moonlord wrote:@pat: I can't explain the notation better than PWrong, so I'll let him do it . If you take a rubber cylinder and stick the bases together, bending the cylinder, you get a torus. As for the other spherations, I must admit I don't fullly understand them.
pat wrote:And, if circle = 2 and circle#circle = (21) and circle#circle#circle = ((21)1), and duocylinder = 22, then how is duocylinder#circle = (22) and not (221)?
Additionally, there's nothing in the notation that says whether you're producting with a circle or a sphere or a glome or what-have-you, is there? Is it assumed that if you start with a k-dimensional object, you spherize with a k-dimensional sphere?
Also, I thought that torus products were commutative? But, the toratope list shows circle#sphere as distinct from sphere#circle.
I'm also a bit lost on the "english" versions. How is a toraspherinder (211) related to a spherinder 31? Are the names skewed in the toratopes or is my understanding that far behind?
moonlord wrote:@iNVERTED: Well, it makes some sense now. Therefore, a 3-cube is one of the hypercubes, and so on? Ok, it does make sense.
@pat: I can't explain the notation better than PWrong, so I'll let him do it . If you take a rubber cylinder and stick the bases together, bending the cylinder, you get a torus. As for the other spherations, I must admit I don't fullly understand them.
In particular, the notation "21" for cylinder and "(21)" for torus seem at odds. The torus is not the "spheration" of the cylinder (at least not in any way that I can see).
I'm also a bit lost on the "english" versions. How is a toraspherinder (211) related to a spherinder 31? Are the names skewed in the toratopes or is my understanding that far behind?
And, it seems weird that the 3-torus is written ((21)1). That seems to belie the symmetry that circle#circle#circle has.
Also, I thought that torus products were commutative? But, the toratope list shows circle#sphere as distinct from sphere#circle.
Isn't the cartesian product of two circles a sphere?
EDIT: Now writing, I've figured out about the duocylinder. Being a 22, it's cartesian are x^2 + y^2 = 1 and z^2 + w^2 = 1 which make a 2D figure afterall... In the begining, I've assumed a dimension is common to those two circles...
PWrong wrote:If both are centred on the origin, you get a sphere. If not, you get a torus.
In a sense, it is a special kind of torus. But for classification purposes, I think it's better to keep them separate. The sphere is also a special case of the spherinder, ellipsoid, and countless other shapes, but it's not always worth mentioning this.So shouldn't the sphere be treated as a special kind of torus?
Marek14 wrote:Also, I thought that torus products were commutative? But, the toratope list shows circle#sphere as distinct from sphere#circle.
That's right, they are different. If you slice circle*sphere with a coordinate hyperplane, you get either two separate spheres, or a torus. If you slice sphere*circle, you get either two concentric spheres, or a torus.
PWrong wrote:Here's another definition.
A "beast" is an object that includes a bracket containing two or more sets of brackets. For instance, (22) = ((11)(11)) and ((21)3) are beasts, but ((31)111) is not. Generally, beasts are the hardest objects to visualise.
Now, you can easily write any non-beast rotope with torus products. Start with the innermost bracket and work outwards, counting the number of elements in each bracket.
For example, ((31)111) = 3 # 2 # 4
It's harder to convert a beast into the torus product of spheres. In fact I'm not sure if it's possible.
For instance, the circle#sphere is simply a spherinder, stretched and attached at the ends. The sphere#circle is like a cubinder, wrapped up like a sheet around a ball.
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