Multi-dimensional Fun

Jeff Carreira Blog Posts 5 Comments

Let’s continue our thought about dimensionality in the upbeat spirit of the holidays by visiting one of the great books about multiple dimensions, Edwin Abbott’s 1884 novella “Flatland, A Romance of Many Dimensions.” If you haven’t read it, it is well worth a small investment of time. In the book Abbott describes three different lands: lineland, flatland, and spaceland.

Lineland is a world entirely contained by a single line. The beings of lineland are line segments and the perception of each of these beings is limited to the line which limits their reality. A being in lineland only “sees” the end point of the being in front of them in the line.

Flatland is a land bounded on a flat plane. The beings on flatland can be circles or squares or triangles, but because their perception is limited to the plane upon which they live, they can only see the shape of each other being edge-wise as a line.

Spaceland is a land that contains three-dimensional space. The beings are now spheres, and cones and cubes, but they only see each other in two dimensions. Remember, if were not for the fact that we have two eyes and can compare the view from each, we would only see other human beings in two dimensions as well.

Lineland is a domain bounded by a single dimension, length, flatland by two dimensions of length and width, and spaceland by the three dimensions of length, width and height. In Abbott’s brilliant book he helps us get familiar with the idea of multiple dimensions by exploring not only how life looks from within each dimension, but how life looks from a being of a high dimension. For instance, the linelanders can only see themselves as points, but the flatlanders can see them as the line segments that they are. The flatlanders can only see themselves as lines, but the spacelanders can see them as the circles, squares and triangles that they are.

Abbot goes on to explore what happens when a being from a lower dimension perceives a being from a higher dimension if they happen to intersect their world. A linelander for instance cannot see a flatlander unless the flatlander intersects the line of their world, and even then all the linelander can see is a point. Likewise the flatlander can’t see a spacelander unless the spacelander intersects the plane of their world, and then all the flatlander can see of the spacelander as a line.

To take this a little further, a spherically shaped spacelander could hover above the plane of flatland and watch all of the circles, squares and triangles living their lives from a perspective they could never gain because the spacelander exists in another dimension. If the spacelander should decide to cross through the plane at any giving moment that plane would cut a circle out it’s spherical body. That circle is all of the spacelander that would exist in flatland and all a flatlander would see would be the line of the circle edgewise. To the flatlander it would simply look like another flatlander had appeared magically out of no where.

The illustrations in the book make this easier to imagine, but you are probably getting the idea that the beings of each dimension have perception that is limited to the dimension of their world.  Beings of higher dimensions can clearly see all of the beings of lower dimensions and can intersect those lower dimensions at different spots. The beings of the lower dimensions can only be aware of that portion of the higher being that intersects their lower dimension. If I haven’t confused you completely, you might be getting a sense of it. This multi-dimensional description sounds a great deal like the kind of holonic configuration that Integral Theory describes, which is a great deal like the vision of reality that the American Pragmatists were outlining. A holon is a whole that is also a part of a larger whole. Similarly a lower dimension is complete in itself, while at the same time being encompassed completely by higher dimensions.

How does all this relate to the unfolding of the universe and the problems of mind and matter?

Let us think of the universe metaphorically as a collection of wooden planks and for a long time all we do is lay them out flat side by side. We could expand our universe indefinitely in this way without ever leaving the two dimensions of the flat plane. Let us say that one day after laying planks out for a long time we started to lay some on their side and stake them to create a wall. Now we have just added another dimension that we can grow in. So we can now grow our original plane indefinitely and we can build our wall in our new plane indefinitely. Now we start to build more and more walls. Maybe we build four walls and put a roof on them and create a room.

But when we do that something interesting happens, something extra. Now not only do we have the walls, floors and roofs in our universe, but we now have an object, a room that has an inside and an outside. How did we build this inside and outside? What is the inside made of? Is the inside made of wood like the rest of our universe? Do we now have a dualistic universe that is made up of wood and some mysterious new metaphysical substance called inside? Is this metaphor in analogous to our dilemma of mind and matter? Is it possible that something extra happens in the universe when mind appears so that mind is dependent on matter like the inside is dependent on the wood, but isn’t really built out of it, like the inside of the room isn’t made out of wood?

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Brian
Brian
10 years ago

On Earth as it is in Flatland: “a Woman is a needle; being, so to speak, all point, at least at the two extremities. Add to this the power of making herself practically invisible at will, and you will perceive that a Female, in Flatland, is a creature by no means to be trifled with.”

liesbeth
liesbeth
10 years ago

As far as I understand it now mind is dependend on matter (brain). De Quincey describes Consciousness as Knowing. For knowing A Thing (the inside of the house) mind is needed which is connected with the evolving brain. BEFORE that there is the POTENTIAL to know, subject knows no-thing, which is Intelligence, which is there already in the smalles particle there is, the quantum. This intelligence was explained long time ago by Erwin Laszlo as deriving from previous Universes, which seems very logical. In your example is is both inside/outside. De Quincey brings in Whitehead to explain how Consciousness is… Read more »

Catherine
Catherine
10 years ago

Dear Jeff, all those consideration about dimensionality invariably evoke to me the idea of a “fractal land”. You see, somehow it is not enough to increase dimensionality to acquire depth or perspective. It is one of the things that I reproach to the current Strong Theory; to say live in a universe of 11 dimensions looks like a flat statement. Then after 11 dimensions, will we come to 12,13,14… and what next. We are back to the flatland of creativity. For me to topology of spiritual evolution is a fractal. Meaning it is a space made of universe imbricated one… Read more »

Catherine
Catherine
10 years ago

sorry for the typos!

Chuck R
Chuck R
10 years ago

Did anyone ever read “Sphereland” by Dionys Burger (1965), a sequal to Flatland? Interesting, but not as wonderful and funny like Flatland. Do you physicists have any opinions on action-at-a-distance & Bell’s Theorem? My take is that when two particles seem to be communicating with each other at faster-than-light speeds, it’s because “they” are actually one particle-entity which intersects with our space-time universe in two loci. We can’t see the “connecting” portion because it’s outside our space-time universe. Somewhere in the extra dimensions which String Theory says are all curled up. If a spaceland creature were to push a “V”… Read more »