Is there an algorithm to determine the largest equilateral triangle in a convex polygon?

Find 2 tetrahedrons $ABCD$ and $EFGH$ such that $EFGH$ lies completely inside $ABCD$. The sum of edge lengths of $EFGH$ is strictly greater than the sum of edge lengths of $ABCD$. I am completely stumped on this. Seems very counter intuitive to begin. I now have doubts if a solution exists or not. Source : […]

Let $C_n\subset\mathbb{R}^n$ be the $n$-dimensional cube with side $1$, and let $P_k$ be any $k$-dimensional plane, $k\leq n$. What is the maximal $k$-volume $V_{n,k}$ of the projection of $C_n$ on $P_k$? Quite obviously, the minimal area should be $1$, obtained by taking $C_n = [0,1]^n$ and projecting it on $\{\mathbf{x}\in\mathbb{R}^n|x_{k+1}=\ldots=x_n=0\}$. I think the maximum should […]

We have to find the area of the pink region. As we all know this can be evaluated using limiting its Riemann sum, of which its a standard example. However I want to know if this can be done without using calculus, with directly using geometry. I think it would be very interesting challenge, but […]

I have been reading the paper Spiral hexagonal circle packings in the plane (Alan F. Beardon, Tomasz Dubejko and Kenneth Stephenson, Geometriae Dedicata Volume 49, Issue 1, pp 39-70), which proves that “these ’coherent’ [Doyle] spirals, together with the regular hexagonal packing, give all possible hexagonal circle packings in the plane”. On the obvious reading, […]

I need to show that the union of xy-plane and xz-plane, i.e. the set $S:=\lbrace (x,y,z)\in\mathbb{R}^3 : z=0 \mbox{ or } y=0\rbrace$, is not a surface. Here is my claim, $\textbf{Claim :}$ Suppose $p$ is the point $(0,0,0)$ and $U:=S\cap B(p,\epsilon)$, where $\epsilon > 0$. Then $U$ cannot be homeomorphic to any open set of […]

The game I mentioned in “Navigating though the surface of a hypersphere in a computer game” is taking shape in here. The world is a 3-sphere where everything belongs. In Euclidean coordinates, for every point $x^2 + y^2 + z^2 + w^2 = 1$. We move the objects by applying orthornormal transformations to them. We […]

There is a well known theorem often stated as the angle in a semi-circle being $90$ degrees. To be more accurate, any triangle with one of its sides being a diameter and all vertices on the circle has its angle opposite the diameter being $90$ degrees. The standard proof uses isosceles triangles and is worth […]

I’ve taken a small course in Riemann surfaces, and there is one part that I still don’t understand (and I’ve been unable to find a reference that explains this rigorously and in detail). It is about the construction of a Riemann surface associated to an analytic germ. By analytic germ we mean a couple $(z_0, […]

I am trying to characterize the continuum $\mathcal{C}$ using only the notion of midpoint, i.e. the operation $\mu : \mathcal{C}\times\mathcal{C} \to \mathcal{C}$ assigning to each pair of points the midpoint between them. I thought of defining “unbounded interval” as a set $I \subseteq \mathcal{C}$ such that $I$ as well as $\mathcal{C}\backslash I$ are closed under […]

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