For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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265
votes
16answers
22k views

Is value of $\pi = 4$?

What is wrong with this? SOURCE
329
votes
10answers
358k views

Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real?
43
votes
15answers
13k 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 ...
11
votes
2answers
2k views

Finding an angle within an 80-80-20 isosceles triangle

The following is a geometry puzzle from a math school book. Even though it has been a long time since I finished school, I remember this puzzle quite well, and I don't have a nice solution to it. So ...
34
votes
7answers
14k 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 ...
31
votes
6answers
6k 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?
53
votes
14answers
7k 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! ...
75
votes
21answers
39k views

Software for drawing geometry diagrams

What software do you use to accurately draw geometry diagrams?
7
votes
7answers
1k views

Different definitions of trigonometric functions

In school, we learn that sin is "opposite over hypotenuse" and cos is "adjacent over hypotenuse". Later on, we learn the power series definitions of sin and cos. How can one prove that these two ...
12
votes
8answers
6k views

Why Circle encloses largest Area?

In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all i would ...
20
votes
7answers
3k views

How to prove $\cos \frac{2\pi }{5}=\frac{-1+\sqrt{5}}{4}$?

I would like to find the apothem of a regular pentagon. It follows from $$\cos \dfrac{2\pi }{5}=\dfrac{-1+\sqrt{5}}{4}.$$ But how can this be proved (geometrically or trigonometrically)?
42
votes
6answers
5k views

How come $32.5 = 31.5$?

Below is a visual proof (!) that $32.5 = 31.5$. How could that be?
12
votes
7answers
7k views

Drawing heart in mathematica

It's not really a typical math question. Today, while studying graphs, I suddenly got inquisitive about whether there exists a function that could possibly draw a heart-shaped graph. Out of sheer ...
3
votes
6answers
381 views

Diophantine quartic equation in four variables

Comments from a recent Question, Cyclic quadrilateral with equal area and perimeter, ask about such cases with (positive) integer lengths. Using Brahmagupta's formula for the area of a cyclic ...
5
votes
2answers
615 views

well separated points on sphere

Is there a way to generate k points on a n-sphere, say, $x_1,\dots,x_k$ such that $\min_{ i \neq j } \| x_i - x_j \| $ is as large as possible? Approximate solutions are also OK, I just need well ...
109
votes
3answers
15k 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 ...
7
votes
3answers
6k views

Which tessellation of the sphere yields a constant density of vertices?

One way to tessellate a 3D sphere is by iterated subdivision of an icosahedron. I am wondering whether this method gives a homogeneous surface density of vertices. To the eye, it seems to do so, and ...
5
votes
2answers
788 views

Tranforming 2D outline into 3D plane

I am writing a program where I would like to allow the user to draw 4 connecting lines, such as: And convert this shape into a 3D plane. Is this possible? Is there an existing algorithm to do so? ...
4
votes
3answers
724 views

How quickly we forget - basic trig. Calculate the area of a polygon

I think the easiest way to do this is with trigonometry, but I've forgotten most of the maths I learnt in school. I'm writing a program (for demonstrative purposes) that defines a Shape, and ...
3
votes
2answers
1k views

Finding Intersection of an ellipse with another ellipse when both are rotated

Equation of first ellipse=> $$\dfrac {((x-xFirstEllipseCenterPoint)\cdot \cos(A)+(y-yFirstEllipseCenterPoint)\cdot \sin(A))^2}{(a_1^2)}+\dfrac{((x-xFirstEllipseCenterPoint)\cdot ...
10
votes
6answers
3k views

How to find an ellipse , given 2 passing points and the tangents at them?

Please answer to a question , how to find an ellipse which passes the 2 given points and has the given tangents at them. And one related question is that the given condition can decide just one ...
5
votes
4answers
3k views

Can the Surface Area of a Sphere be found without using Integration?

When we were in school they told us that the Surface Area of a sphere = $4\pi r^2$ Now, when I try to derive it using only high school level mathematics, I am unable to do so. Please help.
56
votes
4answers
6k 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 ...
41
votes
9answers
13k views

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

(Originally asked on MO by AJAY.) What is the geometric, physical, or other meaning of the third derivative of a function at a point? If you have interesting things to say about the meaning of the ...
22
votes
8answers
8k views

Is there an equation to describe regular polygons?

For example, the square can be described with the equation $|x| + |y| = 1$. So is there a general equation that can describe a regular polygon (in the 2D Cartesian plane?), given the number of sides ...
12
votes
1answer
673 views

Nested sequences of balls in a Banach space

This seems to be a fairly easy question but I'm looking for new points of view on it and was wondering if anyone might be able to help (by the way- this question does come from home-work, but I've ...
9
votes
3answers
13k views

Compass-and-straightedge construction of the square root of a given line?

Given A straight line of arbitrary length The ability to construct a straight line in any direction from any starting point with the "unit length", or the length whose square root of its magnitude ...
11
votes
4answers
4k views

How to understand dot product is the angle's cosine?

How can one see that a dot product gives the angle's cosine between two vectors. (assuming they are normalized) Thinking about how to prove this in the most intuitive way resulted in proving a ...
3
votes
2answers
705 views

Using equations to draw out complex objects

How do people come up with equations of curves to draw out complex objects? Some popular examples would include: batman curve & PSY curve. This stackexchange link explains the rationale for the ...
96
votes
6answers
134k 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 ...
51
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 $$V_1=2\qquad$$ $$V_2\approx 3.14$$ ...
18
votes
7answers
4k views

Generate a random direction within a cone

I have a normalized 3D vector giving a direction and an angle that forms a cone around it, something like this: I'd like to generate a random, uniformly distributed normalized vector for a ...
28
votes
5answers
3k 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 ...
5
votes
3answers
12k views

Is there a way to rotate the graph of a function?

Assuming I have the graph of a function $f(x)$ is there function $f_1(f(x))$ that will give me a rotated version of the graph of that function? For example if I plot $\sin(x)$ I will get a sine wave ...
4
votes
2answers
233 views

A Regular Tetrahedron is a cool Polyhedron.

A regular tetrahedron has this property: For any two of its vertices exists a third vertex, which forms a regular triangle with these 2 vertices.(It doesn't necessarily have to be a face of it). Are ...
4
votes
4answers
8k views

How do I convert the distance between two lat/long points into feet/meters?

I've been reading around the net and everything I find is really confusing. I just need a formula that will get me 95% there. I have a tool that outputs the distance between two lat/long points. ...
17
votes
4answers
3k views

The locus of two perpendicular tangents to a given ellipse

For a given ellipse, find the locus of all points P for which the two tangents are perpendicular. I have a trigonometric proof that the locus is a circle, but I'd like a pure (synthetic) geometry ...
15
votes
5answers
1k views

Polynomial approximation of circle or ellipse

Trying again, with a somewhat simpler sounding question, since my previous one (Generalizations of equi-oscillation criterion) got zero response: Let $F:[0,1] \to R^2$ be a parametric polynomial ...
8
votes
3answers
700 views

How to find the center of an ellipse?

I have the following data:- I have two points ($P_1$, $P_2$) that lie somewhere on the ellipse's circumference. I know the angle ($\alpha$) that the major-axis subtends on x-axis. I have both the ...
6
votes
7answers
12k views

Equation of a rectangle

I need to graph a rectangle on the Cartesian coordinate system. Is there an equation for a rectangle? I can't find it anywhere.
15
votes
4answers
7k views

Picking random points in the volume of sphere with uniform probability

I have a sphere of radius $R_{s}$, and I would like to pick random points in its volume with uniform probability. How can I do so while preventing any sort of clustering around poles or the center of ...
6
votes
5answers
572 views

Evaluating $\int_a^b \frac12 r^2\ \mathrm d\theta$ to find the area of an ellipse

I'm finding the area of an ellipse given by $\frac{x^2}{a^2}+\frac{y^2}{b^2} = 1$. I know the answer should be $\pi ab$ (e.g. by Green's theorem). Since we can parameterize the ellipse as ...
104
votes
10answers
14k views

How can a piece of A4 paper be folded in exactly three equal parts?

This is something that always annoys me when putting an A4 letter in a oblong envelope: one has to estimate where to put the creases when folding the letter. I normally start from the bottom and on ...
38
votes
4answers
2k 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 ...
29
votes
6answers
852 views

Prove Existence of a Circle

There are two circles with radius $1$, $c_{A}$ and ${c}_{B}$. They intersect at two points $U$ and $V$. $A$ and $B$ are two regular $n$-gons such that $n > 3$, which are inscribed into $c_{A}$ and ...
23
votes
8answers
2k views

Books on classical geometry

I'm curious to whether you guys have any tips on book concerning classical euclidean geometry. I'd like somewhat of an advanced treatment, around the same level as Coxeter's "Geometry revisited". I'd ...
45
votes
6answers
4k 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 ...
26
votes
4answers
723 views

Geometric explanation of $\sqrt 2 + \sqrt 3 \approx \pi$

Just curious, is there a geometry picture explanation to show that $\sqrt 2 + \sqrt 3 $ is close to $ \pi $?
49
votes
9answers
18k 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 ...
34
votes
3answers
4k 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 $A = \{(U_{\alpha}, \phi_{\alpha})\}$ ...