For questions about visualizing mathematical concepts. This includes questions about visualization of mathematical theorems and proofs without words.
Questions tagged [visualization]
990 questions
1657
votes
89 answers
Visually stunning math concepts which are easy to explain
Since I'm not that good at (as I like to call it) 'die-hard-mathematics', I've always liked concepts like the golden ratio or the dragon curve, which are easy to understand and explain but are mathematically beautiful at the same time.
Do you know…
RBS
- 861
290
votes
14 answers
Help with a prime number spiral which turns 90 degrees at each prime
I awoke with the following puzzle that I would like to investigate, but the answer may require some programming (it may not either). I have asked on the meta site and believe the question to be suitable and hopefully interesting for the…
Karl
- 4,763
207
votes
34 answers
List of interesting math videos / documentaries
This is an offshoot of the question on Fun math outreach/social activities. I have listed a few videos/documentaries I have seen. I would appreciate if people could add on to this list.
$1.$ Story of maths Part1 Part2 Part3 Part4
$2.$ Dangerous…
user17762
192
votes
31 answers
Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction
I recently proved that
$$\sum_{k=1}^n k^3 = \left(\sum_{k=1}^n k \right)^2$$
using mathematical induction. I'm interested if there's an intuitive explanation, or even a combinatorial interpretation of this property. I would also like to see any…
Fernando Martin
- 6,027
165
votes
16 answers
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…
Martin Hurtado
- 1,843
139
votes
24 answers
Visually deceptive "proofs" which are mathematically wrong
Related: Visually stunning math concepts which are easy to explain
Beside the wonderful examples above, there should also be counterexamples, where visually intuitive demonstrations are actually wrong. (e.g. missing square puzzle)
Do you know the…
puzzlet
- 785
125
votes
5 answers
How were 'old-school' mathematics graphics created?
I really enjoy the style of technical diagrams in many mathematics books published in the mid-to-late 20th century. For example, and as a starting point, here is a picture that I just saw today:
Does anybody know how this graphic was created? Were…
TSGM
- 1,263
85
votes
0 answers
Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles?
The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a known regular way to do this? Just playing and not…
2'5 9'2
- 56,991
82
votes
2 answers
Visual proof of $\sum_{n=1}^\infty \frac{1}{n^4} = \frac{\pi^4}{90}$?
In his gorgeous paper "How to compute $\sum \frac{1}{n^2}$ by solving triangles", Mikael Passare offers this idea for proving $\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$:
Proof of equality of square and curved areas is based on another…
VividD
- 16,196
69
votes
4 answers
Algebra: Best mental images
I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on itself. I know that a categorical approach is…
Rachmaninoff
- 2,680
62
votes
7 answers
Visualizing the 4th dimension.
In a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as projections of 4-D objects. How does this helps us visualize 4-D objects?
I searched that we can at least see their 3-D cross-sections. A Tesseract hypercube…
pooja somani
- 2,605
58
votes
15 answers
Interesting math-facts that are visually attractive
To give a talk to 17-18 years old (who have a knack for mathematics) about how interesting mathematics (and more specifically pure mathematics) can be, I wanted to use nice facts accompanied by nice looking visualizations. However, the underlying…
Jan Keersmaekers
- 1,046
53
votes
1 answer
Pattern "inside" prime numbers
Update $(2020)$
I've observed a possible characterization and a possible parametrization of the pattern, and I've additionally rewritten the entire post with more details and better definitions.
It remains to prove the observed possible…
Vepir
- 13,072
53
votes
4 answers
What is the explanation for this visual proof of the sum of squares?
Supposedly the following proves the sum of the first-$n$-squares formula given the sum of the first $n$ numbers formula, but I don't understand it.
Nitin
- 2,988
45
votes
4 answers
Trying to visualize the hierarchy of mathematical spaces
I was inspired by this flowchart of mathematical sets and wanted to try and visualize it, since I internalize math best in that way. This is what I've come up with so far:
Version 1 (old diagram)
Version 2:
Is there anything that I'm missing, or…
Patch
- 4,607