Note: This site is only for linking and organizing my visualization works and is using a free site name.
Go to "Collection" for the main contents of the site.
Go to "Collection" for the main contents of the site.
AboutThis is a gallery to showcase beautiful and meaningful shapes in mathematics and statistics.
"Fractals" "Chaos" and "Diffusions" and a lot of relevant topics are revisited. But the main purpose is to clarify ideas using animations and visualization tools and packages that can be easily reused. For this purpose, the packages must be: - capable of customization for your particular math problem - fast and efficient to handle large amount of iterations - written with clear and readable code with illustrated demos to be reused. For the purpose of illustrating mathematics, we plot not only the trace of the orbits of systems but multiple aspects viewed at different angles(for example, cobweb plots, vector fields, phase portraits, winding numbers,...) to illustrate why math theories are useful in studying their properties. If one is patient enough to learn about the math structures lying inside the problems, one can understand why different setups of the plots can bring insights. My previous packages (some with collaborations) are in R, but most of the new packages (and code demos) to create these graphics are in Julia. Most for generating documents and gifs to be reused in other docs are in Plots.jl. I have implemented some CPU parallelable function in it, and tweaked the grammar of calling GR device to minimize the rendering time. For more time-consuming tasks, I have also tried a combination of CPU parallel and GPU but it is difficult to optimize for the moment. For this reason I save a lot of 3d isomorphs to files to be rerendered whenever needed. For tasks that require more graphics computing, using OpenGL with a GPU greatly improves speed and light effects. There are some functions that uses it, but using OpenGL to take over all the tasks is still a work in progress(in both my Julia and R codes, using Makie.jl and plot3drgl respectively). |
Packages
Tools: R and Julia, with APIs to GR and OpenGL
Reasons:
R has its plotting ecosystems, namely ggplot2, rgl. But there are 2 factors (in my personal view) that ceases the capability of the ecosystem of R: its speed, and the advances of other packages catching up. The main goal of this project is to develop reusable packages and codes to visualize fractals effectively. By 'effectively', it means the plots and animation should be easy to code, tolerable in render speed, and the plots are ready for presentation and print. A more ambitious goal is to create animations with documentation-level quality. Also, consistency and legibility of color palettes are considered.
Since this is a personal project so far, consistency of aesthetic of the plots is occasionally compromised as I find no time to rewrite some of my previous R codes.
I have also submitted more than 10 issues for Plots.jl and Makie.jl on GitHub.
Reasons:
R has its plotting ecosystems, namely ggplot2, rgl. But there are 2 factors (in my personal view) that ceases the capability of the ecosystem of R: its speed, and the advances of other packages catching up. The main goal of this project is to develop reusable packages and codes to visualize fractals effectively. By 'effectively', it means the plots and animation should be easy to code, tolerable in render speed, and the plots are ready for presentation and print. A more ambitious goal is to create animations with documentation-level quality. Also, consistency and legibility of color palettes are considered.
Since this is a personal project so far, consistency of aesthetic of the plots is occasionally compromised as I find no time to rewrite some of my previous R codes.
I have also submitted more than 10 issues for Plots.jl and Makie.jl on GitHub.
shttps://github.com/HaoLi111/SeriesXplorer
R package to visualize Lindenmayer system is (a modification of)
LindenmayeR by B. Hanson https://github.com/bryanhanson/LindenmayeR
And there is a plan to build a fluid page out of it, if you can help, a primitive solution that I wrote is on
https://github.com/bryanhanson/LindenmayeR/issues/4
Sample code for numerical simulation is at
https://github.com/HaoLi111/Julia_Numerical_Recipe
I have also created a jl package, but since it is inferior to DynamicalSystem.jl and DifferentialEquations.jl, the visualization functions may be modified to work with these 2 packages instead.
R package to visualize Lindenmayer system is (a modification of)
LindenmayeR by B. Hanson https://github.com/bryanhanson/LindenmayeR
And there is a plan to build a fluid page out of it, if you can help, a primitive solution that I wrote is on
https://github.com/bryanhanson/LindenmayeR/issues/4
Sample code for numerical simulation is at
https://github.com/HaoLi111/Julia_Numerical_Recipe
I have also created a jl package, but since it is inferior to DynamicalSystem.jl and DifferentialEquations.jl, the visualization functions may be modified to work with these 2 packages instead.
A Glance
Use the search function in the collection page, either by the name / equations of the dynamical systems or by their presence on references.
This show case lists some most commonly known examples revisited.
For the code repository, check the posts in the "collections".
This show case lists some most commonly known examples revisited.
For the code repository, check the posts in the "collections".
More
Thanks:
Prof. L. Tucker( for recomending the book Non-Linear Dynamics and Chaos by S. Strogatz, some of the equations for the fractals come from this book, and the suggestion to visualize from biological related models)
Prof. P. Tabrizian (reader may be interested in his channel https://www.youtube.com/channel/UCoOjTxz-u5zU0W38zMkQIFw The videos are helpful)
for recommending relevant materials, clarifying concepts, commenting on the plots and providing me with problems to work on.
Prof. L. Tucker( for recomending the book Non-Linear Dynamics and Chaos by S. Strogatz, some of the equations for the fractals come from this book, and the suggestion to visualize from biological related models)
Prof. P. Tabrizian (reader may be interested in his channel https://www.youtube.com/channel/UCoOjTxz-u5zU0W38zMkQIFw The videos are helpful)
for recommending relevant materials, clarifying concepts, commenting on the plots and providing me with problems to work on.
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