Newsgroups: sci.math
From: roy@dsbc.icl.co.uk (Roy Lakin)
Subject: Re: Hexaflexagon/ Trihexaflexagon
Date: Wed, 13 Sep 1995 07:48:44 GMT

In article <DEE488.4J3@novell.co.uk> Chris Lusby Taylor <clusbyta@novell.com> writes:
>pdx4d@teleport.com (Kirby Urner) wrote:
>>
>> I have these M.C. Escher paper things of folded triangles which I can

[...]

Sorry to tack on to this, but I tried to reply to an email on this subject,
but was bounced.


S  if MNR F BREUER <s9520155@itu2.sun.ac.z>a is listening, here is a fuller
description of how to make a hexatetraflexagon:

For the benefit of others who read this, take a rectangular piece of paper,
fold into 16 and remove the middle 4 panels.

 
_____________________
| 1  | 2  | 3  | 4  |   okay. Here is a layout of the paper with the
|____|____|____|____|   four middle squares cut out. I labelled the squares
| 5  |         | 6  |   and the back sides of the squares are labelled 1B,
|____| cut out |____|   2B, etc.. (xB is the other side of x).
| 7  |         | 8  |   Could you please describe what faces I should see
|____|_________|____|   after each step?
| 9  | 10 | 11 | 12 |
|____|____|____|____|
 
 
Also xU means upside down (and of course xUB...)
 
Fold back the left side:
 
 
________________
| 2  | 3  | 4  |
|____|____|____|
| 5B |    | 6  |
|____|out |____|
| 7B |    | 8  |
|____|____|____|
| 10 | 11 | 12 |
|____|____|____|
 
Fold up the bottom:
 
 
________________
| 2  | 3  | 4  |
|____|____|____|
| 5B |    | 6  |
|____|____|____|
| 9U |11UB|12UB|
|____|____|____|

Fold back the right:


 
___________
| 2  | 3  |
|____|____|
| 5B | 6B |
|____|____|
| 9U |11UB|
|____|____|

Now the tricky bit, fold down the top, but insert your L finger between
the 2 and 1 panels and twist it forward:


___________
| 2UB| 4B |
|____|____|
| 9U |11UB|
|____|____|

then the first flex can be done by taking the 2UB in the L hand,
the 11UB in the R, then twist the L back and the R forward.
___________
| 7B | 3  |
|____|____|
| 10 | 6B |
|____|____|

and the next flex can be done by taking the 7B in the L hand,
the 6B in the R, then twist the L left and the R right.



and so on.


There is a cycle of 4 positions, but if you flex one way when you should
the other in one of those positions, you find a fifth.

There is a sixth (left as an exercise !-)

Now, if you fold the original paper into 64 squares and remove all but the
outside, you can repeat the folding exercise twice and obtain a
tetrakaidekatetraflexagon!

And the centre can be recycled.


(to flo:)

My 51st but when I first caught flexagons I was a 2nd year
undergraduate.  A group of us got together and had a play around with
them: we even sent a letter to one of the group with an important message
on the 6th flex.

But don't let that stop you exchanging problems.  It will help derust
my memories.


Sorry for the delay.

roy