[hackerspaces] [sfx: Discuss] 3D printing of cool geometrical objects
Mark Adam
dreamingforward at gmail.com
Fri Dec 7 00:59:38 CET 2012
On Thu, Dec 6, 2012 at 5:30 PM, Bruce Sherwood <bruce.sherwood at gmail.com> wrote:
> Glad you like Anton's stuff, Owen. Anton tells me that he can't think of
> anything to add.
>
> As for the tools he uses, he writes Python programs to do the calculations,
> with VPython to visualize the 3D objects. His program also writes out a data
> files to be sent to a 3D printer.
That's pretty cool, I think the general hackerspace community will be
interested in those tools. Thanks for the info!
mark
cc: discuss at lists.hackerspaces.org
****Background context: Bruce Sherwood (maintainer of the uber-cool
VPython 3-d programming environment -- well worth downloading and
checking out):*****
In response to a question, Anton comments that his 2D inventions,
including coffee mugs and tablet and phone pads decorated with them,
can be found at zazzle.com/tamfang/gifts.
Here's what Anton says about the 3D objects. Note that I've included
him in the cc's to this note.
The "Klein bagel", I am assured, is topologically equivalent to the
familiar bottle shape, though I haven't seen a continuous deformation
illustrated.
But it's fairly easy to see that both are equivalent to a rectangle
with opposite sides identified, one pair being reversed.
My bagel form starts with the lemniscate of Bernoulli. Rotate it a
half-turn in a plane that's sweeping a full turn, and you're there.
Of course there are subtleties. How much detail do you want?
https://en.wikipedia.org/wiki/Lemniscate_of_Bernoulli
http://kleinbottle.com/ -- the text is hilarious
For the ribbons on hyperspheres:
Consider two great circles on the surface of the unit hypersphere:
( cos(alpha), sin(alpha), 0, 0 )
( 0, 0, cos(beta), sin(beta) )
A ribbon can be stretched between them, always keeping unit radius:
( cos(gamma)*cos(alpha), cos(gamma)*sin(alpha),
sin(gamma)*cos(alpha), sin(gamma)*sin(alpha) )
More interestingly, the ribbon can wrap around the circles at different speeds:
( cos(gamma)*cos(p*alpha), cos(gamma)*sin(p*alpha),
sin(gamma)*cos(q*alpha), sin(gamma)*cos(q*alpha) )
And then I rotate coordinates so that the ribbon does not pass near
(1,0,0,0) -- because I like it that way -- and do a stereographic
projection:
(X,Y,Z) = (x,y,z)/(1+w)
discarding those parts of the ribbon where w<0, to keep the resulting
figure compact. The scale of the projection at the surface is twice
the scale at the center.
...
Part of the above is a fib: it's the easiest way to describe the
ribbon, but not quite how I generate it in fact; I don't remember
exactly why not. But I've practically persuaded myself that the
following procedure is equivalent.
A point on the ribbon is indeed interpolated between the two circles:
( cos(gamma)*cos(alpha), cos(gamma)*sin(alpha),
sin(gamma)*cos(beta), sin(gamma)*cos(beta) )
gamma itself is ((p*alpha+q*beta) mod (2*pi))/4. The ribs in the
model represent constant alpha and constant beta.
On Wed, Dec 5, 2012 at 8:29 PM, Owen Densmore <owen at backspaces.net> wrote:
Could we get an explanation of the mathematical models behind the
figures? I'm thinking of getting some for christmas presents and
would like to know what the shapes represent in n-space.
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