Thread: "Spherical distortion on 4D to 3D cameras"

As Yahoo groups handles images in strange and mysterious ways, here is a direct link to the image he tried to share:
http://superliminal.com/cube/misc/fisheye.jpg
I like it a lot, though I wonder why the central cube does not share in the distortion.

-Melinda

--------------146E87523A7CABBB8E9B26C9
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: 7bit

On 8/5/2019 4:49 PM, mananself@gmail.com [4D_Cubing] wrote:
> [...] Somehow the rendering of the image has a cartoon-ish style. I think it has to do with the curvy surfaces. Is there anything special about the colors? What did you use to rendered after getting the coordinates?

It looks to me like the squares are subdivided into 11x11 grids and lit individually. Part of the effect might simply be self-shadows which we're not used to seeing.

-Melinda

--------------146E87523A7CABBB8E9B26C9
Content-Type: text/html; charset=utf-8
Content-Transfer-Encoding: 7bit






On 8/5/2019 4:49 PM, mananself@gmail.com [4D_Cubing] wrote:

[...] Somehow the
rendering of the image has a cartoon-ish style. I think it has to
do with the curvy surfaces. Is there anything special about the
colors? What did you use to rendered after getting the
coordinates?

--0000000000006f2164058f6e2560
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable

I love this representation! Would be nice to have it in the software as
well.

On Tue, 6 Aug 2019, 09:50 Melinda Green melinda@superliminal.com
[4D_Cubing], <4D_Cubing@yahoogroups.com> wrote:

>
>
> On 8/5/2019 4:49 PM, mananself@gmail.com [4D_Cubing] wrote:
>
> [...] Somehow the rendering of the image has a cartoon-ish style. I think
> it has to do with the curvy surfaces. Is there anything special about the
> colors? What did you use to rendered after getting the coordinates?
>
>
> It looks to me like the squares are subdivided into 11x11 grids and lit
> individually. Part of the effect might simply be self-shadows which we're
> not used to seeing.
>
> -Melinda
>
>=20
>

--0000000000006f2164058f6e2560
Content-Type: text/html; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable

--0000000000002a3e2a058f8f61d5
Content-Type: text/plain; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable

Cool! I've wanted to see MC4D on the hypersphere for a long time :)

I had a couple thoughts catching up on this thread...

- Perhaps the simplest way to map points (from a square, cube,
hypercube, or any shape really) to a sphere is simply to normalize the
points. This makes them unit length, i.e. radially projects them to the
sphere. I'm not sure how your functions compare to radial projection.


- A natural choice for the 4D camera point 'c' is the north pole of the
sphere <0,0,0,1>, which makes the projection of the sphere "stereographi=
c
projection ".
This projection has the nice property that it is "bijection" with Euclid=
ean
space plus a "point at infinity" (meaning it is both one-to-one and onto=
-
in short, no projected points can crash into each other). Stereographic
projection has many other nice properties as well. I would like to see o=
ne
of your Blender images with the camera placed at the north pole. It look=
s
like you might have picked a 4D projection point off the sphere (?)

Cheers,
Roice


On Wed, Aug 7, 2019 at 6:04 PM programagor@gmail.com [4D_Cubing] <
4D_Cubing@yahoogroups.com> wrote:

>
>
> Greetings again
>
> The whole thing is done in Blender 2.80, which has excellent Python
> integration. The rendering is done using a real-time approximating
> raytracer Eevee, which works really smoothly, but the shadows are not ver=
y
> realistic. The bloom is also turned on, which in combination with the
> highlighted outlines made for the cartoon look, I think.
>
> The W- face does experience a little bit of distortion as well, just not
> as much since it is further from the surface of the unit sphere.
>
> Also, I uploaded all related files here:
> https://git.mckay-bednar.net/jiri/mc4d-hw
>
> And if anyone wants to help out with an electronic version of the MC4D,
> here's a little challenge::
> https://math.stackexchange.com/questions/3316660/how-to-reliably-lay-out-=
continuous-unfolded-diagrams-of-3d-shapes
>
>
> ---In 4D_Cubing@yahoogroups.com, wrote :
>
> I love this representation! Would be nice to have it in the software as
> well.
>
> On Tue, 6 Aug 2019, 09:50 Melinda Green melinda@... [4D_Cubing], <
> 4D_Cubing@yahoogroups.com> wrote:
>
>
>
> On 8/5/2019 4:49 PM, mananself@... [4D_Cubing] wrote:
>
> [...] Somehow the rendering of the image has a cartoon-ish style. I think
> it has to do with the curvy surfaces. Is there anything special about the
> colors? What did you use to rendered after getting the coordinates?
>
>
> It looks to me like the squares are subdivided into 11x11 grids and lit
> individually. Part of the effect might simply be self-shadows which we're
> not used to seeing.
>
> -Melinda
>
>
>
>=20

--0000000000002a3e2a058f8f61d5
Content-Type: text/html; charset="UTF-8"
Content-Transfer-Encoding: quoted-printable

--=_26abb7666e22fe47ca52b336ed4acd57
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: quoted-printable

In case you missed this: [1]https://arxiv.org/abs/1106.5736 [2]

"In this paper, we show that the Rubik's Cube also has a rich
underlying algorithmic structure. Specifically, we show that the n x n
x n Rubik's Cube, as well as the n x n x 1 variant, has a "God's
Number" (diameter of the configuration space) of Theta(n^2/log n)."

When I passed this along to Melinda Green, I thought "n" included
higher dimensionality. It only refers to the number of cublets along
any edge but all in 3 dimensions.

In any case, I would think God's Number could be easily found by
establishing how many twists would be needed to reach any cube
configuration. Reverse that an that should be God's Number.

Regards

John Bailey

-----------------------------------------From: "Melinda Green"=20
To: "John Bailey"
Cc:=20
Sent: Thursday October 10 2019 6:46:30PM
Subject: Re: Shortest solution for any dimension Rubik's Cube.

Dear John,

I skimmed the article and it's very interesting. Please consider
posting it to the mailing list where I'm sure several people will
really enjoy it and help the rest of us to understand it.

Best,
-Melinda

On 10/4/2019 11:35 AM, John Bailey wrote:
In case you missed this: [3]https://arxiv.org/abs/1106.5736 [4]=20

"In this paper, we show that the Rubik's Cube also has a rich
underlying algorithmic structure. Specifically, we show that the n x n
x n Rubik's Cube, as well as the n x n x 1 variant, has a "God's
Number" (diameter of the configuration space) of Theta(n^2/log n)."=20

Regards=20

John=20

=20

Links:
------
[1] https://arxiv.org/abs/1106.5736
[2] https://arxiv.org/abs/1106.5736
[3] https://arxiv.org/abs/1106.5736
[4] https://arxiv.org/abs/1106.5736


--=_26abb7666e22fe47ca52b336ed4acd57
Content-Type: text/html; charset=UTF-8
Content-Transfer-Encoding: quoted-printable

8aa9509924f073b@webmail">In case you missed this: rxiv.org/abs/1106.5736" moz-do-not-send=3D"true">iv.org/abs/1106.5736" target=3D"_blank">https://arxiv.org/abs/1106.5736=

verdana, sans-serif; font-size: 13.608px;">"In this paper, we show that th=
e Rubik's Cube also has a rich underlying algorithmic structure. Specifical=
ly, we show that the n x n x n Rubik's Cube, as well as the n x n x 1 varia=
nt, has a "God's Number" (diameter of the configuration space) of Theta(n^2=
/log n)."

;, helvetica, arial, verdana, sans-serif; font-size: 13.608px;">When I pass=
ed this along to Melinda Green, I thought "n" included higher dimensionalit=
y.  It only refers to the number of cublets along any edge but all in =
3 dimensions.

quot;, helvetica, arial, verdana, sans-serif; font-size: 13.608px;">In any =
case, I would think God's Number could be easily found by establishing how =
many twists would be needed to reach any cube configuration.  Reverse =
that an that should be God's Number.

y: "Lucida Grande", helvetica, arial, verdana, sans-serif; font-s=
ize: 13.608px;">Regards

da Grande", helvetica, arial, verdana, sans-serif; font-size: 13.608px=
;">John Bailey

-----------------------------------------