Let's try complete 6D analysis.
1. Hexasphere (IIIIII), (r)
Volume of pentasphere = 8/15*pi^2*r^5
Using pentasphere slices 8/15*pi^2*(r^2 - x^2)^(5/2)
Volume = 1/6*pi^3*r^6
Surface = pi^3*r^5
2. 51-torus ((IIIII)I), ((R)r)
Volume of hollow pentasphere = 8/15*pi^2*(R + r)^5 - 8/15*pi^2*(R - r)^5 = 16/15*pi^2*r*(5*R^4 + 10*R^2*r^2 + r^4)
Hollow pentasphere slices 16/15*pi^2*sqrt(r^2 - x^2)*(5*R^4 + 10*R^2*(r^2 - x^2) + (r^2 - x^2)^2)
Volume = 1/3*pi^3*r^2*(8*R^4 + 12*R^2*r^2 + r^4)
Surface = 2/3*pi^3*r*(8*R^4 + 24*R^2*r^2 + 3*r^4)
3. 411-ditorus (((IIII)I)I), (((R)r)s)
Volume of hollow 41-torus = 1/2*pi^3*R*(r + s)^2*(4R^2 + 3(r + s)^2) - 1/2*pi^3*R*(r - s)^2*(4R^2 + 3(r - s)^2) = 4*pi^3*R*r*s*(2*R^2 + 3*r^2 + 3*s^2)
Hollow 41-torus slices 4*pi^3*R*r*sqrt(s^2 - x^2)*(2*R^2 + 3*r^2 + 3*(s^2 - x^2))
Volume = 1/2*pi^4*R*r*s^2*(8*R^2 + 12*r^2 + 9*s^2)
Surface = 2*pi^4*R*r*s*(4*R^2 + 6*r^2 + 9*s^2)
4. 3111-tritorus ((((III)I)I)I), ((((R)r)s)t)
Volume of hollow 311-ditorus = pi^3*r*(s + t)^2*(8R^2 + 4r^2 + 3(s + t)^2) - pi^3*r*(s - t)^2*(8R^2 + 4r^2 + 3(s - t)^2) = 8*pi^3*r*s*t*(4*R^2 + 2*r^2 + 3*s^2 + 3*t^2)
Hollow 311-ditorus slices 8*pi^3*r*s*sqrt(t^2 - x^2)*(4*R^2 + 2*r^2 + 3*s^2 + 3*(t^2 - x^2))
Volume = pi^4*r*s*t^2*(16*R^2 + 8*r^2 + 12*s^2 + 9*t^2)
Surface = 4*pi^4*r*s*t*(8*R^2 + 4*r^2 + 6*s^2 + 9*t^2)
5. Tetratorus (((((II)I)I)I)I), (((((R)r)s)t)u)
Volume of hollow tritorus = 8*pi^4*R*r*s*(t + u)^2 - 8*pi^4*R*r*s*(t - u)^2 = 32*pi^4*R*r*s*t*u
Hollow tritorus slices 32*pi^4*R*r*s*t*sqrt(u^2 - x^2)
Volume = 16*pi^5*R*r*s*t*u^2
Surface = 32*pi^5*R*r*s*t*u
Tetratorus as rotation of tritorus:
Volume = 8*pi^4*r*s*t*u^2 * 2*pi*R = 16*pi^5*R*r*s*t*u^2
Surface = 16*pi^4*r*s*t*u * 2*pi*R = 32*pi^5*R*r*s*t*u
6. Tiger ditorus ((((II)(II))I)I), ((((R1)(R2)r)s)t)
Volume of hollow tiger torus = same as hollow tritorus = 32*pi^4*R1*R2*r*s*t
Hollow tiger torus slices 32*pi^4*R1*R2*r*s*sqrt(t^2 - x^2)
Volume = 16*pi^5*R1*R2*r*s*t^2
Surface = 32*pi^5*R1*R2*r*s*t
Tiger ditorus as rotation of tritorus:
Volume = 8*pi^4*R1*r*s*t^2 * 2*pi*R2 = 16*pi^5*R1*R2*r*s*t^2
Surface = 16*pi^4*R1*r*s*t * 2*pi*R2 = 32*pi^5*R1*R2*r*s*t
7. 2211-tritorus ((((II)II)I)I), ((((R)r)s)t)
Volume of hollow 221-ditorus = 2*pi^3*R*(s + t)^2*(4r^2 + (s + t)^2) - 2*pi^3*R*(s - t)^2*(4r^2 + (s - t)^2) = 16*pi^3*R*s*t*(2*r^2 + s^2 + t^2)
Hollow 221-ditorus slices 16*pi^3*R*s*sqrt(t^2 - x^2)*(2*r^2 + s^2 + (t^2 - x^2))
Volume = 2*pi^4*R*s*t^2*(8*r^2 + 4*s^2 + 3*t^2)
Surface = 8*pi^4*R*s*t*(4*r^2 + 2*s^2 + 3*t^2)
2211-tritorus as rotation of 311-ditorus:
Volume = pi^3*s*t^2*(8r^2 + 4s^2 + 3t^2) * 2*pi*R = 2*pi^4*R*s*t^2*(8*r^2 + 4*s^2 + 3*t^2)
Surface = 4*pi^3*s*t*(4r^2 + 2s^2 + 3t^2) * 2*pi*R = 8*pi^4*R*s*t*(4*r^2 + 2*s^2 + 3*t^2)
8. 320-tiger 1-torus (((III)(II))I), (((R1)(R2)r)s)
Volume of hollow 320-tiger = same as hollow 221-ditorus = 16*pi^3*R2*r*s*(2*R1^2 + r^2 + s^2)
Hollow 320-tiger slices 16*pi^3*R2*r*sqrt(s^2 - x^2)*(2*R1^2 + r^2 + (s^2 - x^2))
Volume = 2*pi^4*R2*r*s^2*(8*R1^2 + 4*r^2 + 3*s^2)
Surface = 8*pi^4*R2*r*s*(4*R1^2 + 2*r^2 + 3*s^2)
320-tiger 1-torus as rotation of 311-ditorus:
Volume = pi^3*r*s^2*(8R1^2 + 4r^2 + 3s^2) * 2*pi*R2 = 2*pi^4*R2*r*s^2*(8*R1^2 + 4*r^2 + 3*s^2)
Surface = 4*pi^3*r*s*(4R1^2 + 2r^2 + 3s^2) * 2*pi*R2 = 8*pi^4*R2*r*s*(4*R1^2 + 2*r^2 + 3*s^2)
9. Torus tiger torus ((((II)I)(II))I), ((((R1)r)(R2)s)t)
Volume of hollow torus tiger = same as hollow tritorus/hollow tiger torus = 32*pi^4*R1*r*R2*s*t
Hollow torus tiger slices 32*pi^4*R1*r*R2*s*sqrt(t^2 - x^2)
Volume = 16*pi^5*R1*r*R2*s*t^2
Surface = 32*pi^5*R1*r*R2*s*t
Torus tiger torus as rotation of tiger torus:
Volume = 8*pi^4*r*R2*s*t^2 * 2*pi*R1 = 16*pi^5*R1*r*R2*s*t^2
Surface = 16*pi^4*r*R2*s*t * 2*pi*R1 = 32*pi^5*R1*r*R2*s*t
Torus tiger torus as rotation of tritorus:
Volume = 8*pi^4*R1*r*s*t^2 * 2*pi*R2 = 16*pi^5*R1*r*R2*s*t^2
Surface = 16*pi^4*R1*r*s*t * 2*pi*R2 = 32*pi^5*R1*r*R2*s*t
10. 321-ditorus (((III)II)I), (((R)r)s)
Volume of hollow 32-torus = 4/15*pi^2*(r + s)^3*(5R^2 + 4(r + s)^2) - 4/15*pi^2*(r - s)^3*(5R^2 + 4(r - s)^2) = 8/15*pi^2*s*(15*R^2*r^2 + 5*R^2*s^2 + 20*r^4 + 40*r^2*s^2 + 4*s^4)
Hollow 32-torus slices 8/15*pi^2*sqrt(s^2 - x^2)*(15*R^2*r^2 + 5*R^2*(s^2 - x^2) + 20*r^4 + 40*r^2*(s^2 - x^2) + 4*(s^2 - x^2)^2)
Volume = 1/3*pi^3*s^2*(12*R^2*r^2 + 3*R^2*s^2 + 16*r^4 + 24*r^2*s^2 + 2*s^4)
Surface = 4/3*pi^3*s*(6*R^2*r^2 + 3*R^2*s^2 + 8*r^4 + 24*r^2*s^2 + 3*s^4)
11. 2121-tritorus ((((II)I)II)I), ((((R)r)s)t)
Volume of hollow 212-ditorus = 16/3*pi^3*R*r*(s + t)^3 - 16/3*pi^3*R*r*(s - t)^3 = 32/3*pi^3*R*r*t*(3*s^2 + t^2)
Hollow 212-ditorus slices 32/3*pi^3*R*r*sqrt(t^2 - x^2)*(3*s^2 + (t^2 - x^2))
Volume = 4*pi^4*R*r*t^2*(4*s^2 + t^2)
Surface = 16*pi^4*R*r*t*(2*s^2 + t^2)
2121-tritorus as rotation of 221-ditorus:
Volume = 2*pi^3*r*t^2*(4s^2 + t^2) * 2*pi*R = 4*pi^4*R*r*t^2*(4*s^2 + t^2)
Surface = 8*pi^3*r*t*(2s^2 + t^2) * 2*pi*R = 16*pi^4*R*r*t*(2*s^2 + t^2)
12. 221-tiger 1-torus (((II)(II)I)I), (((R1)(R2)r)s)
Volume of hollow 221-tiger = same as hollow 212-ditorus = 32/3*pi^3*R1*R2*s*(3*r^2 + s^2)
Hollow 221-tiger slices 32/3*pi^3*R1*R2*sqrt(s^2 - x^2)*(3*r^2 + (s^2 - x^2))
Volume = 4*pi^4*R1*R2*s^2*(4*r^2 + s^2)
Surface = 16*pi^4*R1*R2*s*(2*r^2 + s^2)
221-tiger 1-torus as rotation of 221-ditorus:
Volume = 2*pi^3*R1*s^2*(4r^2 + s^2) * 2*pi*R2 = 4*pi^4*R1*R2*s^2*(4*r^2 + s^2)
Surface = 8*pi^3*R1*s*(2r^2 + s^2) * 2*pi*R2 = 16*pi^4*R1*R2*s*(2*r^2 + s^2)
13. 231-ditorus (((II)III)I), (((R)r)s)
Volume of hollow 23-torus = pi^3*R*(r + s)^4 - pi^3*R*(r - s)^4 = 8*pi^3*R*r*s*(r^2 + s^2)
Hollow 23-torus slices 8*pi^3*R*r*sqrt(s^2 - x^2)*(r^2 + (s^2 - x^2))
Volume = pi^4*R*r*s^2*(4*r^2 + 3*s^2)
Surface = 4*pi^4*R*r*s*(2*r^2 + 3*s^2)
231-ditorus as rotation of 41-torus:
Volume = 1/2*pi^3*r*s^2*(4r^2 + 3s^2) * 2*pi*R = pi^4*R*r*s^2*(4*r^2 + 3*s^2)
Surface = 2*pi^3*r*s*(2r^2 + 3s^2) * 2*pi*R = 4*pi^4*R*r*s*(2*r^2 + 3*s^2)
14. 420-tiger ((IIII)(II)), ((R1)(R2)r)
Smearing hollow glome with volume 4*pi^2*R1*y*(R1^2 + y^2)
over circle with circumference 2*pi*(R2 + x)
4*pi^2*R1*sqrt(r^2 - x^2)*(R1^2 + r^2 - x^2) * 2*pi*(R2 + x) = 8*pi^3*R1*(R2 + x)*sqrt(r^2 - x^2)*(R1^2 + r^2 - x^2)
Volume = pi^4*R1*R2*r*(4*R1^2 + 3*r^2)
Surface = 4*pi^4*R1*R2*(2*R1^2 + 3*r^2)
Or:
Smearing ring with area 4*pi*R2*y
over glome with surface 2*pi^2*(R1 + x)^3
4*pi*R2*sqrt(r^2 - x^2) * 2*pi^2*(R1 + x)^3 = 8*pi^3*R2*(R1 + x)^2*sqrt(r^2 - x^2)
Volume = pi^4*R1*R2*r*(4*R1^2 + 3*r^2)
Surface = 4*pi^4*R1*R2*(2*R1^2 + 3*r^2)
420-tiger as rotation of 41-torus:
Volume = 1/2*pi^3*R1*r^2*(4R1^2 + 3r^2) * 2*pi*R2 = pi^4*R1*R2*r*(4*R1^2 + 3*r^2)
Surface = 2*pi^3*R1*r*(2R1^2 + 3r^2) * 2*pi*R2 = 4*pi^4*R1*R2*(2*R1^2 + 3*r^2)
15. 31-torus 20-tiger (((III)I)(II)), (((R1)r)(R2)s)
Smearing torisphere shell with volume 8*pi^2*r*y*(2*R1^2 + r^2 + y^2)
over circle with circumference 2*pi*(R2 + x)
8*pi^2*r*sqrt(s^2 - x^2)*(2*R1^2 + r^2 + s^2 - x^2) * 2*pi*(R2 + x) = 16*pi^3*r*(R2 + x)*sqrt(s^2 - x^2)*(2*R1^2 + r^2 + s^2 - x^2)
Volume = 2*pi^4*r*R2*s^2*(8*R1^2 + 4*r^2 + 3*s^2)
Surface = 8*pi^4*r*R2*s^2*(4*R1^2 + 2*r^2 + 3*s^2)
Or:
Smearing ring with area 4*pi*R2*y
over torisphere with surface 4*pi^2*(r + x)*(2*R1^2 + (r + x)^2)
4*pi*R2*sqrt(s^2 - x^2) * 4*pi^2*(r + x)*(2*R1^2 + (r + x)^2) = 16*pi^3*R2*(r + x)*sqrt(s^2 - x^2)*(2*R1^2 + (r + x)^2)
Volume = 2*pi^4*r*R2*s^2*(8*R1^2 + 4*r^2 + 3*s^2)
Surface = 8*pi^4*r*R2*s^2*(4*R1^2 + 2*r^2 + 3*s^2)
31-torus 20-tiger as rotation of 311-ditorus:
Volume = pi^3*r*s^2*(8R1^2 + 4r^2 + 3s^2) * 2*pi*R2 = 2*pi^4*r*R2*s^2*(8*R1^2 + 4*r^2 + 3*s^2)
Surface = 4*pi^3*r*s*(4R1^2 + 2r^2 + 3s^2) * 2*pi*R2 = 8*pi^4*r*R2*s*(4*R1^2 + 2*r^2 + 3*s^2)
16. Ditorus tiger ((((II)I)I)(II)), ((((R1)r)s)(R2)t)
Smearing ditorus shell with volume 16*pi^3*R1*r*s*y
over circle with circumference 2*pi*(R2 + x)
16*pi^3*R1*r*s*sqrt(t^2 - x^2) * 2*pi*(R2 + x) = 32*pi^4*R1*r*s*(R2 + x)*sqrt(t^2 - x^2)
Volume = 16*pi^5*R1*r*s*R2*t^2
Surface = 32*pi^5*R1*r*s*R2*t
Or:
Smearing ring with area 4*pi*R2*y
over ditorus with surface 8*pi^3*R1*r*(s + x)
4*pi*R2*sqrt(t^2 - x^2) * 8*pi^3*R1*r*(s + x) = 32*pi^4*R1*r*R2*(s + x)*sqrt(t^2 - x^2)
Volume = 16*pi^5*R1*r*s*R2*t^2
Surface = 32*pi^5*R1*r*s*R2*t
Ditorus tiger as rotation of torus tiger:
Volume = 8*pi^4*r*s*R2*t^2 * 2*pi*R1 = 16*pi^5*R1*r*s*R2*t^2
Surface = 16*pi^4*r*s*R2*t * 2*pi*R1 = 32*pi^5*R1*r*s*R2*t
Ditorus tiger as rotation of tritorus:
Volume = 8*pi^4*R1*r*s*t^2 * 2*pi*R2 = 16*pi^5*R1*r*s*R2*t^2
Surface = 16*pi^4*R1*r*s*t * 2*pi*R2 = 32*pi^5*R1*r*s*R2*t
17. Double tiger (((II)(II))(II)), (((R1)(R2)r)(R3)s)
Smearing tiger shell 16*pi^3*R1*R2*r*y
over circle with circumference 2*pi*(R3 + x)
16*pi^3*R1*R2*r*sqrt(s^2 - x^2) * 2*pi*(R3 + x) = 32*pi^4*R1*R2*r*(R3 + x)*sqrt(s^2 - x^2)
Volume = 16*pi^5*R1*R2*r*R3*s^2
Surface = 32*pi^5*R1*R2*r*R3*s
Or:
Smearing ring with area 4*pi*R3*y
over tiger with surface 8*pi^3*R1*R2*(r + x)
4*pi*R3*sqrt(s^2 - x^2) * 8*pi^3*R1*R2*(r + x) = 32*pi^4*R1*R2*R3*(r + x)*sqrt(s^2 - x^2)
Volume = 16*pi^5*R1*R2*r*R3*s^2
Surface = 32*pi^5*R1*R2*r*R3*s
Double tiger as rotation of torus tiger:
Volume = 8*pi^4*R1*r*R3*s^2 * 2*pi*R2 = 16*pi^5*R1*R2*r*R3*s^2
Surface = 16*pi^4*R1*r*R3*s * 2*pi*R2 = 32*pi^5*R1*R2*r*R3*s
Double tiger as rotation of tiger torus:
Volume = 8*pi^4*R1*R2*r*s^2 * 2*pi*R3 = 16*pi^5*R1*R2*r*R3*s^2
Surface = 16*pi^4*R1*R2*r*s * 2*pi*R3 = 32*pi^5*R1*R2*r*R3*s
18. 22-torus 20-tiger (((II)II)(II)), (((R1)r)(R2)s)
Smearing spheritorus shell with volume 16/3*pi^2*R1*y*(3r^2 + y^2)
over circle with circumference 2*pi*(R2 + x)
16/3*pi^2*R1*sqrt(s^2 - x^2)*(3r^2 + s^2 - x^2) * 2*pi*(R2 + x) = 32/3*pi^3*R1*(R2 + x)*sqrt(s^2 - x^2)*(3*r^2 + s^2 - x^2)
Volume = 4*pi^4*R1*R2*s^2*(4*r^2 + s^2)
Surface = 16*pi^4*R1*R2*s*(2*r^2 + s^2)
Or:
Smearing ring with area 4*pi*R2*y
over spheritorus with surface 8*pi^2*R1*(r + x)^2
4*pi*R2*sqrt(s^2 - x^2) * 8*pi^2*R1*(r + x)^2 = 32*pi^3*R1*R2*(r + x)^2*sqrt(s^2 - x^2)
Volume = 4*pi^4*R1*R2*s^2*(4*r^2 + s^2)
Surface = 16*pi^4*R1*R2*s*(2*r^2 + s^2)
22-torus 20-tiger as rotation of 320-tiger:
Volume = 2*pi^3*R2*s^2*(4r^2 + s^2) * 2*pi*R1 = 4*pi^4*R1*R2*s^2*(4*r^2 + s^2)
Surface = 8*pi^3*R2*s*(2r^2 + s^2) * 2*pi*R1 = 16*pi^4*R1*R2*s*(2*r^2 + s^2)
22-torus 20-tiger as rotation of 221-ditorus:
Volume = 2*pi^3*R1*s^2*(4r^2 + s^2) * 2*pi*R2 = 4*pi^4*R1*R2*s^2*(4*r^2 + s^2)
Surface = 8*pi^3*R1*s*(2r^2 + s^2) * 2*pi*R2 = 16*pi^4*R1*R2*s*(2*r^2 + s^2)
19. 42-torus ((IIII)II), ((R)r)
Volume of 41-torus = 1/2*pi^3*R*r^2*(4R^2 + 3r^2)
Using 41-torus slices 1/2*pi^3*R*(r^2 - x^2)*(4R^2 + 3(r^2 - x^2))
Volume = 8/15*pi^3*R*r^3*(5*R^2 + 3*r^2)
Surface = 8*pi^3*R*r^2*(R^2 + r^2)
20. 312-ditorus (((III)I)II), (((R)r)s)
Volume of 311-ditorus = pi^3*r*s^2*(8R^2 + 4r^2 + 3s^2)
Using 311-ditorus slices pi^3*r*(s^2 - x^2)*(8R^2 + 4r^2 + 3(s^2 - x^2))
Volume = 16/15*pi^3*r*s^3*(10*R^2 + 5*r^2 + 3*s^2)
Surface = 16*pi^3*r*s^2*(2*R^2 + r^2 + s^2)
21. 2112-tritorus ((((II)I)I)II), ((((R)r)s)t)
Volume of tritorus = 8*pi^4*R*r*s*t^2
Using tritorus slices 8*pi^4*R*r*s*(t^2 - x^2)
Volume = 32/3*pi^4*R*r*s*t^3
Surface = 32*pi^4*R*r*s*t^2
2112-tritorus as rotation of 212-ditorus:
Volume = 16/3*pi^3*r*s*t^3 * 2*pi*R = 32/3*pi^4*R*r*s*t^3
Surface = 16*pi^3*r*s*t^2 * 2*pi*R = 32*pi^4*R*r*s*t^2
22. 220-tiger 2-torus (((II)(II))II), (((R1)(R2)r)s)
Volume of tiger torus = 8*pi^4*R1*R2*r*s^2
Using tiger torus slices 8*pi^4*R1*R2*r*(s^2 - x^2)
Volume = 32/3*pi^4*R1*R2*r*s^3
Surface = 32*pi^4*R1*R2*r*s^2
220-tiger 2-torus as rotation of 212-ditorus:
Volume = 16/3*pi^3*R1*r*s^3 * 2*pi*R2 = 32/3*pi^4*R1*R2*r*s^3
Surface = 16*pi^3*R1*r*s^2 * 2*pi*R2 = 32*pi^4*R1*R2*r*s^2
23. 222-ditorus (((II)II)II), (((R)r)s)
Volume of 221-ditorus = 2*pi^3*R*s^2*(4r^2 + s^2)
Using 221-ditorus slices 2*pi^3*R*(s^2 - x^2)*(4r^2 + s^2 - x^2)
Volume = 32/15*pi^3*R*s^3*(5*r^2 + s^2)
Surface = 32/3*pi^3*R*s^2*(3*r^2 + s^2)
222-ditorus as rotation of 32-torus:
Volume = 4/15*pi^2*s^3*(5*r^2 + 4*s^2) * 2*pi*R = 8/15*pi^3*R*s^3*(5*r^2 + 4*s^2)
Surface = 4/3*pi^2*s^2*(3*r^2 + 4*s^2) * 2*pi*R = 8/3*pi^3*R*s^2*(3*r^2 + 4*s^2)
Note the discrepancy. There is probably a mistake somewhere, but I'm having problems finding it.
Can someone try to find the problem? I think I'm done for today
Groups:
Toratopes can be separated into groups that share a single set volume/surface formulas per dimension (and in each dimension it's just previous dimension * 2*pi*(new radius)). Each group is formed by a basic toratope (which is not a nonbisecting rotation of anything, therefore doesn't contain the (II) string) and all higher-dimensional toratopes formed by nonbisecting rotations.
Up to 6D, the groups are:
Point/circle group: Circle | torus | ditorus, tiger | tritorus, tiger torus, torus tiger | tetratorus, tiger ditorus, torus tiger torus, ditorus tiger, double tiger, duotorus tiger
Sphere group: Sphere | spheritorus | 212-ditorus, 221-tiger | 2112-tritorus, 220-tiger 2-torus, 21-torus 21-tiger, triger
Glome group: Glome | 23-torus | 213-ditorus, 222-tiger
Torisphere group: Torisphere | 221-ditorus, 320-tiger | 2121-tritorus, 221-tiger 1-torus, 22-torus 20-tiger, 21-torus 30-tiger
Pentasphere group: Pentasphere | 24-torus
41-torus group: 41-torus | 231-ditorus, 420-tiger
311-ditorus group: 311-ditorus | 2211-tritorus, 320-tiger 1-torus, 31-torus 20-tiger
32-torus group: 32-torus | 222-ditorus, 321-tiger
Hexasphere group: Hexasphere
51-torus grou*p: 51-torus
411-ditorus group: 411-ditorus
3111-tritorus group: 3111-tritorus
321-ditorus group: 321-ditorus
42-torus group: 42-torus
312-ditorus group: 312-ditorus
330-tiger group: 330-tiger
33-torus group: 33-torus