Known Dehn Invariant Zero Tetrahedra

Known Dehn Invariant Zero Last updated June 2022

We present a list of tetrahedra that are known to the author as of the last update of this webpage. In particular, we list the sextuple of dihedral edges in the order (12,34,13,24,14,23). The same goes for edge lengths. If you are aware of Dehn invariant zero tetrahedra that are not listed in this webpage, please share them with me myself (preferably the dihedral angles) and I will update this webpage and acknowledge your contribution.

First Hill Family

This family is parametrized by a single parameter α. For brevity, we denote s=sin⁡(α),c=cos⁡(α). Dihedral AnglesEdge Lengths(α,α,π3,π−2α,π2,π2)(s,s,s,3c,1,1)(−α+2π3,−α+2π3,α+π6,−α+5π6,α,α)(s,s,s−3c2+1,s−3c2+1,s+3c2,s+3c2)(−α+2π3,−α+2π3,2α−π3,π3,π2,π2)(s+3c2,s+3c2,3s−3c2,s+3c2,1,1) Colored in red is the orbit element commonly used to define the family in literature.

Second Hill Family

This family is parametrized by a single realparameter α. For brevity, we denote s=sin⁡(α),c=cos⁡(α) and β=cos−1⁡(12cot⁡α). Dihedral AnglesEdge Lengths(α,β,−α+π2,π3,π−β,π2)(2s,f,2s,3c,f,2)(β,α,α2−β2+2π3,−α2−β2+5π6,−α2−β2+2π3,−α2+β2+π6)(2s,f,f2+s−3c2+1,f2−s+3c2+1,f2+s+3c2−1,−f2+s+3c2+1)(α2−β+3π4,−α2+3π4,α+β2−π12,β2+π12,−α2+β2+π6,−α2−β2+2π3)(f−s+1,s+1,f2+2s−3c2,f2+3c2,−f2+s+3c2+1,f2+s+3c2−1)(−α2+3π4,α2−β+3π4,−α+π2,π3,α2+β−π4,α2+π4)(s+1,f−s+1,2s,3c,f+s−1,s+1)(−β2+5π12,−α+β2+5π12,β2+π12,α+β2−π12,π−β,π2)(f2+3c2,−f2+2s+3c2,f2+3c2,f2+2s−3c2,f,2)(−β2+5π12,−α+β2+5π12,α2−β2+2π3,−α2−β2+5π6,α2+β−π4,α2+π4)(−f2+2s+3c2,f2+3c2,f2+s−3c2+1,f2−s+3c2+1,f+s−1,s+1) Colored in red is the orbit element commonly used to define the family in literature.

Third Hill Family

This family is parametrized by a single realparameter α. For brevity, we denote s=sin⁡(α),c=cos⁡(α),r=sin2⁡(α)+2 and γ=cos−1⁡(13cos⁡α). Dihedral AnglesEdge Lengths(α,γ,π6,π−2α,π−γ,π2)(2s,r,23c,s,r,2)(−3α2+γ2+7π12,−α2−γ2+7π12,−α2+γ2+5π12,3α2+γ2−5π12,π−γ,π2)(−r2+3s2+3c,r2−s2+3c,r2+3s2−3c,r2+s2+3c,2,r)(α2−γ+3π4,−α2+3π4,π−2α,π6,α2+γ−π4,α2+π4)(s+1,r−s+1,s,23c,s+1,r+s−1)(α2−γ+3π4,−α2+3π4,3α2+γ2−5π12,−α2+γ2+5π12,−α−γ2+5π6,−α+γ2+π3)(r−s+1,s+1,r2+3s2−3c,r2+s2+3c,r2+s2+3c−1,−r2+s2+3c+1)(α,γ,α−γ2+π3,−α−γ2+7π6,−α−γ2+5π6,−α+γ2+π3)(r,2s,r2+s2−3c+1,r2−s2+3c+1,r2+s2+3c−1,−r2+s2+3c+1)(−α2−γ2+7π12,−3α2+γ2+7π12,α−γ2+π3,−α−γ2+7π6,α2+π4,α2+γ−π4)(r2−s2+3c,−r2+3s2+3c,r2−s2+3c+1,r2+s2−3c+1,s+1,r+s−1) Colored in red is the orbit element commonly used to define the family in literature.

New Family

For any real parameter t>4 , let s=12(cos−1⁡3(t+4)(t−2)t+2+cos−1⁡3(t−4)(t+2)t−2+cos−1⁡t2−28(t−2)(t+2)+cos−1⁡(t−4)(t+4)2(t−2)(t+2)) Then the new family and its Regge orbit is given as follows. Dihedral AnglesEdge Lengths(cos−1⁡3(t+4)(t−2)t+2,cos−1⁡3(t−4)(t+2)t−2,cos−1⁡3(t+4)(t−2)t+2,cos−1⁡3(t−4)(t+2)t−2,cos−1⁡t2−28(t−2)(t+2),cos−1⁡(t−4)(t+4)2(t−2)(t+2))(t+1,t−1,t+1,t−1,t,6)(cos−1⁡3(t+4)(t−2)t+2,cos−1⁡3(t−4)(t+2)t−2,s−cos−1⁡3(t+4)(t−2)t+2,s−cos−1⁡3(t−4)(t+2)t−2,s−cos−1⁡t2−28(t−2)(t+2),s−cos−1⁡(t−4)(t+4)2(t−2)(t+2))(t+1,t−1,t2+2,t2+4,t2+3,3t2−3)

Sporadic Cases

Newly discovered sporadic tetrahedra

(11,9,26,22,30,16)(11,10,24,22,30,14)(12,8,25,23,30,16)(13,7,22,21,23,24)(13,11,21,17,27,12)(13,12,16,8,14,16)(14,10,20,18,27,12)(15,13,18,10,19,9)(16,4,27,25,30,24)(16,4,28,26,29,23)(16,16,29,25,16,30)(17,3,24,22,26,22)(17,3,25,21,25,23)(18,14,27,27,16,30)(19,17,21,15,7,29)(20,7,27,25,28,29)(20,11,21,11,30,9)(20,15,23,19,17,29)(20,15,25,21,15,27)(21,11,24,15,26,5)(21,15,26,18,30,26)(21,19,26,14,26,29)(22,14,25,19,30,26)(22,14,27,14,17,27)(23,9,23,23,25,29)(23,9,25,21,27,27)(23,13,26,18,14,27)(23,21,30,13,29,30)(25,22,29,9,30,27)(13,11,13,12,11,19)(11,9,26,22,30,16)(11,10,24,22,30,14)(12,8,25,23,30,16)(13,7,22,21,23,24)(13,11,21,17,27,12)(13,12,16,8,14,16)(14,10,20,18,27,12)(15,13,18,10,19,9)(16,4,27,25,30,24)(16,4,28,26,29,23)(16,16,29,25,16,30)(17,3,24,22,26,22)(17,3,25,21,25,23)(18,14,27,27,16,30)(19,17,21,15,7,29)(20,7,27,25,28,29)(20,11,21,11,30,9)(20,15,23,19,17,29)(20,15,25,21,15,27)(21,11,24,15,26,5)(21,15,26,18,30,26)(21,19,26,14,26,29)(22,14,25,19,30,26)(22,14,27,14,17,27)(23,9,23,23,25,29)(23,9,25,21,27,27)(23,13,26,18,14,27)(23,21,30,13,29,30)(25,22,29,9,30,27)(13,11,13,12,11,19)
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