# Tagged Questions

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

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### What is the fourth dimension of a Tesseract?

Is the fourth dimension of the Tesseract time? That is why it is represented as a moving 3D structure on Wikipedia? I am asking because I have trouble understanding what it is.
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### Books on the visual/graphical aspects of geometry

Are there any books providing a general overview of the visual/graphical aspects of geometry? For example, Tilings (e.g. hyperbolic) and tessellations Plane/space filling shapes/objects (e.g. Escher'...
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### Explaining Spin(3)

I’m going to discuss the action of Spin(3) on Euclidean vectors. This thing has several alternative names: “versors”/“rotation quaternions”, “quaternionic adjoint representation”, “quaternion action ...
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### Looking for intuïtive explanation why contour integral of $\frac{dz}{z}$equals $2\pi i$ in complex analysis

$$\oint \frac{dz}z = 2\pi i$$ I've seen the derivation of it using the parametrisation. Since this result is used all the time in my complex analysis course, i'd like to understand this intuïtively....
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### How to visualize implicit functions

I have a task of visualizing few implicit functions. Firstly lets say I have the following function of $N$: $$\epsilon = \sqrt{\frac{8}{N}\ln \left( \frac{4(2N)^{50}}{0.05} \right)}$$ Now this is ...
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### Parallel Lines Intersecting in the Projective Plane

My question is about visualizing projective space, in particular the real projective plane $\mathbb{P}^2(\mathbb{R})$. I know there are different ways to define this space, but in each we can say that ...
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### If $z = f(x, y)$, then why are $\partial_x z$ and $\partial_y z$ functions of x and y also? [Stewart P905]

This is Figure 5 from P905 which appears to show this, but Stewart doesn't write this explicitly or explain. I'm interested in an informal, intuitive explanation please. I'm not interested in a ...
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### Pictures for Expectation

Is there a good way to visualize the formula: $$E(x) = \int_{0}^{\infty} 1 - F(X) \,\mathrm{d}x$$ ? for positive continuous random variables? I understand the formula as far as basic calculus ...
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### Computer-aided study of elementary geometry

As a beginning student of elementary (euclidean plane) geometry, so far, I have gotten the impression that there are two major approaches to geometries: naive vs axiomatic. Being a humanities student ...
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### Penrose tilings as a cross section of a $5$-dimensional regular tiling

Could somebody explain to me how a penrose tiling , which is not periodic, can be a cross section of a regular tiling in $5$ dimensions, which is periodic? It does not make sense to me how a periodic ...
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### Determine Cross Product with Left Hand vs Right Hand

If I perceive http://en.wikipedia.org/wiki/Cross_product correctly, then to determine the cross product With a right hand, let: the 1st vector in the cross product = your index finger = in red ...
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### Geometric intuition behind subspaces in $\mathbb C^n$

While learning elementary linear algebra one develops a great deal of geometric intuition in $\mathbb R^n$. It helps to see the forest for the trees and leads through proofs. After meeting ...
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### Intuition. Equivalence of Characterization of Limits and Continuity (Abbott p106 t4.2.3, p110 t4.3.2)

What are the intuitions of these equivalences? Not questioning about proofs or any rigour. I question both equivalences jointly because they look similar. And Are there any figures? Abbott ...
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### Factorial of 1,e+80

Recently I started being very fascinated in logistics, and out of the blue came the question into my head, what is the factorial of the amount of atoms in the observeable universe, which is said to be ...
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### Proof strategy for $(=>)$: If $g \circ f = id_A$, then f onto $\iff$ g 1-1. [Chartrand 3Ed P239 9.72]
For nonempty sets A and B and functions f : A → B and g : B → A, suppose that $g \circ f =$ the identity function on A. $(♦)$ (d) $(=>)$ Assume that $f$ is onto. This means there exist \$a_1, ...