Building 4D Polytopes

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Offline Syntopia

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« on: March 28, 2018, 11:57:39 AM »
I have written a blog post on 'Building 4D Polytopes'. This actually goes back to a Fractal Forums discussion back in 2012, where Knighty came up with a clever scheme for ray marching polytopes. It also discusses the mathematics behind Jenn 3D (a tool for creating explicit geometry).

It is probably my most ambitious post yet - lots of interactive WebGL components - which also means it not very mobile friendly, I'm afraid.

The post can be found here:
https://syntopia.github.io/Polytopia/polytopes.html

Please let me know if you have ideas for improvements or corrections!




Linkback: https://fractalforums.org/fractal-mathematics-and-new-theories/28/building-4d-polytopes/1106/

Offline claude

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« Reply #1 on: March 28, 2018, 06:30:58 PM »
In the first example with the 3 cube reflections I only saw 2 (the red was invisible as it was edge-on to the camera) and took me a while to realize I could rotate it by clicking on it.  Maybe have the default rotation of that example be less axis-aligned?

The iterative construction of the coordinates of the normals reminds me of the Schlaefli determinant mentioned in Coxeter's Regular Polytopes, which I used in my own code: https://code.mathr.co.uk/reflex

Offline Syntopia

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« Reply #2 on: March 28, 2018, 10:24:43 PM »
Good idea - I think I will see if I can make the figures rotate a bit to make it more clear they are interactive.

Do you have any links to screenshots or similar of your code, Claude?

Offline claude

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« Reply #3 on: March 28, 2018, 11:53:18 PM »

Offline mclarekin

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« Reply #4 on: March 29, 2018, 12:19:37 PM »
Thankyou, I find this really interesting.  I will need to read it a few more times, but i understand enough to know that i like it :)

The interactive  "Putting it all together" raymarching is real cool on Firefox.

These shapes i recognize from linear type fractals (e.g. knightys PseudoKlenian).

So I  tam guessing that i can try implementing this maths  as  parameters for constructing/morphing fractals.

And thank you for your blogs and Fragmentarium, they have taught me a lot of things

Offline knighty

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« Reply #5 on: March 29, 2018, 02:03:09 PM »
Super, hyper cool. Thank you, particularly for the explanations about the world problem for groups.

Offline Syntopia

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« Reply #6 on: March 30, 2018, 12:46:54 AM »
Thanks, Knighty - the cleverness is all yours (and Fritz Obermeyers) - I just try to disseminate it!

Btw, I discovered that the order of the U,V,W,(T) parameters did not match the Coxeter-diagrams. I have changed the order, so that it should match now, and now I also show the Coxeter-diagram and modification name (all 16 possible) based on the choice of parameters.


Offline mclarekin

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« Reply #7 on: March 30, 2018, 01:29:56 AM »
Quote
The distance estimator tells us how large steps we are allowed to march.

Possibly reword the above line for better English? :)

The distance estimator tells us how large a step we are allowed to march.

The distance estimator predicts how large a step we are allowed to march.

The distance estimator calculates an estimation of how large a step we can march without overstepping the surface.

Offline mclarekin

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« Reply #8 on: March 30, 2018, 02:59:50 AM »
Quote
That is a choice: this corresponds to 'inflating' the cube onto the sphere, making the edges live on the great circle arcs.

That is a choice: this corresponds to 'inflating' the cube onto the sphere, making the edges lie on the great circle arcs.  I think??

Offline talamhCothrom

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« Reply #9 on: February 28, 2019, 10:39:52 PM »
I know this thread is nearly a year old at this stage but I just wanted to say thanks to Syntopia for creating his blog post. It has introduced me the intrigue of the fourth dimension and has really opened up a new world for me!


The iterative construction of the coordinates of the normals reminds me of the Schlaefli determinant mentioned in Coxeter's Regular Polytopes, which I used in my own code: https://code.mathr.co.uk/reflex


Regarding the above, I've been trying to understand the construction of the normal coordinates but I can't seem to figure out how they are derived. How are the normalisation constants calculated? From where are the cos() terms derived? Are they contained in a rotation matrix? 

I'm attempting to tackle Coxeter's 'Regular Polytopes' as mentioned by claude which I hope will uncover the answers to my questions eventually but I have to say it's tough work and I'm not confident I'll get to the end of it any time soon.

Would someone be able to point me in the right direction for gaining some insight into these values? Or is there some foundation work I need to do before I can fully understand the underlying math?

Either way thanks again for this great work!


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