Hexads of series of regular maps

β°' is the hemi-di-4n+2-gon.
β° is the hemi-4n+2-hosohedron.
Αn ≥ 0,     no. of edges is odd
nameSchläfli formulaVertices, Faces, EdgesmV,mFGenus
α{ 2n+1, 4n+2 }21, 2, 2n+14n+2,2n+1n
α'{ 4n+2, 2n+1 }22, 1, 2n+12n+1,4n+2
α'°{ 2, 2n+1 }4n+22, 2n+1, 2n+12n+1,10
α'°'{ 2n+1, 2 }4n+22n+1, 2, 2n+11,2n+1
α°'{ 4n+2, 2 }2n+12n+1, 1, 2n+1 1,4n+21
α°{ 2, 4n+2 }2n+11, 2n+1, 2n+14n+2,1
α'° is the 2n+1-hosohedron.
α'°' is the di-2n+1-gon.
α°' is the hemi-di-4n+2-gon.
α° is the hemi-4n+2-hosohedron.


γ° is the 2n+1-hosohedron.
γ°' is the di-2n+1-gon.
Βn > 0,     no. of edges is even
nameSchläfli formulaVertices, Faces, EdgesmV,mFGenus
β{ 4n, 4n }21, 1, 2n4n,4nn
β°{ 2, 4n }4n1, 2n, 2n4n,11
β°'{ 4n, 2 }4n2n, 1, 2nn
Γn ≥ 0,     no. of edges is even
nameSchläfli formulaVertices, Faces, EdgesmV,mFGenus
γ{ 2n+2, 2n+2 }22, 2, 2n+22n+2,2n+2n
γ°{ 2, 2n+2 }2n+22, 2n+2, 2n+22n+2,10
γ°'{ 2n+2, 2 }2n+22n+2, 2, 2n+21,2n+2
 
Δn = ±1 modulo 6,     no. of edges is odd
nameSchläfli formulaVertices, Faces, EdgesmV,mFGenus
δ{ 6, 3n }2n2, n, 3n3n,3n
δ'{ 3n, 6 }2nn, 2, 3n3,3n
δ'°{ 2n, 6 }3nn, 3, 3n3,n2n-1
δ'°'{ 6, 2n }3n3, n, 3nn,3
δ°'{ 3n, 2n }63, 2, 3nn,3n(n-1)*3/2
δ°{ 2n, 3n }62, 3, 3n3n,n
Εn = ±2 modulo 6,     no. of edges is even
nameSchläfli formulaVertices, Faces, EdgesmV,mFGenus
ε{ 6, 3n }n2, n, 3n3n,3n
ε'{ 3n, 6 }nn, 2, 3n3,3n
ε'°{ n, 6 }3nn, 6, 3n3,n/2n-2
ε'°'{ 6, n }3n6, n, 3nn/2,3
ε°'{ 3n, n }66, 2, 3nn/2,3n(n-2)*3/2
ε°{ n, 3n }62, 6, 3n3n,n/2
 
Ζn even,     no. of edges modulo 8 is 4
nameSchläfli formulaVertices, Faces, EdgesmV,mFGenus
ζ{ 4, 2n+2 }4n+44, 2n+2, 4n+4n+1,2n
ζ'{ 2n+2, 4 }4n+42n+2, 4, 4n+42,n+1
ζ'°{ 4n+4, 4 }2n+22n+2, 2, 4n+42,4n+4n+1
ζ'°'{ 4, 4n+4 }2n+22, 2n+2, 4n+44n+4,2
ζ°'{ 2n+2, 4n+4 }42, 4, 4n+44n+4,n+12n
ζ°{ 4n+4, 2n+2 }44, 2, 4n+4n+1,4n+4
Ηn even and positive,     no. of edges modulo 8 is 0
nameSchläfli formulaVertices, Faces, EdgesmV, mFGenus
η{ 4, 4n }4n2, 2n, 4n4n, 2n
η'{ 4n, 4 }4n2n, 4, 4n2,4n
η°{ 4n, 4n }42, 2, 4n4n,4n2n-1
Θn even,     no. of edges modulo 8 is 0
nameSchläfli formulaVertices, Faces, EdgesmV, mFGenus
θ{ 4, 2n }2n4, 2n, 4nn, 2n-1
θ'{ 2n, 4 }2n2n, 4, 4n2, n
θ°{ 2n, 2n }44, 4, 4nn, n2n-3
 
 
Ιn > 0,     no. of edges modulo 16 is 0
nameSchläfli formulaVertices, Faces, EdgesmV, mFGenus
ι{ 4, 8n }44, 8n, 16n4n, 14n-1
ι'{ 8n, 4 }48n, 4, 16n1, 4n
ι°{ 4, 4 }8n4, 4, 16n4n, 4n8n-3
 
For λ2, mV=mF=2
Κn odd,     no. of edges modulo 4 is 2
nameSchläfli formulaVertices, Faces, EdgesmV,mFGenus
κ { 4, 2n }4 2n, n2, 2n2 2,1 n2-2n+2
κ' { 2n, 4 }4 n2, 2n,2n2 1,2
κ'° { 4, 4 }2n    {n,0) n2, n2, 2n2 1,1 1
For κ2, mF=4.
For κ'2, mV=4.
For κ'°2, mV=mF=4.
 
For μ2, mF=4.
For μ'2, mV=4.
For μ'°2, mV=mF=4.
Λn>0, n even,     no. of edges modulo 8 is 0
nameSchläfli formulaVertices, Faces, EdgesmV,mFGenus
λ { 4, 2n }4 2n, n2, 2n2 2,1 n2/2-n+1
λ' { 2n, 4 }4 n2, 2n,2n2 1,2
λ'° { 4, 4 }2n    (n,0) n2, n2, 2n2 1,1 1
Μn>0, n even,     no. of edges modulo 16 is 0
nameSchläfli formulaVertices, Faces, EdgesmV,mFGenus
μ { 4, 2n }4 4n, 2n2, 4n2 1,1 (n-1)2
μ' { 2n, 4 }4 2n2, 4n, 4n2 1,1
μ'° { 4, 4 }2n    (n,n) 2n2, 2n2, 4n2 1,1 1
 
Νn > 0,     no. of edges modulo 6 is 0
nameSchläfli formulaVertices, Faces, EdgesmV, mFGenus
ν { 4, 3n }3n 4, 3n, 6n n,2 3n-2
ν' { 3n, 4 }3n 3n, 4, 6n 2,n
ν° { 3n, 3n }4 4, 4, 6n n, n 3n-3

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