4d snowflakes

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

4d snowflakes

Postby anderscolingustafson » Wed Feb 19, 2014 10:34 pm

I was just thinking about snowflakes in 3d and how they are flat. I was just wondering what would a 4d snowflake look like? I mean would a 4d snowflake always have n-1 dimensions or would snowflakes only be flat in odd numbers of dimensions?
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
anderscolingustafson
Tetronian
 
Posts: 316
Joined: Mon Mar 22, 2010 6:39 pm

Re: 4d snowflakes

Postby ICN5D » Wed Feb 19, 2014 11:34 pm

Well, "flat" is very relative. If we have a 2D snowflake, then according to us 3D-ers, it is flat. If we have a 3D snowflake, it would then be flat according to a 4D-er. A 3D snowflake has 3 dimensions, that is, the ones you are referring to. It not only grows along a plane, but has thickness as well. A 4D snowflake would then be a 4D version of this crystalline growth pattern. It's slices would be an entire 3D snowflake, and moving along the height would transform the lattice into whatever it's structure is along 4D. There really is no "flatness" with odd dimensions, only when you make n-1 slices of any n-D shape.

But, I think I see what you mean now. You're looking for a repeated pattern with thickness vs other linear extensions. A 3D snowflake has more 2D size than its 3D thickness. This would also be identical to a 4D snowflake, its 3D extensions would have greater size than its 4D thickness. This attribute can be in any dimension you want, even, odd, you will always have the ability to have a thin plate-like shape. However, with 4D, you can have a 2D plate combined with a 2D thickness, along 3 and 4D. Getting into even higher D, you will have a combinatorial explosion of thickness vs thinness.
It is by will alone, I set my donuts in motion
ICN5D
Pentonian
 
Posts: 1135
Joined: Mon Jul 28, 2008 4:25 am
Location: the Land of Flowers

Re: 4d snowflakes

Postby wendy » Thu Feb 20, 2014 7:05 am

Snow-flakes are generally flat, because the more rounded form of it is called 'hail'.

In any case, one might suppose that one of the symmetries, like semi-cubic = tetrahedral or octahedral, might be the dominate form. The rhombic dodecahedron is also useful for offering relatively attractive shapes too.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: 4d snowflakes

Postby Keiji » Thu Feb 20, 2014 7:10 am

I wonder if snowflakes in a 4D world would even be 3D? There's no reason for them not to be 2D in any dimension.

After all, the hexagonal structure is two-dimensional. A 3D structure would be more limited, maybe something icosahedral?
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: 4d snowflakes

Postby ICN5D » Thu Feb 20, 2014 7:37 am

Hmm. Snowflakes seem to have a radial-type symmetry. They grow from the center, and expand outwards. So, I guess a possible 4-flake could be a spherical version of this. Instead of growing along a plane, it would expand outward spherically, but still forming the complex fractals.
It is by will alone, I set my donuts in motion
ICN5D
Pentonian
 
Posts: 1135
Joined: Mon Jul 28, 2008 4:25 am
Location: the Land of Flowers

Re: 4d snowflakes

Postby Keiji » Thu Feb 20, 2014 8:24 am

Spherical symmetry means there is no structure, though. Snowflakes generally have hexagonal symmetry. There is no uniform analog of the hexagon in 3D, that's why I was considering the icosahedron.
User avatar
Keiji
Administrator
 
Posts: 1984
Joined: Mon Nov 10, 2003 6:33 pm
Location: Torquay, England

Re: 4d snowflakes

Postby wendy » Thu Feb 20, 2014 10:08 am

One might suppose that an atom in 3d is represented by a little cube or some other tiler. (rhombic dodeca etc).

When a crystal forms, these settle on a 'seed', which grows larger. But even though the cells might be cubes, the large scale figures that come from the figure depend on which of the different planes in the lattice is subject to being less absorbant or better fracture. For example, in the cube model, a brag plane of (1,0,0) gives larger cubes. But a plane at (1,1,0) is more likely to lead to rhombo-dodecahedra, and (1,1,1) to octahedra.

Water in ice might be imagined as being hexagonal prisms. This will form layers of some strength, but the layers tend to be less prone to growing. So a snowflake is more likely to add to the rim than the body, and end up being flatish.

Anything that is flat, is less likely to fall fast, because it "divides the air". A hedrid snowflake does not divide the air. You need a chorid or 3d one to do that. If the particular form is not a 3*1 lattice (eg a rhombic-dodecahedron prism), then you won't get a leaf-like structure. Instead, you will get hail-stones. If it isn't space-dividing, it won't flutter down. It will drop lile a pen.

Since there are not really many 3d primitives (only the A4 = B4 = C4 etc), one might suspect something like cubic or pyritohedral symmetry.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: 4d snowflakes

Postby ICN5D » Thu Feb 20, 2014 4:43 pm

I remember learning how water molecules are tiny magnets with a 108 degrees between the magnetic poles. When solidifying at atmospheric pressure, they arrange into hexagonal patterns, responsible for lowering density. Then, of course, there is such a substance as ice-7, called " hot ice ", which is barometrically solidified, not thermodynamically. It's structure is different as in non-hexagonal.

Actually, this reminds me of an old article in Sky & Telescope that talked about halos and parahelial arcs, and other refractions. The focus was on other ices and their different crystal structure. Take Mars for example: with carbon dioxide ice crystals in the atmosphere, the halos and sundogs are very cool and more complex. I think it referred to a simple ray tracing program called " Halo ", not the video game :) . It allowed you to custom design crystals and general arrangement patterns, like vertical columns, flat arrays, angle of the sun, etc. So, I wonder what halos would come from 4D crystal structures. Most likely a 3D array of focal points in a 4D sky. I suppose one could plug in a CRF polychora ice crystal and derive a halo from it.

Here's the website : http://www.atoptics.co.uk/halo/howwork.htm
It is by will alone, I set my donuts in motion
ICN5D
Pentonian
 
Posts: 1135
Joined: Mon Jul 28, 2008 4:25 am
Location: the Land of Flowers

Re: 4d snowflakes

Postby anderscolingustafson » Fri Feb 21, 2014 6:50 am

In 3d there are basically two ways to form the shape of a snow flake. One is to put two triangles together,

Image

the other is to attach six triangles to a hexagon.

Image

The 3d closest 3d equivalent of a hexagon is an icosahedron and the 3d equivalent of a triangle is a tetrahedron. In 3d if you put two tetrahedrons together you get a different shape than if you put tetrahedrons on the surface of an icosahedron. So in 4d if snow flakes were 3d shapes there would be two shapes for the 4d equivalents of the shape of a snow flake, one coming from laying two tetrahedrons on each other and the other from putting tetrahedrons around an icosahedron.
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
anderscolingustafson
Tetronian
 
Posts: 316
Joined: Mon Mar 22, 2010 6:39 pm

Re: 4d snowflakes

Postby Prashantkrishnan » Sat Jan 17, 2015 6:28 am

Neither tetrahedra nor icosahedra tile space... So the snowflakes would not be arranged in a well-defined manner in those shapes. There would be gaps in between. Wendy's model of cubic snowflakes seems most convincing. But to think of it another way, is it necessary that snowflakes should be able to tile space?
People may consider as God the beings of finite higher dimensions,
though in truth, God has infinite dimensions
User avatar
Prashantkrishnan
Trionian
 
Posts: 114
Joined: Mon Jan 13, 2014 5:37 pm
Location: Kochi, Kerala, India

Re: 4d snowflakes

Postby ubersketch » Sun Dec 24, 2017 2:20 pm

anderscolingustafson wrote:In 3d there are basically two ways to form the shape of a snow flake. One is to put two triangles together,

Image

the other is to attach six triangles to a hexagon.

Image

The 3d closest 3d equivalent of a hexagon is an icosahedron and the 3d equivalent of a triangle is a tetrahedron. In 3d if you put two tetrahedrons together you get a different shape than if you put tetrahedrons on the surface of an icosahedron. So in 4d if snow flakes were 3d shapes there would be two shapes for the 4d equivalents of the shape of a snow flake, one coming from laying two tetrahedrons on each other and the other from putting tetrahedrons around an icosahedron.

Correction, the best candidate for 3d solid equivalent of hexagon would be truncated tetrahedron, however if anything, a 3d snowflake would have a truncated octahedral symmetry or be based on the Weaire-Phelan Structure (Weaire-Phelanic).
gwa
discord is spiritbackup#1797
User avatar
ubersketch
Trionian
 
Posts: 159
Joined: Thu Nov 30, 2017 12:00 am


Return to Higher Spatial Dimensions

Who is online

Users browsing this forum: No registered users and 9 guests