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.

Reference:

https://en.wikipedia.org/wiki/Geometry

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The degree of an algebraic curve in higher dimensions

This might be a very simple question but I can't seem to find a precise definition of the degree of an algebraic curve, if such can even be defined. In the plane, the degree of an algebraic curve is clear; it is simply the degree of the defining…
syxiao
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Cube stack problem

How many distinct patterns are possible if you omit (a) 1 piece, (b) 2 pieces and (c) 3 pieces from a cube originally consisting of 27 smaller and equally sized cubes?
ValX
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Proof for SSS Congruence?

I'm hoping that someone can provide a method for deducing the commonly known SSS congruence postulate? The postulate states If the three sides of one triangle are pair-wise congruent to the three sides of another triangle, then the two triangles…
Nick H
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How do I find the middle(1/2), 1/3, 1/4, etc, of a line?

Similar to this question: How to calculate the middle of a line? where it's explained how to find the middle of a line (x,y), so that's half the line 1/2, but I also need to find one third of the line, one fourth, and so on. I tried dividing by 4…
01AutoMonkey
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Maximum number of cubes within a radius

Last night I was mining obsidian in Minecraft, which takes a long time (15 seconds for each block). As a result, I would hold down the left mouse button with my left hand while I did something else. In order to maximize the usefulness of this…
El'endia Starman
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Can the Banach-Tarski Paradox be extended to an arbitrary number of duplications?

In this question, I recently asked if there were free subgroups of rank 3 or higher of the group of rotations in $\mathbb{R}^3$. From the answers, it follows that any free subgroup of rank 2 admits subgroups of arbitrary countable rank. My question…
Rachel
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Find the perimeter of any triangle given the three altitude lengths

The altitude lengths are 12, 15 and 20. I would like a process rather than just a single solution.
Chris Johnson
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Characterization of Convex Polygons

John Lee's Axiomatic Geometry has an interesting characterization of convex and non-convex vertices for polygons. Let $P$ be a polygon. Consider a ray emanating from a vertex of $P$ which does not intercept any other vertices of $P$. Call the ray…
EuYu
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What's the average distance between two discs in the plane?

Consider two discs in the plane of radius $r$ and $s$, with centers separated by a distance $l$. If we choose a point uniformly at random from each disc, what is the expected distance between the two points? (This is a generalization of this…
patricksurry
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Find an angle of an isosceles triangle

$\triangle ABC$ is an isosceles triangle such that $AB=AC$ and $\angle BAC$=$20^\circ$. And a point D is on $\overline{AC}$ so that AD=BC, , How to find $\angle{DBC}$? I could not get how to use the condition $AD=BC$ , How do I use the condition…
MS.Kim
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Cutting a sandwich with a crust

Let $S$ be a simple closed curve in ${\Bbb R}^2$ enclosing a convex region $I$. Must there exist a straight line which cuts $S$ into two pieces of equal length and also cuts $I$ into two regions of equal area? If so, how can such a line be…
MJD
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On the problem $1$ of Putnam $2009$

(This is adapted from problem $1$ of Putnam $2009$) Find all values of $n$ such that the following is true: There is a non-constant function $f : \Bbb{R}^2 \longrightarrow \Bbb{Z}_2$ such that for any regular $n$-gon $A_1...A_n$, $f(A_1) + \cdots +…
pin2
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Is there such a thing as the "edge-face dual" of a polyhedron, and is the "edge-face dual" of a cube a rhombic dodecahedron?

The dual of a polyhedron is a polyhedron where the vertices of one correspond to the faces of the other, and vice versa. Is there always a similar correspondence between a pair of polyhedra where the edges of one correspond to the faces of the…
John Calsbeek
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What should be the proportions of a three sided coin?

A classical coin has almost no chances of ending its course on the side when tossed. A round pencil with both ends flat has no chance of ending its course on the tip, when tossed. What would be the proportions of a cylinder that has 1/3 chances to…
SteeveDroz
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Is there an established notation, either modern or historical, for any unit of measure which is then further subdivided into 360 degrees or parts?

This question about notation is simple as dirt, but would be useful for me regardless, because of some work that I'm doing in music theory. Basically, while there's a notation for subdividing the degree into arcminutes and arcseconds, so that "180…
Mike Battaglia
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