shape, congruence, similarity, transformations, properties of classes of figures, points, lines, angles
269
votes
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?
157
votes
13answers
11k views
79
votes
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 ...
78
votes
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.
...
76
votes
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 ...
71
votes
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 ...
70
votes
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
...
61
votes
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 ...
58
votes
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 ...
57
votes
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 ...
48
votes
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 ...
44
votes
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! ...
44
votes
19answers
14k views
Software for drawing geometry diagrams
What software do you use to accurately draw geometry diagrams?
43
votes
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 ...
40
votes
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
$$V_1=2\qquad$$
$$V_2\approx 3.14$$
...
36
votes
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? ...
36
votes
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 ...
34
votes
6answers
3k views
34
votes
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 ...
33
votes
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 ...
32
votes
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 ...
31
votes
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 ...
31
votes
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 ...
28
votes
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 $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...
27
votes
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 ...
26
votes
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 ...
25
votes
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 ...
24
votes
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 ...
24
votes
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 ...
24
votes
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 ...
23
votes
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 ...
23
votes
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 ...
22
votes
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 ...
22
votes
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 ...
22
votes
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{ ...
22
votes
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 ...
22
votes
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 ...
22
votes
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 ...
21
votes
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 ...
21
votes
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 ...
21
votes
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 ...
21
votes
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 ...
20
votes
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 ...
20
votes
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 ...
20
votes
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) ...
20
votes
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 ...
20
votes
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 ...
19
votes
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?
19
votes
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 ...
18
votes
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, ...
