Articles of geometry

Normed-Space; bound needed for $||x|| + ||y|| – ||x+y||$

Given x and y, is there any way we can express $||x|| + ||y|| – ||x+y||$ in terms of $||y-x||$? Even a bound where $||x|| + ||y|| – ||x+y|| \leq f(||y-x||)$ for some $f(\cdot)$ would be desirable. Geometrically, ||x|| and ||y|| could be two sides of a parallelogram and then $||x+y||$ and $||y-x||$ would be […]

Geometric intuition: Seeing the regions in double integrals

Context: solving double integrals. I had the formula $$x^2+y^2=1-x-y$$ yet I could not see what shape it had. This is even more true with 3D pictures like $$2x^2+2y^2 \le 1+z^2.$$ Is there a summary somewhere of shapes to learn, so that I can get this.

Number of reflection symmetries of a basketball

Excerpt from John Horton Conway, The Symmetries of Things, pg. 12. Basketballs have two planes of reflective symmetry, as do tennis balls. I read this sentence and it immediately struck me as incorrect: from my understanding of the pattern of lines on a basketball, there are three planes of reflective symmetry. Two correspond to the […]

Stereographic projection (Theorem that circles on the sphere get mapped to circles on the plane)

I’m trying to understand the proof of the theorem (given in the link) that states “Stereographic projection maps circles of the unit sphere, which do not contain the north pole, to circles in the complex plane.” Link to the proof In the proof it states “In order to obtain an equation for the projection points […]

structure and properties of a function inherited from its integrals

Let $f:[0,1]^2 \rightarrow \{0,1\}$, $f_B(b) = \int_0^1 f(b, s)\; ds$ and $f_S(s) = \int_0^1 f(b, s)\; db$, such that $f_B$ is non-decreasing and $f_S$ is non-increasing. What can we infer about $f$ from the monotonicity of $f_B$ and $f_S$? For instance, the following class of functions satisfies the monotonicity constraints \begin{align*} f(b,s) = \begin{cases} 1 […]

What is the analogon of the hexagonal grid in 3-dimensional space? Rhombic dodecahedral honeycomb?

Conjecture: The optimal way to divide 3-space into pieces of equal volume with the least total surface area is the rhombic dodecahedral honeycomb. Reasoning: “(The rhombic dodecahedral honeycomb) is the Voronoi diagram of the face-centered cubic sphere-packing, which is the densest possible packing of equal spheres in ordinary space.” (Wikipedia) This resembles very closely how […]

Geometries (Euclidean and Projective)

We can think of Euclidean Geometry and Cartesian (Coordinate) Geometry as equivalent, in the sense that some proposition is true in Euclidean Geometry iff it’s true in Coordinate Geometry. It makes sense to me to think of Coordinate Geometry as Euclidean Geometry “arithmetized”, or even as a model of Euclidean Geometry. I’m comfortable with it […]

Puzzle : cable in the field

Here is an interesting puzzle taken from There is a straight cable buried under a unit square field. You must dig one or >more ditches to locate the buried cable. Where should you dig to guarantee >finding the cable and to minimize digging? For example you could dig an X shape >for total ditch […]

A cone inscribed in a sphere

I have question regarding the possibility of inscribing a cone in a sphere. Essentially, I am looking for some proper definition of that. When dealing with polyhedra, the matter is simple: each of the polyhedron’s vertices must lie on the sphere. However, what conditions need to be met in order for a cone to be […]

Beautiful triangle problem

Circle, inscribed in $ABC$, touches $BC, CA, AB$ in points $A’, B’, C’$. $AA’ BB’, CC’$ intersect at $G$. Circumcircle of $GA’B’$ crosses the second time lines $AC$ and $BC$ at $C_A$ and $C_B$. Points $ A_B, A_C, B_C,B_A, C_A, C_B$ are concyclic. Looks straightforward, but I’m struggling to get something out of the given. […]