Recent content by Philip

No, this isn't how you would define a shape. You would have to go about entirely different means of deriving the equation. What mfb is trying to tell you is that you could define a chair in 3D, as a product of several rectangular prisms, arranged into a chair-like object. Actually, for a...

With respect to what, though? That's like saying a cube in 4D space is 180°. What exactly are you measuring? Where are you getting the 45° from? Can you share this formula with us? How can it be 60° in 2D? It's not a spatial warping, that deals with non-euclidean (hyperbolic, spherical)...

I'm not sure why you go from a 180° for a 1D line, to a 90° angle within a 2D square, to 45° for a cube. For n-dimensional cubes, all 1D edges meet at the vertices, at 90° angles of each other. The 1D line has only one edge and two vertices, so there are no angles between anything, because...

Well, after a LOT of playing around, experimenting, and searching for some kind of meaning here, I finally found it. The circles are in the complex plane, and the roots of z, as stated above, are the simplest possible expression, other than something like ##z = -\sqrt{(b-a)(b+a)} , z =...

Yes, that's a good point. It does reduce to a product of two quadratic equations for a circle. I guess what I'm trying to understand, is by considering this 2D solution of a torus: https://www.desmos.com/calculator/qmk3bia8ia If solving for x, can we say these circles lie in the complex plane...

Thanks for your reply! Assuming a > b, I'm considering the ring torus. Placed at origin, the z-axis will sit in the center of the hole. Translating the torus by the value of 'a' , along x or y, will make z intersect the ring, making two points. More precisely, translate by 'a' along x (where...

The equation for a torus defined implicitly is, $$(\sqrt{x^{2} + y^{2}} -a)^{2} + z^{2} = b^{2}$$ When solving for the z-axis in the torus equation, we get complex solutions, from the empty intersection: $$z = - \sqrt{b^{2} - a^{2}}$$ $$z = \sqrt{b^{2} - a^{2}}$$ I was told by someone that...

Check out Graham's number: http://en.wikipedia.org/wiki/Graham%27s_number , it's right up there.

Ah, yes, higher dimensional objects. These entities are strange, aren't they? So unimaginable, they defy intuition and common 3D sense. I know the feeling. Even though it's impossible to represent a true 4D object in it's full 4D glory, we can still cancel out a dimension and view it in 3D...

It's the other equation I wrote that used complex numbers, not the standard implicit form of (sqrt(x^2 + y^2) - a)^2 +z^2 -b^2 = 0 I used the function (x-1-3)(x+1-3)(x-1+3)(x+1+3) + (y-1-3)(y+1-3)(y-1+3)(y+1+3) + (z-1-3i)(z+1-3i)(z-1+3i)(z+1+3i) + 2((xy)^2 + (xz)^2 + (yz)^2) - 2(3^4 + 1^4)...

Well, that's the thing. When I put those proposed four solutions for z in the torus equation, as graphed in above image, it came out correct. The imaginary numbers define the hole, and the empty space. I understand how it seems wrong. I'm not sure how to properly prove any of it to you. Other...

Ah, probably what you were looking for was, full torus equation (sqrt(x^2 + y^2) - a)^2 +z^2 -b^2 = 0 set x, y to 0, making equation for z-intercepts (-a)^2 + z^2 - b^2 = 0 (-a)^2 + z^2 = b^2 z^2 = b^2 - (-a)^2 z = ± sqrt(b^2 - (-a)^2) (apologies if this flow is wrong, if so, how should it...

I came across this strange relationship when deriving the degree-4 equation for a torus. First thing that comes to mind is the 'Freshman's Dream'. Apparently, it was pure coincidence that they are equal. But, I don't believe in coincidences when it comes to a math expression. There is something...

You'll find the max function in a program like mathematica. It's a different way to express a surface, as a different kind of equation. Probably, I'm not sure. I learned this from self study. I don't think I've seen those implicits anywhere else, other than the forum I helped develop them on.

This is true, negative length does not apply. For all functions above, the value of 'a' and 'b' must be positive. Anything negative, and shape will not appear. Only the coordinate dimensions are using the absolute values, not the size parameter 'a' and/or 'b' . These functions are the long-hand...