Questions tagged [geometry]

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

Geometry is one of the classical disciplines of math. It is derived from two Latin words, "geo" + "metron" meaning earth & measurement. Thus it is concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs. Since its earliest days, geometry has served as a practical guide for measuring lengths, areas, and volumes, and geometry is still used for this purpose today. Geometry is important because the world is made up of different shapes and spaces.

Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics.

Sub-fields of Contemporary Geometry:

$1.\quad$ Algebraic Geometry – is a branch of geometry studying zeroes of multivariate polynomials. It includes the linear and polynomial algebraic equations used for finding these sets of zeros. The applications of algebraic geometry include cryptography, string theory, etc.

$2.\quad$ Discrete Geometry – is concerned with the relative positions of simple geometric objects, such as points, lines, triangles, circles etc.

$3.\quad$ Differential Geometry – uses techniques of algebra and calculus for problem-solving. The applications of differential geometry include general relativity in physics, etc.

$4.\quad$ Euclidean Geometry – The study of plane and solid figures on the basis of axioms and theorems including points, lines, planes, angles, congruence, similarity, solid figures. It has a wide range of applications in computer science, modern mathematics problem solving, crystallography etc.

$5.\quad$ Convex Geometry – includes convex shapes in Euclidean space using techniques of real analysis. It has application in optimization and functional analysis in number theory.

$6.\quad$ Topology – is concerned with properties of space under continuous mapping. Its application includes consideration of compactness, completeness, continuity, filters, function spaces, grills, clusters and bunches, hyperspace topologies, initial and final structures, metric spaces, metrization, nets, proximal continuity, proximity spaces, separation axioms, and uniform spaces.

$7.\quad$ Plane Geometry – This wing of geometry deals with flat shapes which can be drawn on a piece of paper. These include lines, circles & triangles of two dimensions.

$8.\quad$ Solid Geometry – It deals with $3$-dimensional objects like cubes, prisms, cylinders & spheres.


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The staircase paradox, or why $\pi\ne4$

What is wrong with this proof? Is $\pi=4?$
Pratik Deoghare
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Is this Batman equation for real?

HardOCP has an image with an equation which apparently draws the Batman logo. Is this for real? Batman Equation in text form: \begin{align} &\left(\left(\frac x7\right)^2\sqrt{\frac{||x|-3|}{|x|-3}}+\left(\frac…
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V.I. Arnold says Russian students can't solve this problem, but American students can -- why?

In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is 10 inches, the altitude dropped onto it is 6 inches. Find the area of the triangle. American…
Eli Rose
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What is the maximum volume that can be contained by a sheet of paper?

I was writing some exercises about the AM-GM inequality and I got carried away by the following (pretty nontrivial, I believe) question: Q: By properly folding a common $210mm\times 297mm$ sheet of paper, what is the maximum amount of water such…
Jack D'Aurizio
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Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers

An exam for high school students had the following problem: Let the point $E$ be the midpoint of the line segment $AD$ on the square $ABCD$. Then let a circle be determined by the points $E$, $B$ and $C$ as shown on the diagram. Which of the…
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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 eye estimate where to fold. Then I turn the letter…
Nicky Hekster
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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 Venn diagram using only circles as we could…
Larry Wang
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Identification of a quadrilateral as a trapezoid, rectangle, or square

Yesterday I was tutoring a student, and the following question arose (number 76): My student believed the answer to be J: square. I reasoned with her that the information given only allows us to conclude that the top and bottom sides are parallel,…
The Chaz 2.0
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What's the intuition behind Pythagoras' theorem?

Today we learned about Pythagoras' theorem. Sadly, I can't understand the logic behind it. $A^{2} + B^{2} = C^{2}$ $C^{2} = (5 \text{ cm})^2 + (7 \text{ cm})^2$ $C^{2} = 25 \text{ cm}^2 + 49 \text{ cm}^2$ $C^{2} = 74 \text{ cm}^2$ ${x} =…
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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 reaction was "0" or "undefined". But my son wrote…
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Software for drawing geometry diagrams

What software do you use to accurately draw geometry diagrams?
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Why is the Penrose triangle "impossible"?

I remember seeing this shape as a kid in school and at that time it was pretty obvious to me that it was "impossible". Now I looked at it again and I can't see why it is impossible anymore.. Why can't an object like the one represented in the…
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How to distinguish between walking on a sphere and walking on a torus?

Imagine that you're a flatlander walking in your world. How could you be able to distinguish between your world being a sphere versus a torus? I can't see the difference from this point of view. If you are interested, this question arose while I was…
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What's new in higher dimensions?

This is a very speculative/soft question; please keep this in mind when reading it. Here "higher" means "greater than 3". What I am wondering about is what new geometrical phenomena are there in higher dimensions. When I say new I mean phenomena…
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Slice of pizza with no crust

The following question came up at a conference and a solution took a while to find. Puzzle. Find a way of cutting a pizza into finitely many congruent pieces such that at least one piece of pizza has no crust on it. We can make this more…
Dan Rust
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