pages tagged mathematics

Drawing polyhedra with textured faces in Mathematica

Tue, 16 May 2017 23:59:39 +0000

In Mathematica and Wolfram Language, with the feature called Texture it's easy to draw 3D solids such as polyhedra with image textures on the surface such as the faces of polyhedra. When playing with this, I drew a rhombic hexecontahedron (properties), the logo of Wolfram|Alpha with national flags on its faces:

☝ Rhombic hexecontahedron wrapped with the 60 most populous countries' national flags, one different flag drawn on each of the 60 rhombic faces. (It's somewhat odd that the most populous countries happen to have a lot of red and green colors in their national flags.)

The code is in my Git repository.

Remarks:

It was a bit tricky to understand what the option VertexTextureCoordinates is and does, and how to set it up for what I'm trying to draw. It turns out for putting the national flags on the faces of a rhombic hexecontahedron, since both are (usually) isomorphic to a square, VertexTextureCoordinates -> {{0, 0}, {1, 0}, {1, 1}, {0, 1}} is would work as it aligns the four vertices of a rectangular flag to the four vertices of a rhombic face.

Something left to be improved:

Avoid flags that's not isomorphic to the face of the given polyhedron;
Compute the suitable values of VertexTextureCoordinates to reduce or even avoid distortion of the flags;
Find a reasonable way to align national flags of shapes not isomorphic to the shape of a face of a given polyhedron, such that the main feature of the national flag wrapped on a polyhedron face is well recognizable.

Category theory note

Tue, 11 Apr 2017 07:18:07 +0000

Draft

multigraph
quiver graph: directed multigraph, 有向伪图
quiver, loop arrows from and to the same vertex
object, vertex in a graph
from directed multigraph to categories
morphism, or arrow
identity morphism
composition of morphisms
There can be multiple different loops on one object, but one of them must be the identity morphism. Why?
endofunctor
Haskell languge is a category, Hask, with the Haskell language types as the objects and Haskell-langauge functions are morphisms.
'Hask' is a Cartesian closed category
n-category
monoids for input and output in Haskell language

Topics

Haskell and WL
category theory and WL

References

Bartosz Milewsk, "Category Theory for Programmers," 2014 - 2017, https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/.
Eugenia Cheng's research papers on category theory
'TheCatsters' Youtube channel
- Brent Yorgey's index to 'TheCatsters' videos
Tom Leinster, "Basic Category Theory," 2016, arXiv:1612.09375v1 [math.CT]
Tom Leinster's research papers

Compute sample variance of data stream

Tue, 06 Oct 2015 19:39:59 +0000

Sample variance of a data stream $\left \{ X_1, X_2, \cdots, X_N, X_{N+1}, \cdots \right \}$ can be computed without saving the data points individually, but only sample mean and sample variance values for the current sample size $N$:

$$ \begin{align} s² N &= \frac{1}{N} \sum {i=1}^N \left( X i - \bar{X} N \right)² \\ &= \frac{1}{N} \sum {i=1}^N \left( X i² - \bar{X} _N² \right)~, \end{align} $$

where

$$ \begin{equation} {\bar{X}} N = \frac{1}{N} \sum {i=1}^N X _i~. \end{equation} $$

Store $N$, $\bar{X} N$, and $s N²$. When datum $X _{N+1}$ is obtained, the values can be updated as the following:

$$ \begin{equation} \bar{X} {N+1} = \frac{1}{N+1} \left( N \bar{X} N + X _{N+1} \right)~, \end{equation} $$

$$ \begin{align} s² {N+1} &= \frac{1}{N+1} \sum {i=1}^{N+1} \left( X i² - \bar{X} {N+1}^2\right) \\ &= \frac{1}{N+1} \sum {i=1}^{N+1}X i² - \bar{X} {N+1}^2 \\ &= \frac{1}{N+1} \left(N s N² + \bar{X} N² + X {N+1}^2 \right)- \bar{X} _{N+1}^2~. \end{align} $$

1.1.4 summary on basic properties of real number system

Thu, 16 Jul 2015 22:37:46 +0000

實數系是一個連續的有序域。