Message #2274

From: Roice Nelson <roice3@gmail.com>
Subject: Re: [MC4D] Re: Regular abstract polytopes based on {5,3,4} and {4,3,5}
Date: Wed, 13 Jun 2012 20:36:56 -0500

Hi Nan,

I like the {3,5,3}. That’s a lot of edges going into each vertex!

There is a fantastic two page explanation about what the world would look
like from inside S3 in Thurston’s book "Three Dimensional Geometry and
Topology<http://www.amazon.com/gp/product/0691083045/ref=as_li_ss_tl?ie=UTF8&tag=gravit-20&linkCode=as2&camp=1789&creative=390957&creativeASIN=0691083045>".
The section is "Example 1.4.2 (the three-sphere from inside)", and can be
read in isolation. He uses dimensional analogy to provide intuition on
your question about seeing yourself in each direction. He leads you up to
this:

In the background of everything else, you see an image of yourself, turned
> inside out on a great hollow sphere, wih the back of your head in front of
> you.

You can read the section on google
books<http://books.google.com/books?id=9kkuP3lsEFQC&lpg=PP1&dq=three%20dimensional%20geometry%20and%20topology&pg=PA32#v=snippet&q=three-sphere%20&f=false>
(starts
on page 32). Do check it out! After rereading it just now, I suspect the
youtube video you sent has some inaccuracies when it comes to how edge
thicknesses are displayed.

Also, I should have thought to mention this earlier as well, but check out
geometrygames.org, specifically the "CurvedSpaces" program. It will let
you navigate through spherical and hyperbolic spaces, from inside the space.

Even though there are existing youtube videos and programs, S3 would still
be a nice addition to your applet. It’s great to be able to run this stuff
right on a web page, without anything to install.

Cheers,
Roice

On Wed, Jun 13, 2012 at 7:26 PM, schuma <mananself@gmail.com> wrote:

> The {3,5,3} has been added last night.
>
> I was also thinking of adding "navigating inside 3-sphere" to the
> 3-hyperbolic space, where we can see, for example, the spherical hypercube
> when we are inside the 3-sphere. Then I found this playlist on youtube,
> showing the 120-cell, 600-cell and 24-cell in this way.
>
>
> http://www.youtube.com/watch?v=_RCAlhVlsWY&feature=bf_prev&list=PL09DF17B94CA3C6FD
>
> An interesting phenomenon is the retracting red edges. They just come when
> you assume light travels along the great circles of the hypersphere.
> Ideally, if there’s no limit of sight, you can see your back no matter
> which direction you are looking at, because light can travel a circle to
> your eyes. I really wonder what it means by saying "I can see myself in
> each direction". Does anyone have an idea?
>
> Given that these videos exist, should I include the regular spherical
> polytopes in my applet?
>
> Nan
>
>