Articles of geometry

Largest Equilateral Triangle in a Polygon

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

Can a tetrahedron lying completely inside another tetrahedron have a larger sum of edge lengths?

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 : […]

Volume of the projection of the unit cube on a hyperplane

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 […]

Area under parabola using geometry

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 […]

Hexagonal circle packings in the plane

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, […]

Union of two self-intersecting planes is not a surface

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 […]

How to project the surface of a hypersphere into the full volume of a sphere?

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 […]

Proof of Angle in a Semi-Circle is of $90$ degrees

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 […]

About the Riemann surface associated to an analytic germ

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, […]

Defining “interval” using the notion of midpoint only

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 […]