4
votes
0answers
48 views

Name for a body that can be completely described using its silhouettes

I'm shooting blind over here because I have no background in this field of mathematics. I assume that if you have a body (in $\mathbb{R}^3$), you can call it convex if any segment from one point ...
9
votes
2answers
248 views

Why is a straight line the shortest distance between two points?

The first application I was shown of the calculus of variations was proving that the shortest distance between two points is a straight line. Define a functional measuring the length of a curve ...
3
votes
0answers
71 views

Pythagorean like theorem for general spaces

There are laws, for example Pythagorean theorem, for calculating distance of two points, say $d(a,b)$, using third point and knowing $d(a,c)$ and $d(c,b)$ in vector spaces. My question is that for ...
1
vote
1answer
58 views

I'm planning on taking an Introduction to Geometry class next term. What should I expect?

For reference, I'm heading in to my second year of my bachelor of Computer Science. I'm required to take a second year level math course and I was thinking about taking Introduction to Geometry. It ...
2
votes
4answers
194 views

What is the most fundamental trigonometric function: cosine or sine? [closed]

$$\cos(\theta) = \sin \left(\tfrac{\pi}{2} - \theta\right)$$ $$\sin(\theta) = \cos \left(\tfrac{\pi}{2} - \theta\right)$$ Both are the same entity. But is sine the copy of cosine, or is cosine the ...
4
votes
0answers
134 views

How to balance learning and researching as a new PhD student?

As a new PhD student, how to balance learning and researching? I am in Australia and here we don't have any course in PhD period. I know I need to learn something about my programme, but sometimes ...
7
votes
3answers
89 views

Why do we say $n$ distinct points?

" Let's say we have $n$ distinct points... " , you see this every time you open a geometry textbook. Why not just $n$ points ? If the points are not distinct, they are not exactly $n$ points, are they ...
2
votes
3answers
93 views

How to define the magnitude of a rotation in $\mathbb{R}^n$?

Is there some well established way on how to quantify rotations in $\mathbb{R}^n$? To say which rotation is greater and which is smaller? In $\mathbb{R}^2$ the rotation is characterized by a single ...
5
votes
2answers
61 views

Geometry after Khan academy's tutorials

I always liked geometry at school, so by the side of my normal studies I've been going through the Khan academy videos. Could anyone suggest some good books that takes these geometry topics further? ...
3
votes
2answers
63 views

What is the most elementary but still correct according to the most rigorous standard proof of the isoperimetric inequality?

Can you write the most elementary proof of the isoperimetric inequality (but still correct according to the most rigorous standard )? $$l^2> 4πA$$
6
votes
1answer
186 views

Very interesting graph!

I found a VERY interesting graph on http://www.xamuel.com/graphs-of-implicit-equations/. It looked very, very cool, but the equation of the graph is so simple! Here is the image of the graph: My ...
12
votes
2answers
306 views

What problems are easier to solve in a higher dimension, i.e. 3D vs 2D?

I'd be interested in knowing if there are any problems that are easier to solve in a higher dimension, i.e. using solutions in a higher dimension that don't have an equally optimal counterpart in a ...
1
vote
1answer
56 views

curves bounding discs

I'm interested in the following question. Please forgive me if my question is lacking in precision. I'm not a mathematician, and need some help getting started: If I have a smooth, simple curve ...
2
votes
2answers
156 views

Which small area of mathematics had fully developed already and thus no more research in this area?

Which small area of mathematics had fully developed already and thus no more research in this area? For example, no more PHD research in Euclidean Geometry anymore.
4
votes
1answer
110 views

What is synthetic geometry?

What is synthetic geometry? Could you provide a short (i.e. a paragraph or two, not much longer) explanation in general elementary terms? In particular, I hope to be able to understand the contrast ...
10
votes
1answer
343 views

Thinking outside of the box

You want to draw a circle with a 4 inch radius. A trivial task for you and your trusty compass. When you go to grab your compass which has not had much love for a while you find it is rusted shut; ...
-2
votes
1answer
108 views

Insistence on classical geometry [closed]

This post may surprise and even irritate some people here but I am sure that if any sort of discussion springs out of it, we will get to read some interesting opinions. For the rest of the post, if I ...
9
votes
0answers
183 views

Big geometry grad schools - for an average applicant

What are some schools that have a lot of geometry going on, but that might accept some middle-of-the-range applicants? Let me add some context... I left grad school (UC Davis) with an MS in 2012 ...
0
votes
2answers
71 views

metric preserving transformations (isometry)

Maybe it will be very general question but i wonder what is the importance of metric preserving transformations? Where can we use this concept in mathematics?
0
votes
1answer
46 views

Adding together curves or shapes to approximate something more complex

I'm looking for proper terminology / references for the following sort of problem: Say we have some one-dimensional curve like $y = 10$ defined over the real valued domain $[0,1]$, and we ask, how ...
0
votes
1answer
56 views

Norm, Euclidean Space and Distance

To a complete layman, how would you define the following terms intuitively? $norm$ , $euclidean$ $space$ , and $euclidean$ $distance$ ? Note: I have tagged Linear Algebra and Probability Theory ...
10
votes
5answers
301 views

A question of H.G. Wells' mathematics

H.G Wells' short story The Plattner Story is about a man who somehow ends up being "inverted" from left to right. So his heart has moved from left to right, his brain, and any other asymmetries ...
7
votes
2answers
227 views

What does it mean to be $0.9-$Dimension?

We can visualize $1\mathrm D$, $2\mathrm D$, $3\mathrm D$ and we can think of a higher $M-\mathrm {Dimension}$ where $M\in \mathbb N^+$as a vector. But I recently learned that there are non integer ...
1
vote
1answer
161 views

How is algebra and geometry related to calculus?

Probably a too trivial question but here goes: What is the relationship of algebra and geometry with calculus? Are they pre-required knowledge for calculus, is calculus a different type of theory on ...
2
votes
2answers
125 views

Alternative model of Euclidean geometry

I'm planning to teach high-school geometry. As usual, this will be by building from axioms. (The axioms used are AFAICT particular to the book I've been assigned, but they're some combination of ...
4
votes
3answers
204 views

Geometry and Physics

I have to do a presentation on Geometry and Physics. I am asking it here (rather than physics.se) because I have to focus on Geometry More than Physics. The intended audience is Undergraduate Seniors ...
18
votes
5answers
751 views

Is there a deep reason why $(3, 4, 5)$ is pythagorean? [closed]

The triple $(3, 4, 5)$ is a pythagorean triple - it satisfies $a^2 + b^2 = c^2$ and, equivalently, its components are the lengths of the sides of a right triangle in the Euclidean plane. But of ...
3
votes
0answers
164 views

Imagining four or higher dimensions and the difference to imagining three dimensions

I’m very interested in how people envision four or higher dimensions. And I’m especially interested in how geometers and topologists who actually work in four dimensions do. Now I know of the video ...
2
votes
1answer
43 views

The best softwares to understand the intersections of the 3D objects in the Euclidean space

What is the best software (Easy to follow and clear graphics) to draw the intersections between two spheres, Two spheres and a pyramid, for example. The centre and the radius of the spheres are given ...
5
votes
2answers
128 views

Why generalize the Euclidean metric?

It is well known that the Euclidean metric can be generalized to $\Bbb R^n$ by $\sqrt{(x_1-x'_1)^2+\cdots + (x_n-x'_n)^2}$, and that under this generalization it is still a metric and satisfies ...
5
votes
1answer
81 views

Geometric explanations of approximations of $\pi$

Does any fast modern algorithm for approximating $\pi$ have a geometric interpretation as $\int \sqrt{1 - x^2}$ does?
8
votes
2answers
460 views

Connection between algebraic geometry and high school geometry.

if there is one thing that going to math competitions has taught me it is that I suck at high school olympiad level geometry. However I often find solace in the fact that not a lot of mathematicians ...
2
votes
0answers
169 views

Draw curves and planes in latex [closed]

I want to submit a paper and the only thing that's left to do is to add some graphics. My deadline is on Wednesday/Thursday. What I want to do is some basic curve/plane drawings. Here is an image: ...
3
votes
4answers
1k views

Why do we draw the $xyz$ coordinate system like this?

Usually people (including, for instance, Calculus teachers) draw the $xyz$ coordinate system in such a way that the $y$ and $z$ axes are perpendicular to each other: Imagine I actually got three ...
3
votes
4answers
152 views

What is a vector?

What is a vector? As the question says what is a vector and what are its uses or, I mean, when should we use vectors? Is this a branch of geometry or algebra or trigonometry?
8
votes
2answers
1k views

Geometric intuition behind gradient, divergence and curl

I learned vector analysis and multivariate calculus about two years ago and right now I need to brush it up once again. So while trying to wrap my head around different terms and concepts in vector ...
1
vote
1answer
126 views

Book recommendation request for geometric bodies (cube, pyramid, prism etc.)

Can anyone recommend books that deal with geometric bodies (cube, pyramid, prism etc.)? I haven't been able to find any.
2
votes
3answers
470 views

Number of points on line segment

I know the line segment have a infinite number of points, but i know that exist different kinds of infinity ( $\aleph_0 $). My question is there same number of points on segment of line and entire ...
3
votes
0answers
67 views

Books on n-dimensional euclidian geometry?

Could anyone please recommend some books on n-dimensional Euclidean geometry? Thanks!
17
votes
5answers
748 views

What Mathematics questions can be better solved with concepts from Physics?

Over the years, I've seen several questions in mathematics that can be solved using concepts borrowed from Physics. Having seen these question, I'm interested to find out what other mathematics ...
22
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 ...
4
votes
4answers
354 views

Strategies for arranging coins in a circle

Suppose you have $n$ identical circular coins and you would like to arrange them on the table so that their edges touch and their centers lie on a circle. Mathematically, there is no trouble. "Just" ...
10
votes
3answers
411 views

Elementary Geometry Nomenclature: why so bad?

A long-ish wall of text, and I apologize. Some background: when I was a first-year university student, my chemistry professor was lecturing and was trying to find the word to describe a shape. A ...
20
votes
8answers
2k views

I'm teaching a college geometry course. What should I cover?

I've been asked to teach "Foundations of Geometry" at the University of South Carolina. Apparently, professors in the past have all done very different things, and I have a lot of choice in the ...
2
votes
1answer
163 views

Why should coordinate transformations be reversible?

Intuitively I understand why coordinate transformation should be reversible. New coordinates should cover the same area covered by the initial coordinates, i.e. there should be one-to-one mapping. ...
16
votes
15answers
2k views

Explaining Horizontal Shifting and Scaling

I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect. For example, ...
2
votes
3answers
431 views

How to fill up the gap between a typical advanced undergraduate algebraic curve course and High school basic geometry/precalculus course?

Based on this question i asked recently: A question about geometry of plane curve books, i think it is too advance for a HS student/ typical second or third year undergraduate math majors to read on ...
13
votes
2answers
310 views

Does every closed curve contain the vertices of a square?

This is the question on Futility Closet Is there really no answer?
4
votes
1answer
155 views

Intuition of projective plane and space

What is the geometric intuition of projective plane and space? I can understand affine plane and 3 dimension affine space, for higher dimension, at least I can imagine it similarly as the 2,3 ...
16
votes
6answers
646 views

Implication and Interpretation of Banach Tarski

As I understand, the Banach-Tarski paradox says a ball in 3-space may be decomposed into finitely many pieces and reassembled into two balls each of the same size as the original. Despite being called ...