Forth Dimension: Tetraspace

[edn]

Hello. New to this site. Known about 10 + 1 dimensions for a couple of years, but got lost in mentally transcending from 3rd to higher. Just read the above topic about the Dimensional Analogy in which it states that if a 3d nail were driven through (presumably perpendicularly) 2d Flatland, that a 2D Flatlander would see this as a "circle." I can't visualize this at all, unless the Flatlander were able to pop above or below his 2d plane and see the "ring" nature of the circle, just as us RealmLanders can visualize. Just as a Flatlander could not view his "square" safe as a "square" by other than inference (to turn it and measure the sides to know that they are all the same length), the Flatlander could not view the "cross section" of the nail as a circle, except by turning it around and being able to infer its circularity. Then the author went on to the "analogy" part by stating, if a 4d nail (a tetranail) were driven into the 3d world, the RealmLander woud perceive it as a "Sphere." I can understand the authors intention in jumping from a circle up to the next dimensional level and stating that the RealmLander would visualize the tetranail as a "sphere," but, for the life of me I cannot visualize this. Perhaps it is because I cannot visualize a 4d Tetranail, that I cannot visualize how it would appear as a sphere in a 3d world.
Any comments would be appreciated.
Thanks
Edn

[Keiji]

As 3D beings, we see in two dimensions so when we look at a sphere, we see a circle but we know it's a sphere because of depth perception. A 2D being would know a circle was a circle by seeing it in 1D and using depth perception in the same way.

[edn]

As 3D beings, we see in two dimensions so when we look at a sphere, we see a circle but we know it's a sphere because of depth perception. A 2D being would know a circle was a circle by seeing it in 1D and using depth perception in the same way.

Dear Keiji: Thanks for the reply. It seems to me that when I look at a golf ball I see a "sphere" not a "circle" which I must then infer is a "sphere" because of depth perception." Also, if I were a 2D being in a 2D world, with 2D vision, I would see a "line", the length of which would be the diameter of the circle, and then due to depth perception, I would see that it is a "curved" line, with the "center" of the "line" being closer to me than the two "ends" of the line, and hence I would know it was a "circle" or at least a "curved" object of some geometric shape. Please comment. Thanks, Edn.

[wendy]

One should understand that we in three dimensions, really do see two-dimensions, and use clues in colouring and shading to create the illusion of depth. see, eg visual illusions.

Were one to drive a hyper-object through a space, then occupants of the space would only see the interruption in their space. For example, a two-dimensional being would see only a circle, while we would see a sphere. This is because for 2d, one can see only x,y when w=z=0. while in 3d, we see x,y,z where w=0. So, the exact shape of what is seen is how the thing is orientated in space, and exactly where the section is driven.

A nail, driven through a peice of paper, yields its cross-section - ie a circle. In 2d, a nail is indeed a hyper-solid, since it is implemented a higher dimension. The hedrobour (2d being), sees the cross section as a solid in that space, which the illusions of depth conclude that becomes a circle. Likewise, we see such a thing as a sphere, (the length being extended in the direction w).

When one wants to look at dimensions, one needs to consider whether the dimension is relative to solid, to surface, or to a point. We are essentially solid animals, living in essentially a 2d world with limited height.

Consider for example, the following:

line = 1d (in bus-line, bee-line, queue, etc) [all are connections]
line = (s-2)d in dead-line, line in the sand, front-line, [all are divisions of surface]
edge = (s-2)d in "to the edge", 'the edge of the earths', etc.

face = S-2 d in "facing", "facet",

Generally, if the function of something is to 'divide', then its dimensionality is relative to solid space, while a thing to "unite" is relative to to a point.

A knife divides: specifically, the product of the blade and time (the sweep) gives a dividing (s-1) dimensions. A knife would in four dimensions, have a two-dimensional blade.

Roads and railways unite, so their dimension is unchanged to teraspace. correspondingly, they remain 1d, and become s-3 d, which no longer divides surface: roads still exist in 4d, but the city is not divided into blocks.

Likewise, in 2d, a snake (1d) is a flying thing (s-1)d = foil. This supprising fact is revealed in the book 'the planiverse'.

[ParadoxJuice]

As 3D beings, we see in two dimensions so when we look at a sphere, we see a circle but we know it's a sphere because of depth perception. A 2D being would know a circle was a circle by seeing it in 1D and using depth perception in the same way.

Elementary, my dear Watson. Well, maybe not, but this is still right. Usually, when people here talk about Flatlanders, it is assumed that they see just like us and they see one dimension at a time but use depth perception to understand things in 2D, just as we see in 2D and use depth perception to understand things in 3D. Therefore, it is assumed that a 4D being would see in 3D and use depth perception to understand things in 4D.

However, this does assume that 4D beings are like us. They could be tentacle monsters from Mars that use sonar to see, just like bats. All the answers seem to be systematic-we compare 2D to 3D, and use the differences in that to compare 3D to 4D.

But wait, I just noticed something. A nail isn't a circle, it has all those spirals on its side. So, a Flatlander wouldn't see a perfect circle, but one with a protrusion on it. So, we wouldn't see a sphere, but a sphere with a spherical protrusion on it. Strange...

[Keiji]

A nail isn't a circle, it has all those spirals on its side.

That's not a nail, it's a screw.