Let X be a compact (or periodic) union of integer translates of unit cubes such that the interior of X is connected. (If it makes any difference, suppose that the dimension n of X is 3.) I am interested in finding a minimal (or at least reasonably small) and preferably “nice” decomposition of X into … Read more
Can shapes determined by some number of points? From an amazing theorem in plane curves geometry we know that vertices of triangles similar to arbitrary triangle $T$ is dense on every closed jordan curve in a plane, so if $J$ be such curve and $A,B,C$ be 3 noncollinear points on plane then at least one … Read more
The icosahedron and dodecahedron are well known to share symmetry groups. This partially accounts for the fact that one can form a type of compound of the two where each of the vertices in the icosahedron are the points, on the same sphere as the vertices of the dodecahedron, which are at a maximal distance … Read more
Since each finite group G can be considered as a subgroup of the symmetric group, by Cayley’s theorem, we might see the elements of G as permutations π. Consider for each π∈G the set: X(π):={(π(i),π(j))|1≤i<j≤|G|} Then the Jaccard similarity-kernel, which is positive definite, is: J(π,π′):=|X(π)∩X(π′)||X(π)∪X(π′)| We can consider the matrix M=(J(g,h)g,h∈G) ordered somehow by an … Read more
Let x1,…,xk∈Rd be k unit vectors in d dimensional Euclidean space, and let S=span(x1,…,xk) be a linear subspace defined by these points. Let P∈Rℓ×d be a ℓ×d random projection matrix as specified by the Johnson-Lindenstrauss lemma, for ℓ≈O(logk/ϵ2). Let y∈Rd be an arbitrary unit vector, and define d(y,S)=min to be the distance between y to … Read more
Closed. This question needs details or clarity. It is not currently accepting answers. Want to improve this question? Add details and clarify the problem by editing this post. Closed 3 years ago. Improve this question There is a standard construction of a regular heptadecagon by H.W. Richmond (1893) (https://en.wikipedia.org/wiki/Heptadecagon ) (As anyone knows, it was … Read more
In 2D one cannot construct Pythagorean triples x2+y2=m2 (x,y,m∈Z) that lie on every line through the origin (e.g., a Pythagorean triple with x=y would require √2 to be rational). What happens when moving to planes in 3D? Given a,b,c∈Z can one find x,y,z,m∈Z such that m≠0, x2+y2+z2=m2, and ax+by+cz=0? I would be happy with a … Read more
My question: I am looking for a proof of problem as following: Introduction: When I research a theorem as following: Theorem 1: Let ABC be a triangle, let (S) be a circumconic of ABC, let P be a point on the plane. Let the lines AP,BP,CP meet the conic again at A′,B′,C′. Let D be … Read more
I draw a cyclic spherical hexagon and I check by geogebra that Pascal’s theorem is true in this case. My question 1. Is there simple proof for this? My question 2. Can we change the circle on sphere by the curve like “conic” in plane? My question 3. If we use orthogonal projection to project … Read more
Na, Nb, Nc are the outer Nagel points. A’B’C’ is the contact triangle. I claim that lines A’B’, A’C’, B’C’ always cut the sides of the triangle NaNbNc at six points corresponding to an unknown circle. Surprisingly its center is not defined in the ETC: Is it safe to say that this is actually the … Read more
