circle # square vs square # circle

Discussion of shapes with curves and holes in various dimensions.

circle # square vs square # circle

Postby Secret » Sat Jan 14, 2012 10:48 am

when reading the toracubinder and the toraspherinder
I have the below thought (which my vague memory told me that it has been discussed before (something about skeletons), but I forgot)

Ok we are currently in 3D

Assume I have two toriod objects
A is made by circle # square (i.e. fold a cuboid into a loop)
B is made by square # circle (i.e. bend a cylinder 3 times and then join the circular ends)

Although it is obvious that A=/=B (By observation)
How should we name them in order to distinguish them?

Now we are in 4D
Similar scenario is also met
High dimension wiki:
Toracubinder (Circle # sphere) vs Toraspherinder (Sphere # circle)

Actually I think toracubinder is incorrect
e.g. Your call the cubinder that way because there is at least one realmic view it is a cube
But there is nothing cubic in the "toracubinder"!
In addition it is folded using a spherinder not a cubinder

Thus I think what is called toracubinder in the wiki should be named as toraspherinder
But then we have a problem of don't know how to name the weird (sphere # circle) toroid

I then try to check wikipedia for the meaning of the word "torus"
Then it gives me "cushion" thus it does not help

IMO the true toracubinder is the torinder (choping away the (circle#square) cell from the cubindrical swock and fold won't work as you will end up stretching and deforming the cylinders (more specifically, the circular ridges become elliptical ridges and thery won't join because you always end up having two curved 1D portions failed to coincide completely) regardless on which direction you stretch it) (I might be wrong here...)

However it is possible to take a tesseract swock, chop an opposite pair of cubical cells off, and then roll it up to from a weird cubinder like object but with the cylinders replaced by an extruded anulus (i.e. you get a trihose)


In conclusion I think a new naming scheme is needed to distinguish the A#B shapes from the B#A shapes
Corrected all instances of "cublinder" to "cubinder". I don't usually bug people about spelling, but since you linked to the wiki page for it in this very post... ~Keiji
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Re: circle # square vs square # circle

Postby wendy » Sun Jan 15, 2012 8:28 am

The equation given for the torocylinder is a (sphere-circle)-comb, while the torocubinder is given as a ((anulus × circle ) crind. The two are different.

The second is an analus-circle crind product. A crind product of X, Y, Z might be represented by overlays of prisms of xX, yY, zZ, ... where x,y,z, ... is the coordinates of a sphere. In the case of an analus, we see that placing an analus-line crind, would give an circle (a+r, a+r, a), with a second ellipsoid (a-r,a-r,a) removed. At the point (0,0,1) the two ellipsoids meet, so four surfaces come together.

The full figure for the torocubinder here is then rss(a+r, a+r, a, a) - rss(a-r,a-r,a,a). which meets at a line-thin junction at (0,0,z,w).
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Re: circle # square vs square # circle

Postby Secret » Sun Jan 15, 2012 12:52 pm

wendy wrote:The equation given for the torocylinder is a (sphere-circle)-comb, while the torocubinder is given as a ((anulus × circle ) crind. The two are different.

The second is an analus-circle crind product. A crind product of X, Y, Z might be represented by overlays of prisms of xX, yY, zZ, ... where x,y,z, ... is the coordinates of a sphere. In the case of an analus, we see that placing an analus-line crind, would give an circle (a+r, a+r, a), with a second ellipsoid (a-r,a-r,a) removed. At the point (0,0,1) the two ellipsoids meet, so four surfaces come together.

The full figure for the torocubinder here is then rss(a+r, a+r, a, a) - rss(a-r,a-r,a,a). which meets at a line-thin junction at (0,0,z,w).


I think I'm still a bit confused on the crind product, maybe its because I have not learnt vector calculus yet or that my matrix knowledge is still preliminary, thus I failed to manipulate the coordinates to soemthing I can understand easily

I'm also confused about the torocylinder you give about also
http://teamikaria.com/hddb/dl/ZG7347AWKYGB46VJ77QW2YA91F.jpg
Unless I misunderstood the comb product I can't seem to find any way to fold that cylinder into a torinder

Do you mean torocylinder = (Circle x line) # circle? (# is spheration)
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Re: circle # square vs square # circle

Postby Keiji » Mon Jan 16, 2012 8:46 am

Secret, please read the page on four-dimensional torii, this may answer many of your questions.

As that page explains, both forms are made by folding a spherinder; the outside fold (bending the line of extrusion into a circle) forms a toracubinder while the inside fold (folding through the line of extrusion) forms a toraspherinder. Thus they are not named after the shape they are folded up from (since they're both folded up from the same shape), but instead are named because the toratopic notation corresponds with the shape they are named after (see list of toratopes). I learnt this the hard way many years ago.
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Re: circle # square vs square # circle

Postby wendy » Mon Jan 16, 2012 8:50 am

The various radiant products are derived in this manner.

You take a direction, like (3,4), and calculate its radial coordinate as where the ray from (0,0) to (3,4) crosses the surface. For a square of the standard coordinate, it's (+/- 1, +/- 1), the ray crosses at (3/4, 1). A standard circle is radius 1, gives (3/5, 4/5). The standard rhomb has vertices at (1,0), (0,1), etc, gives (3/7, 4/7).

For a radiant product, the surface is always at distance '1'. This makes the polytope into a kind of function where the centre is 0, and the surface is always 1. Any point greater than 1 is outside, less than 1 is inside. You then set up a figure up so that its elements are mutually perpendicular at the centre, and then for a general point, its coordinates are, eg (a,b), where eg a is the radius in x,y,z, and b is the radius in (u,v,w). The general coordinate is then calculated by applying a function to a,b.

The prism-product is then r= max(a,b), the tegum-product is then r = sum(a,b), and the crind is r = rss(a,b), ie r² = a²+b².

For our point a,b, then max(a,b) = 4, sum(a,b) = 7, and rss(a,b) = 5. For a standard line, of coordinate (1) to (-1), centre 0, we see that the values of the coordinates of 3,4 gives a radius of 4, 7 and 5, and so the surface crosses the ray (3,4) at (3/4, 4/4), or (3/7, 4/7), or (3/5, 4/5).

Likewise, for a cylinder, = prism(crind(x,y), z) gives max(rss(x,y), z), the point 3,4,5 would then cross the surface at max(rss(3,4),5) = max(5,5) = 5. This is at the notional point (3/5, 4/5, 1).
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Re: circle # square vs square # circle

Postby Secret » Mon Jan 16, 2012 11:20 am

Keiji wrote:[...]Thus they are not named after the shape they are folded up from (since they're both folded up from the same shape), but instead are named because the toratopic notation corresponds with the shape they are named after (see list of toratopes). I learnt this the hard way many years ago.


Yes you are right
This answers my question on how to fold the tiger

The problem is when I visit the Toratopic notation there are still things yet to be clarify
e.g. Toracubinder = ((II)II)
From that page I know
()=spheration , from this forum I know spheration is to puff all the points on the shape into circles
|= digon
and total number of |= dimension of shape

But what does the lines sticking together mean e.g. ||
and what does () sticking to lines mean (e.g. (|)|)?

Also you missed the torinder in that page


wendy wrote:
[...]You then set up a figure up so that its elements are mutually perpendicular at the centre, and then for a general point, its coordinates are, eg (a,b), where eg a is the radius in x,y,z, and b is the radius in (u,v,w).[...]

For our point a,b, then max(a,b) = 4, sum(a,b) = 7, and rss(a,b) = 5. For a standard line, of coordinate (1) to (-1), centre 0, we see that the values of the coordinates of 3,4 gives a radius of 4, 7 and 5, and so the surface crosses the ray (3,4) at (3/4, 4/4), or (3/7, 4/7), or (3/5, 4/5).

Likewise, for a cylinder, = prism(crind(x,y), z) gives max(rss(x,y), z), the point 3,4,5 would then cross the surface at max(rss(3,4),5) = max(5,5) = 5. This is at the notional point (3/5, 4/5, 1).

I don't understand the bolded part, therefore I don't understand how dividing the coordinate of (a,b) for a ray passing a,b must equal to the intersection point of the ray passing through (a,b) and the surface of the polytope under consideration

Also in the cylinder case above, do you mean I trace out the cylinder by considering that looking from the xz plane the figure looks like a circle (thus using rss(a,b) to trace out this circle). When I switched to the xy plane the figure look like a square (thus applying max(c,d) to trace out the square, but because two of the sides of this square in the view are actually circles, I need to make c=rss(a,b)?
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Re: circle # square vs square # circle

Postby Keiji » Mon Jan 16, 2012 1:10 pm

Secret wrote:But what does the lines sticking together mean e.g. ||
and what does () sticking to lines mean (e.g. (|)|)?


See Rotope#Notations, although this works with the obsolete group and digit notations you can easily transform toratopic notation into these.

Secret wrote:Also you missed the torinder in that page


No I didn't, the torinder is an open toratope so it's in the left column instead of the right one.
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Re: circle # square vs square # circle

Postby wendy » Tue Jan 17, 2012 7:31 am

I'll try to explain radiant products with examples.

Consider the square, whose vertices are (1,1), (1,-1), (-1,-1) and (-1,1). It is bounded by the equation max(abs(x), abs(y)) = 1.

You can then plot in space, all sorts of squares, like (2,2), (2,-2), (-2,-2), and (-2,2), bounded by the equation max(abs(x), abs(y) = 2.

Every point in this space, then has a radial coordinate equal to the maximum absolute coordinate, so the point (3,5) falls first on a square (5,5) etc.

We now take a line, (1), (-1), where the general radiant coordinate is abs(z).

A product of two figures, say a square and line, goes with the square in (X,Y,0), and the line in (0,0,Z). When we calculate the loose radiant functions, this gives a point-pair (a,b), where a = max(abs(x),abs(y)), and b = abs(z). The surface of the radiant product is then a function of (a,b) by product. a is derived entirely from x,y and b from z. So for the point 3,4,5. here, a=4, b=5.

The intersection of cylinders is the crind product, so we calculate rss(a,b), ie r² = max(x², y²)+z² , which we set the surface to 1.

A ray passing from a point (0,0,0) to (3,4,5) is evaluated using this formula (r² = ..) gives max(9,16)+25 = 41. The intersect at the surface is then at a point of 3/sqrt(41), 4/sqrt(41), 5/sqrt(41).
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