Articles of geometry

Sphere on top of a cone. Maximum volume?

I received an interesting problem which I can not figure how to solve. It follows: An ice cream has the shape of a sphere and a cone like the image below. What is the maximum volume that the sphere can occupy out of the cone? Note that $M$ is the center of the sphere, not […]

Parametric equations for hypocycloid and epicycloid

Suppose that the small circle rolls inside the larger circle and that the point $P$ we follow lies on the circumference of the small circle. If the initial configuration is such that $P$ is at $(a,0)$, find parametric equations for the curve traced by $P$, using angle $t$ from the positive $x$-axis to the center […]

Finding points on two linear lines which are a particular distance apart

I have two linear, skew, 3D lines, and I was wondering how I could find a points on each of the lines whereby the distance between the two points are a particular distance apart? I’m not after the points where the lines are the closest distance apart, nor the furthest distance apart, nor the point […]

Functions determine geometry … Riemannian / metric geometry?

Given a compact Riemannian manifold $(M,g)$, is there a subring of $C^{\infty}(M)$ that determines the isomorphism class of $(M,g)$? (In the same way that $C^{\infty}(M)$ determines the diffeomorphism class via the max spectrum.) Same question for compact metric spaces, more generally. I am not sure what condition to ask for on these functions… maybe something […]

Tarski-like axiomatization of spherical or elliptic geometry

Preamble Tarski’s axioms formalize Euclidean geometry in a first-order theory where the variables range over the points of the space and the primitive notions are betweenness $Bxyz$ (meaning $y$ is on the line segment between $x$ and $z$, inclusive) and congruence $xy\equiv zw$ (meaning the line segment between $x$ and $y$ is the same length […]

How can I pack $45-45-90$ triangles inside an arbitrary shape ?

If I have an arbitrary shape, I would like to fill it only with $45-45-90$ triangles. The aim is to get a Tangram look, so it’s related to this question. Starting with $45-45-90$ triangles would be an amazing start. After the shape if filled I imagine I could pick adjacent triangles, either $2$ or $4$ […]

Least greedy square

There are $n$ squares of $m$ different colors. Squares of the same color are interior disjoint, but squares of different colors may intersect. For every square, define its “greed” as the maximum number of squares of a single color that it intersects. For example, in the figure below, the top-left red square has a greed […]

Prove the opposite angles of a quadrilateral are supplementary implies it is cyclic.

There is a well-known theorem that a cyclic quadrilateral (its vertices all lie on the same circle) has supplementary opposite angles. I have a feeling the converse is true, but I don’t know how to prove it. The converse states: If a quadrilateral’s opposite angles are supplementary then it is cyclic. Should I approach this […]

Why does the amoeba shrink to its skeleton when we go to infinity?

Let $f\in\mathbb{C}[X_1^{\pm1},\ldots,X_n^{\pm1}]$ a Laurent polynomial. Let $\mathrm{Log}:(\mathbb{C}\setminus\{0\})^n\to\mathbb{R}^n$ defined by $\mathrm{Log}(z_1,\ldots,z_n)=(\log|z_1|,\ldots,\log|z_n|)$. We call amoeba of $f$ the set $\mathcal{A}_f=\mathrm{Log}(f^{-1}(0))$, i.e. the image of the variety defined by $f$ under the logarithm. It can be shown that there exists a convex subdivisions of $\mathbb{R}^n$ such that, each $n$-dimensional cell contains exactly one connected component of $\mathbb{R}^n\setminus\mathcal{A}_f$. Then […]

Rigorous synthetic geometry without Hilbert axiomatics

Are there books or article that develop (or sketch the main points) of Euclidean geometry without fudging the hard parts such as angle measure, but might at times use coordinates, calculus or other means so as to maintain rigor or avoid the detail involved in Hilbert-type axiomatizations? I am aware of Hilbert’s foundations and the […]