The Fourth Dimension

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December 24th : The Fourth Dimension

Quote of the Day

"To
The Inhabitants of SPACE IN GENERAL
And H. C. IN PARTICULAR
This Work is Dedicated
By a Humble Native of Flatland
In the Hope that
Even as he was Initiated into the Mysteries
Of THREE Dimensions
Having been previously conversant
With ONLY TWO
So the Citizens of that Celestial Region
May aspire yet higher and higher
To the Secrets of FOUR FIVE OR EVEN SIX Dimensions
Thereby contributing
To the Enlargement of THE IMAGINATION
And the possible Development
Of that most rare and excellent Gift of MODESTY
Among the Superior Races
Of SOLID HUMANITY"

-Dedication of Edwin A. Abbott's, Flatland: A romance of many dimensions.

Thoughts for the Day

Yesterday, we contemplated the unintuitive geometry of direction on a sphere and how this might apply to the 'beginning' of our universe. We pretended to move back and forth around the place where the direction, past, ends. We used this thought-experiment to think about how time started, one of the Big Questions of metaphysics. Today we will consider another such Big Question. 'What is a direction and how many directions can there be?' Edwin A. Abbott's Flatland is a story about a square from a two dimensional universe called Flatland. The square has the most extraordinary encounter with the inhabitants of Lineland, a one-dimensional universe, and Spaceland, a three-dimensional universe. What Abbott was trying to do with this story was educate his contemporaries about multiple dimensions and what they were.

It is worth reading Flatland. It is available on the web at http://www.geom.uiuc.edu/~banchoff/Flatland/.

From a reading of soft-science-fiction or fantasy, one might come to the conclusion that dimensions are places. Authors of these works often speak of travelling to another dimension. But dimensions are not places and you can not travel to them any more than you can travel to Up-Down. Up and Down are a pair of directions that are opposite. Travel Up for a while and then turn 180 degrees and you will be travelling down. This is true of all dimensions. Imagine you are floating in outer-space, holding your body straight as if you are standing up. Based on the attitude of your body in space you could define 6 directions all at 90 or 180 degrees to all of the other 5. These would be three pairs, Up and Down, Left and Right, Backward and Foreward. Each of these pairs of directions is a dimension and each is at 90 degrees to the other two.

In the flatland story, the inhabitants of one-dimensional Lineland only know forward-backward. They can imagine no other direction. The inhabitants are short line segments and they believe that the points at their ends as their outsides and their lengths as their insides. The King of Lineland says, "I know not what you mean by 'right' and 'left.' But I deny that you saw these things. For how could you see the Line, that is to say the inside, of any Man?" In traveling to Spaceland, the square is utterly confused for he can no more understand Up and Down than the King of Lineland can understand Right and Left. The King of Lineland has no words for Left and Right and the Square has no words for Up or Down. The implication is that we have no words for the fourth dimension of space or the two directions in it.

But the fourth dimension of space is a geometrical concept and, as such, the implications of a fourth dimension for a three dimensional creature will be almost exactly the same as the implications of a third dimension for a two dimensional creature.

A line-segment living in Lineland can't touch any part except the ends of his compatriots, and therefore, believes that his length is inside him and cannot imagine his inside being touched from the side. To the line-segment, the length of a line is inside the line inaccessible to any other object.

A square living in Flatland can't touch any part except the perimeters of his compatriots, and therefore, believes that his area is inside him and cannot imagine his inside being touched from above or below. To the square, the area of a shape is inside the shape inaccessible to any other object.

A human living in a three-dimensional universe, can't touch any part of another human other than their surface area (skin), and therefore, believes that his volume (guts) are inside him and cannot imagine his inside being touched from the fourth dimension. But from the fourth dimension every atom of our insides would be visible just as every point along a line's length or on a square's area is visible to us.

The sphere and the square are talking in Flatland:

"Sphere. Tell me, Mr. Mathematician; if a Point moves [forward], and leaves a luminous wake, what name would you give to the wake?

"I. A straight Line.

...

"Sphere. Now conceive the [forward] straight Line moving parallel to itself, [left] and [right], so that every point in it leaves behind it the wake of a straight Line. What name will you give to the Figure thereby formed? We will suppose that it moves through a distance equal to the original straight Line. - What name, I say?

"I. A Square.

...

"Sphere. Now stretch your imagination a little, and conceive a Square in Flatland, moving parallel to itself upward.

"I. What? Foreward?

"Sphere. No, not [foreward]; upward; out of Flatland altogether.

"If it moved [foreward], the [backward] points in the Square would have to move through the positions previously occupied by the [foreward] points. But that is not my meaning.

"I mean that every Point in you - for you are a Square and will serve the purpose of my illustration - every Point in you, that is to say in what you call your inside, is to pass upwards through Space in such a way that no Point shall pass through the position previously occupied by any other Point; but each Point shall describe a straight Line of its own. This is all in accordance with Analogy; surely it must be clear to you."

It is not clear to the square and so the square gets angry. The sphere was, of course, describing a cube. Try not to be like the square as I extend the sphere's logic to a hyper-cube, a four dimensional object which is like a cube but more so. Lets call the two directions of the fourth dimension fourthward and anti-fourthward.

Contemplation for the Day

Let us imagine

Do this contemplation with a few sheets of paper, a pencil and a light source such as a desk lamp. If you can have some solid cubes (perhaps a Rubic's Cube or some 6-sided dice that might help you also. Allow yourself to draw out your thoughts to the best of your ability, without judging your drawing ability or your imagination. Be gentle with yourself.

Stretch your imagination a little, and conceive a cube moving parallel to itself (that is to positions which are parallel to its current position) fourthward.

Imagine that every point in the cube, that is its entire inside, is to pass forthward through 4-space in such a way that no point shall pass through the position previously occupied by any other point but each point shall describe a straight line of its own. This is all in accordance with Analogy.

Is it clear to you. Of course not. Your mind having been naturally selected to picture things in three dimensions and being practiced in picturing things in three dimensions does not have the ability to picture things in four-dimensions.

So try a different analogy, imagine a cube floating above a piece of paper on a sunny day. What does it's shadow look like? Two squares with the vertices connected and the space in between filled in. See the shape below but with the different parts slightly different shades just for illustration.

ShadowOfACube.png

Going down one dimension, imagine a square floating above a wire on a sunny day. To actually drop a dimension altogether, the square, the wire and the sun must all be in same plane. As the illustration below shows two sides of the square can be taken to caste separate line segment shadows on the wire with the space between filled in.

ShadowOfASquare.png

Going up one dimension, imagine a hypercube floating fourthward from our universe. Further fourthward from the hypercube is a four-dimensional light source (a hyper-Sun). Illuminating a patch of our universe. In the first example, the sun illuminated the surface of the paper; you can imagine that if the paper were large enough (perhaps a few square light years, the sun would really only illuminate a circular patch of the paper. Likewise but plus one dimension, the hyper-Sun would illuminate a spherical patch of our universe; a glowing ball of space illuminated from nowhere that we could see. In the midst of that glowing ball of space would be a region that would not be illuminated; the shadow of the hyper-cube. It would be two cubic regions of space (not squares; cubes) near to each other with the edges of the cubes connected by flat planes and the space between the connections filled in with shadow.

You may not be able to picture the hyper-cube but you can still picture its consequences if you use analogy and imagination.

Gently contemplate this idea. Your mind is limited but by careful analysis you can describe things beyond your imagination.

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