Definition of toratopes through implicit equations

Discussion of shapes with curves and holes in various dimensions.

Definition of toratopes through implicit equations

I'm trying to inductively define toratope notation using implicit equations.

As well all know, to find the equation for ((III)(II)), we take the equations for (III) and (II) and sort of "Pythagoras them together".

(III): sqrt(x^2 + y^2 + z^2) - r_1 = 0
(II): sqrt(x^2 + y^2) - r_2 = 0

((III)(II)): sqrt((sqrt(x^2 + y^2 + z^2) - r_1)^2 + (sqrt(w^2 + t^2) - r_2)^2) - r_3 = 0.

I thought this would work for everything. But the problem I've just run into is that if you follow this rule naively for intervals, it doesn't work. Take the basic torus ((II)I).

(II): sqrt(x^2 + y^2) - r_2 = 0
I: sqrt(x^2) -r_2 = |x| - r_2 = 0

If we combine this we get

((II)I): sqrt((sqrt(x^2 + y^2) - r_1)^2 + (|z| - r_2)^2) - r_3 = 0,
which is actually a pair of torii. It only degenerates to a torus when we let r_2 = 0. What we actually want is

((II)I): sqrt((sqrt(x^2 + y^2) - r_1)^2 + z^2) - r_3 = 0.

So we have a special case to put in the definition. Can anyone think of an elegant way to deal with this?

PWrong
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Re: Definition of toratopes through implicit equations

PWrong wrote:I'm trying to inductively define toratope notation using implicit equations.

As well all know, to find the equation for ((III)(II)), we take the equations for (III) and (II) and sort of "Pythagoras them together".

(III): sqrt(x^2 + y^2 + z^2) - r_1 = 0
(II): sqrt(x^2 + y^2) - r_2 = 0

((III)(II)): sqrt((sqrt(x^2 + y^2 + z^2) - r_1)^2 + (sqrt(w^2 + t^2) - r_2)^2) - r_3 = 0.

I thought this would work for everything. But the problem I've just run into is that if you follow this rule naively for intervals, it doesn't work. Take the basic torus ((II)I).

(II): sqrt(x^2 + y^2) - r_2 = 0
I: sqrt(x^2) -r_2 = |x| - r_2 = 0

If we combine this we get

((II)I): sqrt((sqrt(x^2 + y^2) - r_1)^2 + (|z| - r_2)^2) - r_3 = 0,
which is actually a pair of torii. It only degenerates to a torus when we let r_2 = 0. What we actually want is

((II)I): sqrt((sqrt(x^2 + y^2) - r_1)^2 + z^2) - r_3 = 0.

So we have a special case to put in the definition. Can anyone think of an elegant way to deal with this?

It Looks that you are mixing up the simple "I" (interval) with the thing "(I)" (one dimensional sphere = interval endpoints).
Only the latter would have the individual equation, you're providing above.

In fact you would have the same "exceptional need" already when defining the n-D balls themselve, i.e. when doing the transition from "I...I" to "(I...I)". Right?

--- rk
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Re: Definition of toratopes through implicit equations

The thing is that toratopes as set forward in these threads are rather much the same as brick-product polytopes, except instead of the three coherent products, they are using two forms of the comb-product. They widely open the kind of holes available, but there are much more fierce creatures out there, even if one restricts oneself to pancake holes.

For example, the tiger ((ii)(ii)) can be constructed as a torus-swirl-prism. It has two torus-shaped holes in it. Holes by Hopf fibulation give rise to torus-shaped things, and a dodecahedron-shell with 12 holes in its faces, gives rise to a tiger-like thing with 12 torus-shaped holes. You can even have a single torus-shaped hole, by removing just one of the faces of a duocylinder.

The simplest definition of toratopes is that they're "right combs". The comb product can be implemented in series or parallel, the latter gives rise to the tigers. The reasons that the bracket-notation and hence the formulas work, is because they are right combs. The difference between say (((ii)i)i) and ((ii)(ii)), is that while both are torus rotated, the axes of rotation is parallel with a spoke in the first case, and with the axle in the second. There is no general discussion of the oblique case (eg a circle, which has been "spherated" using a torus at an angle).

The comb product is a pondering product, which means that you in the end, share an axis between two elements. This means, that even in the right case, you are going to end up with (r^2 + z^2) as a generating term.
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Re: Definition of toratopes through implicit equations

We now look for equations for the toratopes. There are as many parameters as there are open brackets, and as many coordinates as there are vertical lines.

A sphere might be written as c1 = rss(z, y, x, ...), where RSS is the root-sum-square.

If we mean write this as a shell function, we put c2 = ssh(c1, z, y, x, ..), = abs(c1 - rss(z,y,x,...)). What this does is to convert space as being 0 at the surface of a sphere of radius c1, and the ssh() is the distance above or below this. The shell-radius is c2, the thickness is then 2c2.

A torus is then c2 = rss(ssh(c1, z,y),x).

A tiger is c3 = rss(ssh(c1, z,y),ssh(c2, x,w)), A tri-torus is c3 = rss(ssh(c2, ssh(c1,z,y), x), w)

Spherating then replaces c = rss(£), with c' = ssh(c, £) £ is a list of radii.

The hose-operator replaces an axis, eg y, with ssh(c, z, y).

One notes here the implication is c3 < c2 < c1, or in the case of the tiger, c3 < min(c2, c1). Generally, for a chain-comb, use <. for a tiger-comb, a min() split is called for.

Note this supports 'hose combs', by replacing a coordinate by an ssh term. So, eg

tire of section circle, c2 = rss(ssh(c1, z, y), x).
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Re: Definition of toratopes through implicit equations

Pwrong wrote:(II): sqrt(x^2 + y^2) - r_2 = 0
I: sqrt(x^2) -r_2 = |x| - r_2 = 0

If we combine this we get

((II)I): sqrt((sqrt(x^2 + y^2) - r_1)^2 + (|z| - r_2)^2) - r_3 = 0,
which is actually a pair of torii. It only degenerates to a torus when we let r_2 = 0. What we actually want is

((II)I): sqrt((sqrt(x^2 + y^2) - r_1)^2 + z^2) - r_3 = 0.

So we have a special case to put in the definition. Can anyone think of an elegant way to deal with this?

It seems like what you're after is a surface of revolution. Take circle in xz plane:

x^2 + z^2 - R2^2 = 0

perform non-bisecting rotation along plane xy around z-axis, replaces x with (sqrt(x^2 + y^2) - R1)^2

equation for torus ((II)I)
(sqrt(x^2 + y^2) - R1)^2 + z^2 - R2^2 = 0

And for bisecting rotations, add a dimension for plane xy :

equation for sphere (III)
x^2 + y^2 + z^2 - R1^2 = 0

From this method, all five toratopes can be derived from a 2-sphere and a torus.

Sphere : (III) : x^2 + y^2 + z^2 - R1^2 = 0

Bisecting xw rotation into glome (IIII) : x^2 + y^2 + z^2 + w^2 - R1^2 = 0

non-bisecting xw into spheritorus ((II)II) : (sqrt(x^2 + w^2) - R1)^2 + y^2 + z^2 - R2^2 = 0

Torus : ((II)I) : (sqrt(x^2 + y^2) - R1)^2 + z^2 - R2^2 = 0

Bisecting xw rotation into torisphere ((III)I) = (sqrt(x^2 + y^2 + w^2) - R1)^2 + z^2 - R2^2 = 0

Bisecting zw rotation into spheritorus ((II)II) = (sqrt(x^2 + y^2) - R1)^2 + z^2 + w^2 - R2^2 = 0

non-bisecting xw into ditorus (((II)I)I) = (sqrt((sqrt(x^2 + w^2) - R1)^2 + y^2) - R2)^2 + z^2 - R3^2 = 0
- replaces x with (sqrt(x^2 + w^2) - R1)^2

non-bisecting zw into tiger ((II)(II)) = (sqrt(x^2 + y^2) - R1a)^2 + (sqrt(z^2 + w^2) - R1b)^2 - R2^2 = 0
- replaces z with (sqrt(z^2 + w^2) - R1)^2
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