A direct link to the above video is at http://www.youtube.com/watch?v=oDNO6vv1SjE
Another eight-dimensional shape that we've looked at with this project is Garrett Lisi's E8 Rotation. Lisi created a huge stir in the physics world with a new proposal for what he calls An Exceptionally Simple Theory of Everything that uses "E8" - a complex, eight-dimensional pattern with 248 points.
As visually appealing as this shape (pictured below) might be, its implications are startling: Lisi has demonstrated that there is a way to place the various forces and elementary particles (including their possible quantum spin values) on E8's 248 points. Rotating Lisi's model in various ways reveals the explanation for a variety of interactions, some of which (like the clustering of quarks into families of three) are natural outcomes from this structure. Some points in his model are currently occupied by particles which have been theorized but which have yet to be seen, and there is hope that the Large Hadron Collider in Switzerland may some day reveal some of those particles.
Whether Garrett Lisi's theory proves the existence of higher dimensions or not is open to interpretation - Lisi himself says this geometric pattern, although it is based upon an 8 dimensional construct, could be fully realized within our 4D space-time without requiring additional dimensions. On the other hand, in a New Scientist magazine article, string theorist Sabine Hossenfelder (of the Perimeter Institute for Theoretical Physics) points out that this could be complimentary to string theory, which she says also uses E8 to describe the Calabi-Yau manifold, the extra-dimensional shape that string theorists says our universe is derived from.
Lisi also acknowledges that this is not a finished theory but a work in progress, and that there are still some deficiencies in his model that need further refinement. For me, this dimensional connection is fascinating because with this project I've insisted that you can't have any physical expressions of matter in anything beyond the 8th dimension.
Why Do We Need the 8th Dimension?
A commonly asked question about this project is, "if our universe and every other universe can be thought of as "points", or perhaps "positions" within a multiverse landscape, and in this way of visualizing the dimensions we get to that landscape by the time we're at the seventh dimension, then why do we need to think about any more dimensions beyond that?
To my way of thinking, you can express the same concern about every definition of every single dimension, and the arguments have to keep coming back to "what are we missing within this current dimension?".
For instance, people say that the third spatial dimension is really all you need, because no matter what other universe you imagine it should have a third dimensional expression.
Or there are people who say all we need is space-time, because by the time you have the third dimension plus time, which gives the third dimension a way to change from state to state, then you can imagine every single possible universe strung end to end within infinite space-time, and because that string of possible states is infinite you will eventually get to every possible universe.
Or you can have people who say space-time plus its probabilistic outcomes (which Everett's Many Worlds Interpretation says are "orthogonal" or "at right angles" to space-time, leading to my conclusion that these branches are in the fifth dimension) gives you everything you need to imagine every possible universe.
From there, I've proposed that the branches for our universe which no amount of chance or choice will allow us to select (like the branches where dinosaurs never became extinct or the ones where I died in a car accident last year) are in the sixth dimension, and this uses the same logical reasoning that defines any spatial dimension - there always needs to be a "new degree of freedom" added for us to be able to call what we're talking about a new dimension, and the new dimension allows us to get to something that was unavailable from the previous dimensions.
Likewise, I've proposed that those other universes with different basic physical laws are located in other positions within the seventh dimension and above, and this is why our universe, constrained at its position within that multiverse landscape never "wanders off" into one of those other universes.
(The rotating shapes below are known as the Platonic Solids. Pictured here are the tetrahedron, the cube or hexahedron, the octahedron, the dodecahedron, and the icosahedron. We'll talk about these shapes later on in this entry.)
So, why isn't the seventh dimension as far as we need to go? We have to continue with the logic we've used from the outset. Imagine a point of infinite size in the sixth dimension, and it's much the same as imagining the indeterminate point we started from with our thought experiment - the point encompasses the whole dimension. If we can think of a different point that is not subsumed by the first point, then the point-line-plane postulate tells us we've found a way to get to the "next dimension up": those two points define a line in the seventh dimension.
What if the second point we just imagined represents a universe where the strength of gravity is different from ours? Then the line that passes through those two points is like a one-dimensional line, and that line extends through all possible values for gravity. Different positions on that line would include "impossible" universes which could not have come into existence because the strength of gravity was above or below a certain threshold needed to allow a physical universe to express itself, and such imaginary universes might be at many places along the line, interspersed with universes that were able to cohere into more organized states.
What about a universe with a different value for the speed of light (or whatever physical constant you care to imagine)? That's not on the line we just drew. We could erase the other point we drew, then place a new point for a universe with a different speed of light, and the new line that passes through those two points would be all possible realities resulting from different values for the speed of light.
But what if I want to consider both lines simultaneously? It can't be done without entering the next dimension up, the eighth dimension. So with the logic of the point-line-plane postulate we have a way of thinking of the eighth dimension as being like a plane, and our universe can be viewed as a point in the eighth dimension with those two lines or any other lines representing other different-initial-conditions universe passing through our point within that plane.
Remember this: with the point-line-plane postulate, the point you start from is in dimension x, the line to a different point is in dimension x+1, and the plane defined by a third point not on that line is in dimension x+2. Having worked through three dimensions, nothing moves or changes about the first point, we are just adding dimensions that allow us to view that starting point from different perspectives. Thinking of our universe as a "largest possible point" encompassing the phase space of the sixth dimension, then, doesn't mean that we can't view that unchanging point from the widening perspective of the seventh or eighth dimension. Quite the contrary: this logic is the accepted approach to visualizing spatial dimensions. Likewise, the idealized shapes we've started each of these entries with (like the octeract at the start of this entry) have an underlying symmetry, where each of the points are equidistant to their adjacent ones in the same way that the adjacent points of a 3D cube are equidistant when viewed within the third dimension. This means that the 8D octeract, like any of the other hypercube shapes, can be placed within a hypersphere with the same number of dimensions, and the outer points of the hypercube are pushed apart from the center of the hypersphere symmetrically: the point of indeterminate size can become a sphere occupying as little or as much of the dimension it's placed within. At its largest possible expression, this point becomes a finite but unbounded hypersphere occupying the entire phase space of the dimension it's within. We can also look at the other Platonic Solids we've been looking at here with this same idea of symmetric points pushed out from a central position in mind, and this includes shapes like the nested tetrahedra we see below left, which can be used to create the dodecahedron at the right.
Speaking of phase space, here's a link to a recent New Scientist article about a new paper published by Lee Smolin and others at arxiv.org, proposing that all possible universes could be contained within an eight-dimensional phase space. To follow the logic we've been pursuing here, the six-dimensional phase space for our own universe (or any specific universe with locked-in physical laws) would be a subset of an eight-dimensional phase space that encompasses all possible universes. Likewise, it's worth noting that Everett didn't limit his Theory of the Universal Wavefunction to just our unique universe: so while the phase space of a unique different-initial-conditions universe such as ours could be encompassed by the sixth dimension, the logic we've followed here shows us that Everett's theory ultimately should be considered from the phase space of the eighth dimension to include the wave function of all those other possible universes.
So what's beyond the eighth dimension? The simplest way to take our thought experiment to the next dimension up now is to ask this: how would you jump from the universe with a different value for gravity to the universe with the different speed of light without passing through the intermediate possibilities? To get to that addition degree of freedom, you need the next dimension up. For our 2D flatlander ant, that was the third dimension. For our fly in the fifth-dimensional garden hose analogy, that freedom to flit from location to location was in the sixth dimension. And for all these different universes considered from the eighth dimension, where do you think we'd achieve this additional degree of freedom?
Next: Imagining the Ninth Dimension
Imagining the Seventh Dimension
Imagining the Sixth Dimension
Imagining the Fifth Dimension
Imagining the Fourth Dimension
Imagining the Third Dimension
Imagining the Second Dimension