Welcome to everyone who has come here from Rudy Rucker’s blog, from Vimeo, or from my LKL talk. I have attempted here to give a a bit more explanation of my recent 4D rotation animations, in as non-technical a way as possible.

I have done some experiments with 3D inversion and this developed out of that. Now if you dont know it there is this fantastic video by Douglas Arnold and Jonathan Rogness called Moebius transformations revealed. Its beautifully done and I really recommend you watch it.

The key point for the following discussion is at around 1:55
I was very impressed by how clear and intuitive this made the notion of inversion in 2D. So I got to thinking about something similar for my inversions in 3D.
For this I’ve used something called the 3-sphere.

(Now its important to note that what in everyday language is called a sphere is referred to by mathematicians as a 2-sphere. A 1-sphere is a circle and a 3-sphere (or hypersphere) is an object which lives in 4-dimensional space just as an everyday 2-sphere lives in 3D space.)

So I wrote some code which uses an inverse stereographic projection (often described in 3D, but it generalises naturally to higher dimensions) to project Euclidean 3-space (ie normal, everyday 3D space) onto the 3-Sphere (A space of non-Euclidean geometry, a curved space like those of Einstein’s relativity).
I then performed a 4-dimensional rotation on this 3-Sphere while stereographically projecting back down to Euclidean 3-space. Just as the ‘Moebius transformations revealed’ video projects rotation in 3D to the 2D plane, my animations project rotation in 4D to a 3D space so that we can see it.

[…] one then has lines and circles about which the space revolves. I began to explore this a bit in my post here, using stereographic projection in 4D. Things rapidly become rather confusing, with the […]

Perhaps if the atomic structures of the material that forms some hypothetical newly evolved cortex of the brain – if this atomic material were to contain an axis lying in a fourth direction; let’s say a .0000006 seconds into the future and .0000005 seconds into the past, and with particles at these distances, then information of four space might be able to be processed into perceptual interpretation mediated through the 4-D neurons of this cortex. -scriAlphi

Daniel, tks for the reply. As a matter of fact I have the book, but haven’t read it in it’s entirety. I submitted this idea as a result of having read a discription of a science fiction pseudochemical as described by Isaac Asimov called thiotimeline or something like that. This possessed those very properties i described but not as part of any cortex. But were reactively chained to extend this miniscule penetration into the future to up to 2 seconds. I’ll serch for the article or any reference to thiotimoline (or something like that) in google. -schriAlphi

[…] on rheotomic surfaces Those coming from BoingBoing (!!!) in search of Gnarl might also enjoy my 4D rotation animations or experimants with Cellular Automata. Possibly related posts: (automatically generated)Rheotomic […]

Hi. This is C. Ernesto S. Lindgren from Brazil. Nice videos. May I suggest you to take a look at http://randaljbishop.com/AboutUs.html and http://www.youtube.com/watch?v=VDOSGrB95Mg? Randy´s video was based on the book that I wrote with Professor Steve Slaby, Princeton University, back in 1968. I have some other intresting thing which is a 4D environment consisting of four 3D Euclidean spaces exchanging matter and antimatter. Speed of the particles ought to be graeter than light. There would be 96 transfers so with the neutrino is one down and 95 to go. It was presented in a paper at 12th ICGG, Salvador, Brazil, 2006. Best regards.

Your blog is pretty cool to me and your topics are very relevant. I was browsing around and came across something you might find interesting. I was guilty of 3 of them with my sites. “99% of blog owners are committing these five errors”. http://is.gd/FDoL2T You will be suprised how fast they are to fix.

Like Alex (#12), I would like to see this ported to OpenGL or some other platform that doesn’t require a commercial software license. You said in comment #13 that you planned to post the Rhino code soon. I understand how that goes … no criticism implied. But I would be interested in seeing that code if there’s still a prospect of posting it. Or, a pointer to some math explanations would be better than nothing. Glad to have the link about stereographic projection… but I’m not confident I’d be able to translate that into code.

//getting the coordinates of the input point
double xa = Point.X;
double ya = Point.Y;
double za = Point.Z;
double wa = 0;

//reverse stereographically project to Riemann hypersphere
double xb = 2 * xa / (1 + xa * xa + ya * ya + za * za);
double yb = 2 * ya / (1 + xa * xa + ya * ya + za * za);
double zb = 2 * za / (1 + xa * xa + ya * ya + za * za);
double wb = (-1 + xa * xa + ya * ya + za * za) / (1 + xa * xa + ya * ya + za * za);

Hey there,
I while back I saw your posts here about the Moebius transformation and got completely swept away! It fascinates me how this fundamental transformation so effortlessly swaps inside with outside. It’s like a 3 dimensional “lens”.

So I made this WebGL implementation about a year ago. It’s a GLSL vertex shader.

I’d like to make a modifier in Blender so that I can transform animated scenes in full color. Right now I’ve got some rudimentary stuff happening in Blender’s game engine using the GLSL shader. It’s really fun to “navigate” by rotating and translating the hypersphere. The Moebius transformation turns a model of the human heart into an entire landscape!

It’s very interesting to transform a simple 3D “light cone”. Since the past and future both pass through infinity, they must intersect one another, just as they intersect at the “origin” (“here and now”). Well, at least that’s what it looks like on the screen. It actually looks that any two light cones a finite distance apart just might intersect one another at infinity. But if you asked me to prove it I don’t know if I’m ready for that. Screencaps http://imgur.com/H9HEGo0,XWv9GlS,WPCZCGf

hint: you can use the spacebar, arrows and drag the mouse to control it

Anyway, I really appreciated your posts, thanks a bunch. They have taken me on an adventure! I’ve even got a couple of geometric algebra/calculus books i’ve been slowly working through!

Lots of parents these days are on a budget but this
shouldn’t be a problem when shopping for a stroller because lots of manufacturers offer amazing value for money whilst not compromising on quality. Some parents opt for the pushchairs that have reversible seats. This is considered as a form of investment which can be compared to a car, since it includes the safety of your child along with your convenience.

this is a really nice tool! I installed it, although when I try to run it I get an error, error in running vbs script, line 23… etc.. Does anyone else have this problem, is there a quick-fix?

[…] Projecting Euclidean 3D space onto the hypersphere, and rotating that Non-Euclidean space in 4-dimensions. Read more about it here: 4-Dimensional Rotations […]

[…] Daniel Piker, “4-Dimensional Rotations,” Space Symmetry Structure, accessed February 25, 2017, https://spacesymmetrystructure.wordpress.com/2008/12/11/4-dimensional-rotations/. […]

December 11, 2008 at 7:14 pm

[…] Space Symmetry Structure […]

January 13, 2009 at 6:39 pm

was this made in vvvv? If so care to share the code? If not mybe when I’m not lazy I’ll reuse what you’ve presented and try to code my own… cool stuff

January 13, 2009 at 10:48 pm

Thanks. This is Rhinoscript, not vvvv. Let me know if you do code a version, I’d be happy to help

January 19, 2009 at 4:36 am

[…] Space Symmetry Structure […]

February 6, 2009 at 5:33 pm

[…] one then has lines and circles about which the space revolves. I began to explore this a bit in my post here, using stereographic projection in 4D. Things rapidly become rather confusing, with the […]

March 3, 2009 at 12:15 am

[…] 4-Dimensional Rotations « Space Symmetry Structure (tags: inspiration animation fun visualization math) […]

March 11, 2009 at 12:05 pm

[…] finally, 4-dimensional […]

April 5, 2009 at 12:26 am

Perhaps if the atomic structures of the material that forms some hypothetical newly evolved cortex of the brain – if this atomic material were to contain an axis lying in a fourth direction; let’s say a .0000006 seconds into the future and .0000005 seconds into the past, and with particles at these distances, then information of four space might be able to be processed into perceptual interpretation mediated through the 4-D neurons of this cortex. -scriAlphi

April 5, 2009 at 1:55 pm

schriAlphi – have you read Penrose’s

The Emperor’s New Mind? I think the ideas there are somewhat similar to what you describe.April 5, 2009 at 3:08 pm

Daniel, tks for the reply. As a matter of fact I have the book, but haven’t read it in it’s entirety. I submitted this idea as a result of having read a discription of a science fiction pseudochemical as described by Isaac Asimov called thiotimeline or something like that. This possessed those very properties i described but not as part of any cortex. But were reactively chained to extend this miniscule penetration into the future to up to 2 seconds. I’ll serch for the article or any reference to thiotimoline (or something like that) in google. -schriAlphi

May 23, 2009 at 7:29 pm

[…] on rheotomic surfaces Those coming from BoingBoing (!!!) in search of Gnarl might also enjoy my 4D rotation animations or experimants with Cellular Automata. Possibly related posts: (automatically generated)Rheotomic […]

June 9, 2009 at 10:25 pm

Great work Daniel, I first saw the youtube inversion and wondered about the 3D. Can you share the Rihnoscript for us that want to code in… say OPenGL?

June 10, 2009 at 5:49 pm

Thanks Alex,

Yea, will post the code soon(ish), and maybe a grasshopper version.

I’d love to see this transferred to some other languages, and I’m interested in learning some OpenGL myself

October 4, 2009 at 7:33 pm

[…] Daniel Piker has some very interesting demonstrations of 4-dimensional rotations: […]

November 23, 2009 at 1:02 pm

Wow this is so cool! it’s explained really well too :) thanks

March 20, 2010 at 6:07 am

[…] 4-Dimensional Rotations « Space Symmetry Structure (tags: 3d animation mathematics dimensions) […]

June 10, 2010 at 2:24 pm

[…] paintings inspired by my 4D rotation […]

May 3, 2011 at 10:16 pm

Say a hologram was made tomorrow and you tried to show this 4d horse rotation within that 3d hologram, what would happen?

July 8, 2011 at 3:44 pm

[…] and their blog, equally so 4-Dimensional Rotations « Space Symmetry Structure […]

October 3, 2011 at 10:05 pm

Hi. This is C. Ernesto S. Lindgren from Brazil. Nice videos. May I suggest you to take a look at http://randaljbishop.com/AboutUs.html and http://www.youtube.com/watch?v=VDOSGrB95Mg? Randy´s video was based on the book that I wrote with Professor Steve Slaby, Princeton University, back in 1968. I have some other intresting thing which is a 4D environment consisting of four 3D Euclidean spaces exchanging matter and antimatter. Speed of the particles ought to be graeter than light. There would be 96 transfers so with the neutrino is one down and 95 to go. It was presented in a paper at 12th ICGG, Salvador, Brazil, 2006. Best regards.

December 4, 2011 at 1:59 pm

Daniel Picker,

could you please tell me which course did you take on college to learn how to do those programs you do?

December 9, 2011 at 2:21 pm

Your blog is pretty cool to me and your topics are very relevant. I was browsing around and came across something you might find interesting. I was guilty of 3 of them with my sites. “99% of blog owners are committing these five errors”. http://is.gd/FDoL2T You will be suprised how fast they are to fix.

March 12, 2012 at 7:12 pm

Like Alex (#12), I would like to see this ported to OpenGL or some other platform that doesn’t require a commercial software license. You said in comment #13 that you planned to post the Rhino code soon. I understand how that goes … no criticism implied. But I would be interested in seeing that code if there’s still a prospect of posting it. Or, a pointer to some math explanations would be better than nothing. Glad to have the link about stereographic projection… but I’m not confident I’d be able to translate that into code.

March 22, 2012 at 10:18 pm

Hi all,

Here’s my code :

//getting the coordinates of the input point

double xa = Point.X;

double ya = Point.Y;

double za = Point.Z;

double wa = 0;

//reverse stereographically project to Riemann hypersphere

double xb = 2 * xa / (1 + xa * xa + ya * ya + za * za);

double yb = 2 * ya / (1 + xa * xa + ya * ya + za * za);

double zb = 2 * za / (1 + xa * xa + ya * ya + za * za);

double wb = (-1 + xa * xa + ya * ya + za * za) / (1 + xa * xa + ya * ya + za * za);

//now rotate the hypersphere (use p = q = 1 for isoclinic rotations)

//and vary t between 0 and 2*PI

double xc = +(xb) * (Math.Cos((p) * (t))) + (yb) * (Math.Sin((p) * (t)));

double yc = -(xb) * (Math.Sin((p) * (t))) + (yb) * (Math.Cos((p) * (t)));

double zc = +(zb) * (Math.Cos((q) * (t))) – (wb) * (Math.Sin((q) * (t)));

double wc = +(zb) * (Math.Sin((q) * (t))) + (wb) * (Math.Cos((q) * (t)));

//then project stereographically back to flat 3D

double xd = xc / (1 – wc);

double yd = yc / (1 – wc);

double zd = zc / (1 – wc);

Point3d outPoint = new Point3d(xd, yd, zd);

March 7, 2013 at 2:36 am

Hey there,

I while back I saw your posts here about the Moebius transformation and got completely swept away! It fascinates me how this fundamental transformation so effortlessly swaps inside with outside. It’s like a 3 dimensional “lens”.

So I made this WebGL implementation about a year ago. It’s a GLSL vertex shader.

http://hyperspectives.timbremill.net/moebius/hyperhorse.html

I’d like to make a modifier in Blender so that I can transform animated scenes in full color. Right now I’ve got some rudimentary stuff happening in Blender’s game engine using the GLSL shader. It’s really fun to “navigate” by rotating and translating the hypersphere. The Moebius transformation turns a model of the human heart into an entire landscape!

It’s very interesting to transform a simple 3D “light cone”. Since the past and future both pass through infinity, they must intersect one another, just as they intersect at the “origin” (“here and now”). Well, at least that’s what it looks like on the screen. It actually looks that any two light cones a finite distance apart just might intersect one another at infinity. But if you asked me to prove it I don’t know if I’m ready for that. Screencaps http://imgur.com/H9HEGo0,XWv9GlS,WPCZCGf

hint: you can use the spacebar, arrows and drag the mouse to control it

Anyway, I really appreciated your posts, thanks a bunch. They have taken me on an adventure! I’ve even got a couple of geometric algebra/calculus books i’ve been slowly working through!

March 7, 2013 at 2:39 am

(Just to clarify, the hint is for the WebGL page)

August 10, 2013 at 11:09 pm

Lots of parents these days are on a budget but this

shouldn’t be a problem when shopping for a stroller because lots of manufacturers offer amazing value for money whilst not compromising on quality. Some parents opt for the pushchairs that have reversible seats. This is considered as a form of investment which can be compared to a car, since it includes the safety of your child along with your convenience.

October 23, 2013 at 8:48 am

Dear Daniel,

this is a really nice tool! I installed it, although when I try to run it I get an error, error in running vbs script, line 23… etc.. Does anyone else have this problem, is there a quick-fix?

Have a nice day/Sophia

December 12, 2016 at 1:06 am

[…] Projecting Euclidean 3D space onto the hypersphere, and rotating that Non-Euclidean space in 4-dimensions. Read more about it here: 4-Dimensional Rotations […]

February 28, 2017 at 8:22 am

[…] Daniel Piker, “4-Dimensional Rotations,” Space Symmetry Structure, accessed February 25, 2017, https://spacesymmetrystructure.wordpress.com/2008/12/11/4-dimensional-rotations/. […]