For questions about visualizing mathematical concepts. This includes questions about visualization of mathematical theorems and proofs without words.

# Questions tagged [visualization]

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### Angles between planes and the volume of a tetrahedron

Regarding the tetrahedron OABC in the picture, with sides $BC=10$, $AC=8$, $OA=4$,
$\sin(\angle ACB)=\frac{3}{4}$ and $\triangle ABC \equiv \triangle OBC$. With this, you find that the area of $\triangle ABC=30$.
Moreover, if $AH$ denotes the…

Renato

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### If I have n posts that don't occur at the end of the fence, why are there n + 1 sections?

https://en.wikipedia.org/wiki/Off-by-one_error#Fencepost_error:
More generally, the problem can be stated as follows:
If you have n posts, how many sections are there between them?
The correct answer may be n − 1 if the line of posts is…

NNOX Apps

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### How could one geometrically visualize any given metric space $(X,d)$?

Example. Say $X=\mathbb{R}$ and $d(x,y)=\frac{d_0(x,y)}{1+d_0(x,y)}$ where $d_0(x,y)=|x-y|$ is the Euclidean metric.
The visualization of e.g. $\mathbb{R}$ with Eucledian distance $d_0(x,y)=|x-y|$ is clear, because it coincides with the intuitive…

Yash Goyal

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### What is the growth rate of $h(n)?$

$$h(n) = \#\{ \pi(x)\pi(n-x),x\le n\}$$
What is the growth rate of $h(n)?$
(the notation means find the distinct values of $h(n)$ for each $n \in \Bbb N)$
for example, plotting the point $(12,4)$ corresponds to $n=12$ and $4$ distinct values for…

geocalc33

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### How to think about $|a| \leq b$

Let $a,b \in \mathbb{R}$.
$a \leq |b|$ is equivalent to the expression $-b \leq a \leq b$. Easy, geometrical, elegant, intuitive.
But what about
$|a| \leq b$
Suppose $b \geq 0$, then
$|a| \leq b$ seems to be equivalent to $a \leq b \wedge -a \leq…

Curaçao Hajek

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### How does derivative of definite integral make sense

Derivative is taken at a point and hence is value at a point. But definite integral is the value over a domain. Then how come derivative of definite integral make sense.

lorilori

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### If f' exists and f'(c) > 0 then f'(x) > 0 for all |x - c| < d for some d. (S.A. pp 137 question 5.2.8b)

If $f'$ exists on an open interval, and there is some point $c$ where $f'(c) > 0$,
then there exists a d-neighborhood $\{x \in \mathbb{R} : |x - c| < d\} = V_d(c)$ around c in which $f'(x) > 0$ for all $x \in V_d(c).$
1. How to presage this is…

NNOX Apps

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### Physical significance of the complex numbers

In case of the given statement below $\arg(z+i) =2π/3$ where $z$ is a complex numbers .
Here what is the physical way of seeing $z+i$, I mean does this represents a vector that joins $-i$ to $z$ or vice versa or something new as a whole ? Please…

SUDEEPA GUPTA

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