Penteract (I would prefer to consider the irregular case penteractoid or penteroid or whatever else it may be called as the regular one is a special case of this and can always be inferred from this)
Surteron bulk: 2(lbht + lbhw + lbtw + lhtw + bhtw)
Pentavolume: lbhtw
Tesserinder
Surteron bulk: 2πr(rab + rac + rbc + abc)
Pentavolume: πr2abc
Duocyldyinder
Surteron bulk: 2π2ab{ab + (a + b)h}
Pentavolume: π2a2b2h (where a and b are the two radii)
Cubspherinder
Surteron bulk: 4πr2{ab + (2/3)r(a + b)}
Pentavolume:(4/3)πr3ab
Cylspherinder
Surteron bulk: 4π2ab2{a + (2/3)b} (I'm not sure whether I computed this right, though most probably I did)
Pentavolume: (4/3)π2a2b3 (where a is the radius of the circular portion and b is the radius of the spherical portion)
Glominder
Surteron bulk: π2r3(2h + r)
Pentavolume: (1/2)π2r4h
Pentasphere
Surteron bulk: (8/3)π2r4
Pentavolume: (8/15)π2r5
As for the pentasphere, this is what is given in the wiki:
The hypervolumes of a pentasphere are given by:
total edge length = 0
total surface area = 0
total surcell volume = 0
surteron bulk = π2∕2 · r4
pentavolume = π2∕8 · r5
How could this be? This surteron bulk is not even the derivative of the pentavolume w.r.t. the radius. Unless I am very much mistaken, this is true of any number of dimensions and, in addition, we have these methods (quoted from viewtopic.php?f=25&t=1891):
Klitzing wrote:You either can calculate the volume of the unit hyperball recursively:
or explicitely, but separate for even and odd numbers, by means of
- Code: Select all
V_(2k) = pi^k / k!
V_(2k+1) = 2 . k! . (4 pi)^k / (2k+1)!
--- rk
Whew! That finishes all the rotatera!