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Round Shapes

Round shapes kind of fall out of a description by incidence matrices because several elements are missing there. E.g. a sphere has no edges at all. Also occur bended elements which thus sidestep from a hierarchical description. Therefore we are bound here more to a mere listing of individual shapes.

Mrs. W. Krieger invented a Dynkin type description applicable to severl of those. She uses the link symbol O for these round, i.e. circular elements. The first node of that corresponding component then has to be ringed as usual, resp. in the typewriter friendly linearized notation is given as x. Even so this cannot provide a true relative scaling factor, esp. when being combined with further (linear) edges additionally. On the other hand, within the round component itself, i.e. that Dynkin subsymbol, which uses O linkages only, one will have further node symbols also. There a subsequent o would represent a further dimension with the same curvature, while a node with any different measure y would represent a correspondingly different curvature. Thus xOo = xOx represents a (filled) circle, while xOy represents a (filled) ellipse.

In 2005 the operations of Cartesian product and spheration led to a different toratope notation. P. Wright, M. Četrnáct, and W. Krieger in the sequel settled for an I representing each Cartesian dimension and a surrounding (...) for each spheration. Here each dimension I gets bounded by some maximal value, whereas in spherations (...) the squared sum of all contained entities (I or (...)) does. Thus a square is II, the circle is simply (II), a cylinder is (II)I, and a torus is ((II)I).

In parallel to the former P. Pugeau and Quickfur developped their |,>,o devices. It always is to be read from the left, i.e. the rightmost operation is to be applied to the left "base". (This is why these symbols always start with a |.) The main constituents here are the extrusion |, acting like a prism operator, then the tapering >, acting like a pyramid/cone operator, and finally the spin o. – While the spin does rotate the body described by the left subsymbol around an internal axis, P. Pugeau later added a further (o) in order to make the spin wrt. to an external axis. While (o) here describes an external circular move, (oo) an external spherical one, etc.


---- 2D (up) ----

Image Symbols Name Remarks
 ©
xOo = xOx
(II)
|o
circle, circular disk
|x| = r
surface equation
x12 + x22 = r2
area
π r2
 ©
xOy
ellipse, ellipse area
|x| = a1 < |y| = a2
surface equation
x12/a12 + x22/a22 = 1
area
π a1a2

---- 3D (up) ----

Image /
Links
Symbols Name Remarks
 ©
© ©
xOoOo = xOxox
(III)
|oo
sphere, ball
surface equation
x12 + x22 + x32 = r2
volume
4/3 π r3
           ©
© -
xOoOy = xOxOy
prolate spheroid
|x| < |y|
      ©
© -
xOyOo = xOyOy
oblate spheroid
|x| < |y|
© -
xOyOz
general ellipsoid
|x| = r1 < |y| = r2 < |z| = r3
surface equation
x12/r12 + x22/r22 + x32/r32 = 1
volume
4/3 π r1 r2 r3
 ©
© ©
x xOo
(II)I
|o| = ||o
cylinder
volume equation
x12 + x22 ≤ r2
|x3| ≤ 1/2 l
volume
π r2 l
 ©
© ©
oxOoo&#x
|o> = |>o
cone
volume
1/3 π r2 h
 ©
© ©
-
crind, Steinmetz solid intersection of x xOo with xOo x
volume equation
x12 + x22 ≤ r2
x12 + x32 ≤ r2
volume
16/3 r3
sometimes a sqrt(2) taller variation therefrom is being considered, where the edges become circles
 ©
© ©
((II)I)
|o(o)
torus
r < R
surface equation
(R - sqrt[x12 + x22])2 + x32 = r2
volume
2 π2 R r2

---- 4D (up) ----

Image /
Links
Symbols Name Remarks
© ©
xOoOoOo
(IIII)
|ooo
glome
surface equation
x12 + x22 + x32 + x42 = r2
bulk
1/2 π2 r4
© ©
x xOoOo
(III)I
|oo| = |o|o = ||oo
spherinder prism (of length l) of a sphere (of radius r)
bulk
4/3 π r3 l
- ©
oxOooOoo&#x
|oo> = |o>o = |>oo
sphone cone (of length h) of a sphere
bulk
1/3 π r3 h
 ©
- ©
xx oxOoo&#x
|o>| = |>o|
coninder prism (of length l) of a cone (of height h)
bulk
π r2 h l
©
© ©
x4o xOo
(II)II
|o|| = ||o| = |||o
cubinder duoprism of a square and a circular disk
bulk
π r2 l2
- ©
((II)I)I
|o(o)|
torinder prism (of length l) of a torus
r < R
bulk
2 π2 R r2 l
© ©
xOo xOo
(II)(II)
duocylinder limit of douprism
intersection of x4o xOo with xOo x4o
bulk equation
x12 + x22 ≤ r12
x32 + x42 ≤ r22
bulk
π2 r12 r22
- ©
((III)I)
|o(oo)
torisphere torus with a spherical crossection
r < R
surface equation
(R - sqrt[x12 + x22 + x32])2 + x42 = r2
bulk
8/3 π2 R r3
- ©
((II)II)
|oo(o)
spheritorus
r < R
surface equation
(R - sqrt[x12 + x22])2 + x32 + x42 = r2
- ©
((II)(II))
tiger
r < Rk
surface equation
(R1 - sqrt[x12 + x22])2 + (R2 - sqrt[x32 + x42])2 = r2
- ©
(((II)I)I)
|o(o)(o)
ditorus torus of a torus crossection
ρ < r < R
surface equation
(r - sqrt[(R - sqrt[x12 + x22])2 + x3])2 + x42 = ρ2
bulk
4 π3 R r ρ2



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