqrt[34])))+((183308125 (1/3) (3 Log[3+(Sqrt[34])]+3 Log[5+(Sqrt[34])]))/(384864768 (Sqrt[34])))+((375 (5 Log[3+Sqrt[34]]+3 Log[5+Sqrt[34]]))/(9248 (Sqrt[34])))-((5125 4 (1/3) (3 Log[3+(Sqrt[34])]+3 Log[5+(Sqrt[34])]))/(332928 (Sqrt[34]))),20]

Expand[rxy]

Log[(3-(Sqrt[34]))(5-(Sqrt[34]))//Expand]

N[rxy=(139/612)+((24875 Log[2])/(3773184 (Sqrt[34])))-((10195135 Log[3])/(7546368 (Sqrt[34])))-((422649125 Log[5])/(384864768 (Sqrt[34])))-((3125 Log[17])/(2829888 (Sqrt[34])))-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))+((6948125 (Log[-3+(Sqrt[34])]+Log[-5+(Sqrt[34])]))/(22639104 (Sqrt[34])))+((6125 Log[34-3 (Sqrt[34])])/(314432 (Sqrt[34])))-((3968125 Log[8-(Sqrt[34])])/(22639104 (Sqrt[34])))+((106805 Log[-5+(Sqrt[34])])/(471648 (Sqrt[34])))+((2172125 Log[-3+(Sqrt[34])])/(192432384 (Sqrt[34])))+((5125 Log[2+(Sqrt[34])])/(36992 (Sqrt[34])))+((37787375 Log[3+(Sqrt[34])])/(48108096 (Sqrt[34])))+((9342445 Log[5+(Sqrt[34])])/(11319552 (Sqrt[34])))+((483125 Log[8+(Sqrt[34])])/(22639104 (Sqrt[34])))-((2875 Log[34+3 (Sqrt[34])])/(166464 (Sqrt[34]))),20]

Expand[rxy]

Log[34+3 (Sqrt[34])]//N

Log[(Sqrt[34])((Sqrt[34])-3)]//N

Log[34]/2+Log[3+(Sqrt[34])]//N

Log[34]/2+Log[(Sqrt[34])-3]//N

N[rxy=(139/612)+((24875 Log[2])/(3773184 (Sqrt[34])))-((10195135 Log[3])/(7546368 (Sqrt[34])))-((422649125 Log[5])/(384864768 (Sqrt[34])))-((3125 Log[17])/(2829888 (Sqrt[34])))-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))+((6948125 (Log[-3+(Sqrt[34])]+Log[-5+(Sqrt[34])]))/(22639104 (Sqrt[34])))+((6125 (Log[34]/2+Log[(Sqrt[34])-3]))/(314432 (Sqrt[34])))-((3968125 Log[8-(Sqrt[34])])/(22639104 (Sqrt[34])))+((106805 Log[-5+(Sqrt[34])])/(471648 (Sqrt[34])))+((2172125 Log[-3+(Sqrt[34])])/(192432384 (Sqrt[34])))+((5125 Log[2+(Sqrt[34])])/(36992 (Sqrt[34])))+((37787375 Log[3+(Sqrt[34])])/(48108096 (Sqrt[34])))+((9342445 Log[5+(Sqrt[34])])/(11319552 (Sqrt[34])))+((483125 Log[8+(Sqrt[34])])/(22639104 (Sqrt[34])))-((2875(Log[34]/2+Log[3+(Sqrt[34])]))/(166464 (Sqrt[34]))),20]

Expand[rxy]

ByteCount[(139/612)+((24875 Log[2])/(3773184 (Sqrt[34])))-((10195135 Log[3])/(7546368 (Sqrt[34])))-((422649125 Log[5])/(384864768 (Sqrt[34])))-((3125 Log[17])/(2829888 (Sqrt[34])))+((3125 Log[34])/(2829888 (Sqrt[34])))-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))-((3968125 Log[8-(Sqrt[34])])/(22639104 (Sqrt[34])))+((12074765 Log[-5+(Sqrt[34])])/(22639104 (Sqrt[34])))+((129959375 Log[-3+(Sqrt[34])])/(384864768 (Sqrt[34])))+((5125 Log[2+(Sqrt[34])])/(36992 (Sqrt[34])))+((9239125 Log[3+(Sqrt[34])])/(12027024 (Sqrt[34])))+((9342445 Log[5+(Sqrt[34])])/(11319552 (Sqrt[34])))+((483125 Log[8+(Sqrt[34])])/(22639104 (Sqrt[34])))]

rxy=(139/612)-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))+(Sqrt[34])Expand[(1/34)(((24875 Log[2])/(3773184 ))-((10195135 Log[3])/(7546368 ))-((422649125 Log[5])/(384864768 ))-((3125 Log[17])/(2829888 ))+((3125 Log[34])/(2829888 ))-((3968125 Log[8-(Sqrt[34])])/(22639104 ))+((12074765 Log[-5+(Sqrt[34])])/(22639104 ))+((129959375 Log[-3+(Sqrt[34])])/(384864768 ))+((5125 Log[2+(Sqrt[34])])/36992)+((9239125 Log[3+(Sqrt[34])])/(12027024 ))+((9342445 Log[5+(Sqrt[34])])/(11319552 ))+((483125 Log[8+(Sqrt[34])])/(22639104 )))]

N[rxy=(139/612)-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))+(Sqrt[34]) (((24875 Log[2])/128288256)-((10195135 Log[3])/256576512)-((422649125 Log[5])/13085402112)-((3125 Log[17])/96216192)+((3125 Log[34])/96216192)-((3968125 Log[8-(Sqrt[34])])/769729536)+((12074765 Log[-5+(Sqrt[34])])/769729536)+((129959375 Log[-3+(Sqrt[34])])/13085402112)+((5125 Log[2+(Sqrt[34])])/1257728)+((9239125 Log[3+(Sqrt[34])])/408918816)+((9342445 Log[5+(Sqrt[34])])/384864768)+((483125 Log[8+(Sqrt[34])])/769729536)),20]

N[rxy,20]

ByteCount[rxy]

FullSimplify[rxy]

ByteCount[%]

N[rxy=(139/612)-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))+(Sqrt[34]) (((24875 Log[2])/128288256)-((10195135 Log[3])/256576512)-((422649125 Log[5])/13085402112)-((3125 Log[17])/96216192)+((3125 (Log[2]+Log[17]))/96216192)-((3968125 Log[8-(Sqrt[34])])/769729536)+((12074765 Log[-5+(Sqrt[34])])/769729536)+((129959375 Log[-3+(Sqrt[34])])/13085402112)+((5125 Log[2+(Sqrt[34])])/1257728)+((9239125 Log[3+(Sqrt[34])])/408918816)+((9342445 Log[5+(Sqrt[34])])/384864768)+((483125 Log[8+(Sqrt[34])])/769729536)),20]

Expand[(((24875 Log[2])/128288256)-((10195135 Log[3])/256576512)-((422649125 Log[5])/13085402112)-((3125 Log[17])/96216192)+((3125 (Log[2]+Log[17]))/96216192)-((3968125 Log[8-(Sqrt[34])])/769729536)+((12074765 Log[-5+(Sqrt[34])])/769729536)+((129959375 Log[-3+(Sqrt[34])])/13085402112)+((5125 Log[2+(Sqrt[34])])/1257728)+((9239125 Log[3+(Sqrt[34])])/408918816)+((9342445 Log[5+(Sqrt[34])])/384864768)+((483125 Log[8+(Sqrt[34])])/769729536))]

N[rxy=(139/612)-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))+(Sqrt[34])( ((5125 Log[2])/22639104)-((10195135 Log[3])/256576512)-((422649125 Log[5])/13085402112)-((3968125 Log[8-(Sqrt[34])])/769729536)+((12074765 Log[-5+(Sqrt[34])])/769729536)+((129959375 Log[-3+(Sqrt[34])])/13085402112)+((5125 Log[2+(Sqrt[34])])/1257728)+((9239125 Log[3+(Sqrt[34])])/408918816)+((9342445 Log[5+(Sqrt[34])])/384864768)+((483125 Log[8+(Sqrt[34])])/769729536)),20]

logs=Cases[rxy,Log[_Plus],∞]

c=FindIntegerNullVector[logs]

c.logs

FullSimplify[%]

Log[8-(Sqrt[34])]==(1/2)(Log[-5+(Sqrt[34])]+2 Log[2+(Sqrt[34])]-Log[5+(Sqrt[34])])//FullSimplify

N[rxy=(139/612)-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))+(Sqrt[34])( ((5125 Log[2])/22639104)-((10195135 Log[3])/256576512)-((422649125 Log[5])/13085402112)-((3968125(1/2)(Log[-5+(Sqrt[34])]+2 Log[2+(Sqrt[34])]-Log[5+(Sqrt[34])]))/769729536)+((12074765 Log[-5+(Sqrt[34])])/769729536)+((129959375 Log[-3+(Sqrt[34])])/13085402112)+((5125 Log[2+(Sqrt[34])])/1257728)+((9239125 Log[3+(Sqrt[34])])/408918816)+((9342445 Log[5+(Sqrt[34])])/384864768)+((483125 Log[8+(Sqrt[34])])/769729536)),20]

rxy=Expand[rxy]

logs=Cases[rxy,Log[_Plus],∞]

c=FindIntegerNullVector[logs]

c.logs

Log[8+(Sqrt[34])]==(1/2)(-Log[-3+(Sqrt[34])]+2 Log[2+(Sqrt[34])]+Log[3+(Sqrt[34])])//FullSimplify

N[rxy=(139/612)+((5125 Log[2])/(665856 (Sqrt[34])))-((10195135 Log[3])/(7546368 (Sqrt[34])))-((422649125 Log[5])/(384864768 (Sqrt[34])))-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))+((6727135 Log[-5+(Sqrt[34])])/(15092736 (Sqrt[34])))+((129959375 Log[-3+(Sqrt[34])])/(384864768 (Sqrt[34])))-((831625 Log[2+(Sqrt[34])])/(22639104 (Sqrt[34])))+((9239125 Log[3+(Sqrt[34])])/(12027024 (Sqrt[34])))+((41337905 Log[5+(Sqrt[34])])/(45278208 (Sqrt[34])))+((483125 (1/2)(-Log[-3+(Sqrt[34])]+2 Log[2+(Sqrt[34])]+Log[3+(Sqrt[34])]))/(22639104 (Sqrt[34]))),20]

rxy=Expand[rxy]

rxy=(139/612)+((5125 Log[2])/(665856 (Sqrt[34])))-((10195135 Log[3])/(7546368 (Sqrt[34])))-((422649125 Log[5])/(384864768 (Sqrt[34])))-((Log[1+(Sqrt[2])])/(12 (Sqrt[2])))+((6727135 Log[-5+(Sqrt[34])])/(15092736 (Sqrt[34])))+((83901875 Log[-3+(Sqrt[34])])/(256576512 (Sqrt[34])))-((5125 Log[2+(Sqrt[34])])/(332928 (Sqrt[34])))+((599517125 Log[3+(Sqrt[34])])/(769729536 (Sqrt[34])))+((41337905 Log[5+(Sqrt[34])])/(45278208 (Sqrt[34])))

N[rxy,20]

logs=Cases[rxy,Log[_Plus],∞]

c=FindIntegerNullVector[logs]

CAD

cad=CylindricalDecomposition[PolyhedronData[pname,"RegionFunction"][x,y,z],{x,y,z},"Function"]

((NIntegrate[(Sqrt[(x^2)+(y^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol)

((Integrate[(Sqrt[(x^2)+(y^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol)//Timing

Boole Integrate doing z integral first

{{xmin,xmax},{ymin,ymax},{zmin,zmax}}==CoordinateBounds[p]

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]

(i1=Integrate[Boole[ineq](Sqrt[(x^2)+(y^2)]),{z,zmin,zmax}])//Timing

NIntegrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}]

(i2=Integrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}])//Timing

MeanSquareCylindricalRadius

PolyhedronData[pname,"MeanSquareCylindricalRadius"]

N[%]

Integrate over Polyhedron

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

((NIntegrate[(x^2)+(y^2),{x,y,z}∈p])/vol)//Timing

Divergence theorem

(rxy=MeanSquareCylindricalRadius[p,Debug->True])//FullSimplify//Timing

MeanSphericalRadius

PolyhedronData[pname,"MeanSphericalRadius"]

Integrate over polyhedron

((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/vol)

(rxy=((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)]),{x,y,z}∈p])/vol))//Timing

Divergence theorem [unevaluated integral]

(rxyz=MeanSphericalRadius[p,"Simplification"->FullSimplify,Debug->True])//Timing

FullSimplify[rxyz]//Timing

CAD [probably crashes]

cad=CylindricalDecomposition[PolyhedronData[pname,"RegionFunction"][x,y,z],{x,y,z},"Function"]

((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol)

((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol)//Timing

CAD over 1/4

reg=FullSimplify[PolyhedronData[pname,"RegionFunction"][x,y,z]&&x>0&&y>0]

cad=CylindricalDecomposition[reg,{x,y,z},"Function"]

4((NIntegrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol)

4((Integrate[(Sqrt[(x^2)+(y^2)+(z^2)])Boole[cad],{x,y,z}∈FullRegion[3]])/vol)//Timing

Boole Integrate doing z integral first

{{xmin,xmax},{ymin,ymax},{zmin,zmax}}=CoordinateBounds[p]

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]

(i1=Integrate[Boole[ineq](Sqrt[(x^2)+(y^2)+(z^2)]),{z,zmin,zmax}])//Timing

NIntegrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}]

(i2=Integrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}])//Timing

MeanSquareSphericalRadius

PolyhedronData[pname,"MeanSquareSphericalRadius"]

((NIntegrate[(x^2)+(y^2)+(z^2),{x,y,z}∈p])/vol)//Timing

Integrate

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

Divergence theorem

(rxy=MeanSquareSphericalRadius[p,Debug->True])//FullSimplify//Timing

Boole Integrate doing z integral first

{{xmin,xmax},{ymin,ymax},{zmin,zmax}}==CoordinateBounds[p]

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]

(i1=Integrate[Boole[ineq]((x^2)+(y^2)+(z^2)),{z,zmin,zmax}])//Timing

NIntegrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}]

(i2=Integrate[(i1/vol),{x,xmin,xmax},{y,ymin,ymax}])//Timing

Midsphere

sphere=PolyhedronData[pname,"Midsphere"]

(sphere=Midsphere[p])//Timing

(sphere=Midsphere[p,Method->"PointLineDistance"])//Timing

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

Net

PolyhedronData[pname,"Net"]

Total[Area/@Flatten[PolyhedronData[pname,"Net","Polygon"]]]//ToAlgebraicRoot//Timing

PolyhedronData[pname,"SurfaceArea"]

Polygons

PolyhedronData[pname,"FaceCount"]

Graphics3D/@PolyhedronData[pname,"Polygons"]

Show[%]

Polyhedra

PolyhedronData[pname,"Polyhedra"]

Graphics3D[%]

PolyhedronData[pname,"PolyhedronIndices"]

polygons=PolyhedronData[pname,"Polygons"]

Complement[Range[16],{1,2,3,4,5,6,11,16}]

Graphics3D/@{polygons[[{1,2,3,4,5,6,11,16}]],polygons[[{7,8,9,10,12,13,14,15}]]}

Region

PolyhedronData[pname,"Region"]

RegionFunction

ineq=PolyhedronData[pname,"RegionFunction"][x,y,z]

reg=ImplicitRegion[ineq,{x,y,z}]

Region[reg]

Volume[%]

tet=Lookup[PolyhedronData[pname,"VertexSubsetHulls"],"Tetrahedron"]

tets=ConvexHullRegion[PolyhedronData[pname,"VertexCoordinates"][[#]]]&/@tet

Graphics3D/@tets

ineq=FullSimplify[Or@@(PolyhedronInequalities[#,{x,y,z}]&/@tets)]

reg=ImplicitRegion[ineq,{x,y,z}]

Region[reg]

Volume[reg]

ineq/.{x->#,y->#2,z->#3}

SimplePolyhedron

PolyhedronData[pname,"Simple"]

SimplePolyhedronQ[p]

Skeleton

PolyhedronData[pname,"Skeleton"]

PolyhedronData[pname,"Skeleton","Name"]

With[{gn="TritetrahedralGraph",verts=PolyhedronData[pname,"Vertices"],faces=PolyhedronData[pname,"Faces","List"]},
ToCommonEdges[GraphData[gn],Graph[Range[Length[verts]],UndirectedEdge@@@Union[Sort/@Flatten[Partition[#,2,1,1]&/@faces,1]]]]]//Chop[#,1*^-5]&//N

SurfaceArea

PolyhedronData[pname,"SurfaceArea"]

SurfaceArea[p]//RootReduce//Timing

Total[Area/@PolyhedronData[pname,"Polygons"]]//RootReduce//Timing

VertexCount

PolyhedronData[pname,"VertexCount"]

VertexSubsetHulls

subsethulls=PolyhedronData[pname,"VertexSubsetHulls"]

subsethulls2=Flatten[Function[{pn,l},Rule[pn,#]&/@l]@@@subsethulls]

v=PolyhedronData[pname,"Vertices"];

Rule[#,Graphics3D[{{Green,p},Red,Opacity[.5],ConvexHullRegion[v[[#2]]]}]]&@@@subsethulls2

Platonic

(h=VertexSubsetHulls[p,"Platonic","Skip"->{}])//Timing

Volume

PolyhedronData[pname,"Volume"]

Volume[p]//FullSimplify

PolyhedronVolume[p]//FullSimplify

Volume[PolyhedronData[pname,"Region"]]

Volume[ImplicitRegion[PolyhedronData[pname,"RegionFunction"][x,y,z],{x,y,z}]]

Net construction

s=Sqrt[3]/2;
v={{0,0},{1,0},{2,0},{3,0},{1/2,s},{3/2,s},{5/2,s},{0,2s},{1,2s},{5/2,-s}}

faces=Sort[{{1,2,5},{2,6,5},{2,3,6},{3,7,6},{3,4,7},{5,9,8},{5,6,9},{3,10,4}}]

Graphics[{FaceForm[LightYellow],EdgeForm[Black],Polygon[v,faces],Point[v],MapIndexed[Text[#2[[1]],#,-1.5{1,1},Background->White]&,v]}]