# Tagged Questions

shape, congruence, similarity, transformations, properties of classes of figures, points, lines, angles

9answers
293k views

### Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
13answers
11k views

### Is value of $\pi = 4$?

What is wrong with this? SOURCE
6answers
58k views

### How many sides does a circle have?

My son is in 2nd grade. His math teacher gave the class a quiz, and one question was this: If a triangle has 3 sides, and a rectangle has 4 sides, how many sides does a circle have? My first ...
4answers
1k views

### Hyperbolic critters studying Euclidean geometry

You've spent your whole life in the hyperbolic plane. It's second nature to you that the area of a triangle depends only on its angles, and it seems absurd to suggest that it could ever be otherwise. ...
3answers
4k views

### Can someone explain the math behind tessellation?

Tessellation is fascinating to me, and I've always been amazed by the drawings of M.C.Escher, particularly interesting to me, is how he would've gone about calculating tessellating shapes. In my ...
3answers
6k views

### Why can a Venn diagram for 4+ sets not be constructed using circles?

This page gives a few examples of Venn diagrams for 4 sets. Some examples: Thinking about it for a little, it is impossible to partition the plane into the 16 segments required for a complete 4-set ...
3answers
3k views

### What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into?

What is the smallest number of $45^\circ-60^\circ-75^\circ$ triangles that a square can be divided into? The image below is a flawed example, from http://www.mathpuzzle.com/flawed456075.gif ...
7answers
4k views

### Are non-circular manholes possible?

Circular manholes are great because the cover can not fall down the hole. If the hole were square, the heavy metal cover could fall down the hole and kill some man working down there. Circular ...
4answers
1k views

### Probability that a stick randomly broken in five places can form a tetrahedron

Randomly break a stick in five places. Question: What is the probability that the resulting six pieces can form a tetrahedron? Clearly satisfying the triangle inequality on each face is a necessary ...
1answer
4k views

### Gerrymandering on a high-genus surface/can I use my powers for evil?

Somewhat in contrast to this question. Let's say the Supreme Court has just issued a ruling that the upper and lower roads of an overpass need not be in the same congressional district. This makes ...
4answers
4k views

### Why is a circle in a plane surrounded by 6 other circles

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other number? I'm ...
13answers
3k views

### Why is the volume of a sphere $\frac{4}{3}\pi r^3$?

I learned that the volume of a sphere is $\frac{4}{3}\pi r^3$, but why? The $\pi$ kind of makes sense because its round like a circle, and the $r^3$ because it's 3-D, but $\frac{4}{3}$ is so random! ...
19answers
14k views

### Software for drawing geometry diagrams

What software do you use to accurately draw geometry diagrams?
10answers
5k views

### What's a proof that the angles of a triangle add up to 180°?

Back in grade school, I had a solution involving "folding the triangle" into a rectangle half the area, and seeing that all the angles met at a point. However, now that I'm in university, I'm not ...
3answers
1k views

### Volumes of n-balls: what is so special about n=5?

The volume of an $n$-dimensional ball of radius $1$ is given by the classical formula $$V_n=\frac{\pi^{n/2}}{\Gamma(n/2+1)}.$$ For small values of $n$, we have $$V_1=2\qquad$$ $$V_2\approx 3.14$$ ...
11answers
3k views

### What is a hexagon?

Having a slight parenting anxiety attack and I hate teaching my son something incorrect. Wiktionary tells me that a Hexagon is a polygon with $6$ sides and $6$ angles. Why the $6$ angle requirement? ...
4answers
2k views

### Do circles divide the plane into more regions than lines?

In this post it is mentioned that $n$ straight lines can divide the plane into a maximum number of $(n^{2}+n+2)/2$ different regions. What happens if we use circles instead of lines? That is, what ...
6answers
3k views

### How come $32.5 = 31.5$?

Below is a visual proof (!) that $32.5 = 31.5$. How could that be?
3answers
839 views

### Putting many disks in the unit square

Consider a square of side equal to $1$. Prove that we can place inside the square a finite number of disjoint circles, with different radii of the form $1/k$ with $k$ a positive integer, such that ...
4answers
1k views

### A circle rolls along a parabola

I'm thinking about a circle rolling along a parabola. Would this be a parametric representation? $(t + A\sin (Bt) , Ct^2 + A\cos (Bt) )$ A gives us the radius of the circle, B changes the frequency ...
4answers
1k views

### Why is a full turn of the circle 360°? Why not any other number?

I was just wondering why we have 90° degrees for a perpendicular angle. Why not 100° or any other number? What is the significance of 90° for the perpendicular or 360° for a circle? I didn't ever ...
11answers
4k views

### What is the most elegant proof of the Pythagorean theorem?

The Pythagorean Theorem is one of the most popular to prove by mathematicians, and there are many proofs available (including one from James Garfield). What's the most elegant proof? My favorite ...
8answers
5k views

### What is the meaning of the third derivative of a function at a point

What is the geometric, physical or other meaning of the third derivative of a function at a point? (Originally asked on MO by AJAY) If you have interesting things to say about the meaning of the ...
3answers
2k views

### Why is the Möbius strip not orientable?

I am trying to understand the notion of an orientable manifold. Let M be a smooth n-manifold. We say that M is orientable if and only if there exists an atlas $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
2answers
643 views

### Optimal yarn balls

Winding yarn into a ball suggests some mathematical questions: Under some natural model, what paths should the yarn follow to achieve the most dense ball? One model is that used by Henryk Gerlach ...
4answers
6k views

### Why is Pi equal to 3.14159…?

Wait before you dismiss this as a crank question :) A friend of mine teaches school kids, and the book she uses states something to the following effect: If you divide the circumference of any ...
1answer
831 views

### Dividing a square into equal-area rectangles

How many ways are there to tile an $n\times n$ square with exactly $n$ rectangles, each of which has integer sides and area $n$? The sequence $C(n)$ begins 1, 2, 2, 9, 2, 46, 2, 250, 37. Clearly ...
7answers
8k views

### Why is the volume of a cone one third of the volume of a cylinder?

The volume of a cone with height $h$ and radius $r$ is $\frac{1}{3} \pi r^2 h$, which is exactly one third the volume of the smallest cylinder that it fits inside. This can be proved easily by ...
6answers
5k views

### Why is the derivative of a circle's area its perimeter (and similarly for spheres)?

When differentiated with respect to $r$, the derivative of $\pi r^2$ is $2 \pi r$, which is the circumference of a circle. Similarly, when the formula for a sphere's volume $\frac{4}{3} \pi r^3$ is ...
1answer
744 views

### Ellipse 3-partition: same area and perimeter

Inspired by the question, "How to partition area of an ellipse into odd number of regions?," I ask for a partition an ellipse into three convex pieces, each of which has the same area and the same ...
6answers
7k views

### How many triangles are there?

The question is how many triangles are there in the following picture? I have thought to solve it by creating a formula based on the angles of the lines starting from the bottom of each side. I ...
1answer
486 views

### About Euclid's Elements and modern video games

I just watched this video about Euclid's treatise the Elements. I got introduced to the postulates and a couple of propositions of book I. I really liked this video, I'm not sure if this is because of ...
5answers
2k views

### Pythagorean Theorem Proof Without Words (request for words)

I was intrigued by a book I saw called Proofs without Words. So I bought it, and discovered that the entire book doesn't have any words in it. I figured at least it would have some words explaining ...
5answers
1k views

### Quotient geometries known in popular culture, such as “flat torus = Asteroids video game”

In answering a question I mentioned the Asteroids video game as an example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It ...
4answers
767 views

### Volume of Region in 5D Space

I need to find the volume of the region defined by \begin{align*} a^2+b^2+c^2+d^2&\leq1,\\ a^2+b^2+c^2+e^2&\leq1,\\ a^2+b^2+d^2+e^2&\leq1,\\ a^2+c^2+d^2+e^2&\leq1 &\text{ ...
3answers
1k views

### Have I made a straight line, or a circle?

(Disclaimer: I'm an engineer) Hi everybody, I found this “riddle” posted on the internet: It's meant as a joke, but I do think it deserves an answer :) A bit of background: the orange and blue ...
5answers
988 views

### How to find a random axis or unit vector in 3D?

I would like to generate a random axis or unit vector in 3D. In 2D it would be easy, I could just pick an angle between 0 and 2*Pi and use the unit vector pointing in that direction. But in 3D I ...
3answers
319 views

### Picture of a 4D knot

A knot is a way to put a circle into 3-space $S^1 \to \mathbb R^3$ and these are often visualized as 2D knot diagrams. Can anyone show me a diagram of a nontrivial knotted sphere \$S^2 \to \mathbb ...
3answers
901 views

### Guaranteed Checkmate with Rooks in High-Dimensional Chess

Given an infinite (in all directions), $n$-dimensional chess board $\mathbb{Z^n}$, and a black king. What is the minimum number of white rooks necessary that can guarantee a checkmate in a finite ...
2answers
504 views

### On nonintersecting loxodromes

The (spherical) loxodrome, or the rhumb line, is the curve of constant bearing on the sphere; that is, it is the spherical curve that cuts the meridians of the sphere at a constant angle. A more ...
2answers
321 views

### New twist on a Putnam problem

A recent Putnam problem: Let $f$ be a real-valued function on the plane such that for every square $\square ABCD$ in the plane, $f(A)+f(B)+f(C)+f(D)=0$. Does it follow that $f$ is identically ...
1answer
444 views

### Rolling a Sphere on the Plane

Suppose one starts with a sphere $S$ resting on a ($2$-dimensional) plane $H$ at the origin. A "move" consists of the following: Let $P$ and $Q$ be two points in $H$. Roll the sphere $S$ along a ...
8answers
4k views

### What is the (mathematical) point of geometric constructions?

The ancient discipline of construction by straightedge and compass is both fascinating and entertaining. But what is its significance in a mathematical sense? It is still taught in high school ...
4answers
1k views

### What are the dangers of visual exposition of mathematics?

I've heard several times (such as this one) that it's dangerous to learn/prove/teach mathematics through images. I've also read somewhere that showing mathematics through images helps one's intuition ...
6answers
707 views

### Trying to understand why circle area is not $2 \pi r^2$

I understand the reasoning behind $\pi r^2$ for a circle area however I'd like to know what is wrong with the reasoning below: The area of a square is like a line, the height (one dimension, length) ...
2answers
281 views

### About translating subsets of $\Bbb R^2.$

I'm looking for a pair of sets $A,B$ of points in $\Bbb R^2$ such that $A$ is a union of translated (only translations are allowed) copies of $B;$ $B$ is a union of translated copies of $A;$ $A$ is ...
2answers
311 views

### A strangely connected subset of $\Bbb R^2$

Let $S\subset{\Bbb R}^2$ (or any metric space, but we'll stick with $\Bbb R^2$) and let $x\in S$. Suppose that all sufficiently small circles centered at $x$ intersect $S$ at exactly $n$ points; if ...
5answers
3k views

### Proof that Pi is constant (the same for all circles), without using limits

Is there a proof that the ratio of a circle's diameter and the circumference is the same for all circles, that doesn't involve some kind of limiting process, e.g. a direct geometrical proof?
0answers
345 views

### Ambiguous Curve: can you follow the bicycle?

Let $\alpha:[0,1]\to \mathbb R^2$ be a smooth closed curve parameterized by the arc length. We will think of $\alpha$ like a back track of the wheel of a bicycle. If we suppose that the distance ...
5answers
357 views

### Can you prove why consecutive diagonal intersection points show decreasing fractions inside a rectangle?

When I was in third grade, I was playing with rectangles and diagonal lines, and discovered something very interesting with fractions. I've shown several math teachers and professors over the years, ...
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