Do similar/nearby combinatoric sequences accelerate in the same way? Can various sequences be taken together to meaningfully produce a higher resolution (more continuous) sequence?
seems to consist of one finite and various infinite trees, every graph that's just one cycle, the triangle grid, square grid, and hexagonal grid, and just one more that I can find where each vertex neighbours four triangles and a rhombus
I've not seen this notation, it's a bit of a mouthful, and probably allows all sorts of figures not possible geometrically, but it is very general and kind of what I am looking for at the moment.
I'm thinking a schläfi notation tetrahedron {3, 3}, or a cubic honeycomb {4, 3, 4} (note thease don't need to be able to fit into flat space) and in the archamedian notation truncated cube {8, 8, 3}. How about a (probably hyperbolic) honeycomb made of lets with three truncated cubes two non adjacent octahedrons around each edge notated as something like {{8, 8, 3}, {8, 8, 3}, {3, 3, 3, 3}, {8, 8, 3}, {3, 3, 3, 3}}.
I expect such a theorem to already exist, for if it doesn't, I posit the Intermediate Slice Theorem. It makes statements analogous to the intermediate value theorem.
$d$ is a member of N.
P = $\mathbb{R}^{d}$
Q is an $\mathbb{R}^{d - 1}$ dimensional line in P.
A 0 dimensional line is a point,...
I assume that more intricate formulations based on the "intermediate value theorem" such as the I'm hungry and want a ham sandwich problem, table and planet examples must work in R and R<sup>2</sup>. And presumably therefore in vectors. And even more presumably in tensors and in the complex plane.
2 points moving smoothly (meaning without teleporting) along a line starting at A and B and then moving to each others starting locations have to pass each other.
A very obvious statement, but you can build on it to make the following statements as well.
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A convex shape is a shape where any two points on the shape must be able to be joined with a straight line that stays inside the shape. Convex shapes include rectangles and star shapes and the shape of a comma. The shape of the letter 'c' is not convex, because it's got an alcove.
Constructive Feedback
Constructive Feedback
I expect such a theorem to already exist, for if it doesn't, I posit the Intermediate Slice Theorem. It makes statements analogous to the intermediate value theorem.
$d$ is a member of N.
P = $\mathbb{R}^{d}$
Q is an $\mathbb{R}^{d - 1}$ dimensional line in P.
A 0 dimensional line is a point,...