Message #2314

From: Don Hatch <>
Subject: Re: [MC4D] Re: Hyperbolic Honeycomb {7,3,3}
Date: Thu, 05 Jul 2012 20:29:07 -0400

On Thu, Jul 05, 2012 at 04:39:07PM -0400, I wrote:
> It’s also really strange that this thing is convex.
> That seems, intuitively, to contradict the fact that it’s got fat parts
> and skinny parts separating the fat parts…
> in euclidean space this would be a contradiction,
> but in hyperbolic space, apparently it isn’t.
> It’s an interesting thing to meditate on.
> I wonder if there’s some stronger notion of convexity
> formalizing this intuitive notion. Something like the following
> additional condition:
> whenever you tie a string around the object,
> it should be possible to continuously slide
> the string off the object without stretching it.
> (Your convex object would fail to meet this criterion.)

Actually I think this is a short-lived idea–
a simple cube in H3 (or square in H2)
fails to meet the criterion.
I think this is just one of the discomforts of hyperbolic space
that I just need to accept.

My life seems to resemble hyperbolic space lately…
that is, it’s really hard to tell precisely where I’m headed,
since very slight deviations, or exploring a seemingly small diversion,
seem to result in wild divergences
leading to wildly different neighborhoods and results
which seem entirely unrelated to where I thought I was going.

I guess this is in contrast with spherical space,
in which it doesn’t matter much which way you go, you end up
in the same place anyway.

Contemplating Andrey’s fractal thing (in which my intuition is
continually thwarted, in that what seemed to be a little corner of space
actually is hugely more vast than all of my current experience… repeatedly)
seems to resonate with me in a partly disturbing, partly comforting way.