A Right Turn Into The 4th Dimension
Not too long ago, if you ran afoul of the law, were arrested and deemed a flight risk, you were locked up in the pokey until your trial. Granted, even in the good ol’ days money talked, and your lawyer could probably persuade a judge lenient or dimwitted enough to place you under house arrest. Today, though, when the courts let freedom ring, house arrest means wearing judicial bling – an ankle bracelet – to keep you within police radar range while you hobnob around the neighborhood, visit old haunts and even older friends, and continue to engage in the same illicit behavior that got you arrested in the first place.
But what if house arrest meant you were truly unable to leave the friendly confines of your quaint little crib? Imagine every front, side and back door that once opened to the outside world now only leads you to some other room within your own home. And every window that once held vistas of the Manhattan skyline or the Bronx County courthouse now only lets you peek into some other room of your own home.
Well, all this and more could be yours, penal contestants, if your dream house were suddenly transported from the 3rd dimension into the 4th dimension.
Now, those of you who finished the third grade and are conversant in Einstein’s General Theory of Relativity are no doubt saying, “What the hell are you talking about, you idiot? Time is the 4th dimension! How do you move a house into time?” To which I say, Hold on there, Baba Looie. Let’s think of the 4th dimension as the next logical, geometric construct from the 3rd dimension.
For argument’s sake – and I’m writing this, so it’s my argument – let’s define the first three dimensions geometrically by saying that each dimension exists at a 90˚or right angle to the other. Length is the 1st dimension and width is the 2nd dimension. Width exists at a 90˚ or right angle to length; in other words, if length runs east to west (or west to east for those of you in Los Angeles), then width runs north to south. The 3rd dimension is set at a 90˚ or right angle to both length and width – this is height. As an example, consider a flagpole standing at the corner where Broadway and 96th Street intersect; the neon lights are not as bright at this end of Broadway, so the flagpole should stand out. Broadway represents length, 96th Street represents width, and the flagpole represents height, as well as one more thing to walk into if you’re not paying attention. Where length, width and height all intersect at the same point, we have the three distinct dimensions that define our physical world.
Following this logic, then, the 4th dimension would have to be set at a 90˚ or right angle to all of these three dimensions – length, width and height – simultaneously. Huh?
Let’s go back to the first two dimensions for a moment, shall we? Length and width define a plane, which is a flat surface like, say, a sheet of paper (or, perhaps, the top of one’s head). On this sheet of paper we shall draw a three-dimensional object, such as this cube.
Now, a cube is made up of six faces or squares, and a square, of course, has four equal sides. In this two-dimensional representation, however, we actually only see three sides – the front, the top and the right; we cannot see the side on which the cube sits, nor do we see its left side or its, ahem, back side.
In order to give the above cube the illusion of depth, three lines forming part of the top and right faces of the cube are shortened and set at acute angles to the front face of the cube. Thus, the top and right faces of the cube are not really squares (Got that, daddy-o?), they are trapezoids, i.e., only two of the four sides are parallel. What you are seeing is a two-dimensional representation of a three-dimensional cube; your brain fleshes out the parts unseen. Thus, whenever you see this pancaked version of a cube, you are conditioned to accept it as a three-dimensional object. N’est-ce pas?
Your New Home, Minus The Ceiling
Let’s now imagine how a home would be constructed in the 4th dimension. For years builders have constructed typical (typical?) three-dimensional homes by referring to plans drawn on a two-dimensional plane: a blueprint. To imagine, then, how a fourth-dimensional house would be represented in the 3rd dimension, let’s look from our three-dimensional perspective at a house built in a two-dimensional world.
With grateful acknowledgment to Edwin A. Abbott (1838-1926), let us take a look at a 6-room house in the town of Flatland somewhere in upstate New York, where everything, including the town’s residents, exists in only two dimensions. The house would look something like this:
Clearly, the owner is colorblind or the hardware store had a closeout on paint. In any event, the house is laid out like a ranch house with every room on one level. The rooms are numbered 1 – 6. Each room has four walls, and every wall has a huge sliding glass door (Hey, the owner can do whatever he wants!). Each shared wall leads into an adjacent room; walls that are not shared lead outside the house. Thus, room #1 shares one wall, its south wall, with room #2; the west, north and east walls all lead outside the house. Room #2 is an interior room, sharing all four of its walls with the four adjacent rooms – the north wall is shared with room #1, the south wall is shared with room #3, the west wall is shared with room #5, and the east wall is shared with room #6. Room #3 shares two walls, its north wall with room #2 and its south wall with room #4. Room #4 shares only one wall, its north wall, with room #3. Room #5 shares only its east wall with room #2, and room #6 shares only its west wall with room #2. Everybody got that?
You can enter this house through any room that has a wall facing the outside except room #2, which is in the interior of the house. Rooms 1, 4, 5 and 6 have three walls with access into the house; room #3 has two such walls, the west and east walls. Thank goodness the house comes standard with indoor/outdoor carpeting.
Once inside the house, access to each room is somewhat limited. If you are in room #1, for example, the only way to get to rooms 5 or 6 is to pass through room #2; the same is true if you want to get to room #3. To get to room #4, you have to walk through room #2 and room #3, which at 3:00 AM is not likely to win you any brownie points from anyone who might be asleep there.
Well, We’re Movin’ On Up…
Now let’s “fold” this house into three-dimensional space. We do this by folding along each shared wall, just as you would fold a flat piece of paper with six connected squares into a cube. For those whose opposable thumbs leave them all thumbs, this house is in the shape of a cross, which makes this task rather easy.
First, fold room #4 up – i.e., into three-dimensional space – along its shared wall with room #3. Then fold all four sides of room #2 – i.e., along the walls it shares with rooms 1, 5, 6 and 3 – up into three-dimensional space. Finally, connect the south wall of room #4 with the north wall of room #1 and, voila, we have a cube–er, three-dimensional house.
Now, one way to represent our now three-dimensional house in two-dimensional space is to draw it as a cube, as we did above. If we wish to see all the rooms, though, a combination of trapezoids and rectangles is needed to give the impression that we are looking into a three-dimensional cube.
The figure on the left is a view of our house looking through room #1 back to room #3, the smaller rectangle; room #2 is the base of the cube; rooms 5 and 6 are the sides; and room #4 is the top.
The figure on the right is the house with the sides stretched to make the relationship of each room clearer, as well as more bizarre. In this figure, rooms 1 and 3 are highlighted, with room #1 in the front and room #3 in the back. Since every side of every face of the cube is actually a wall, every wall then is connected to a wall of another room. What this means is that no wall now leads outside the house. No matter what room you are in, regardless of which wall you punch, walking through its sliding glass door will always lead you into another room.
Stairway To Heaven?
Now let’s put our original two-dimensional owner-occupant in room #1. If he (yes, only a man would let someone fold his two-dimensional house into three-dimensional space) walks through the sliding glass door on the north wall, he now enters room #4. When the house existed in its original two-dimensional state – and the owner was somewhat shy about waking his crazed, knife-wielding cousin snoring away in room #3 – he would have decided to exit the house through the sliding glass door on the north wall, and trudge through the mud all the way to the other end of the house until he finally reached room #4. This could be very disconcerting, especially after a late-night burrito and mocha latte snack, as room #4 had the only bathroom.
When our Flatlander looks through a sliding glass door now, regardless of which wall he chooses, he always sees into the room adjacent to that wall. Remember, in the 2nd dimension there is no concept of up or down because those directions only exist in the 3rd dimension. In the 2nd dimension he reached every room of his house by simply walking – or perhaps gliding – straight ahead, or turning left or right. Now in three-dimensional space, however, every wall is connected to another room, and that other room may well be on another level – the second floor or the basement. But as far as our owner-occupant knows, he is still walking on one level as he had always done, albeit now confused as hell.
With his once two-dimensional house now folded into three-dimensional space, our owner-occupant is unable leave the house, as each wall is now connected to another wall, and there is no wall anywhere leading outside the house. His only escape from his house would be to have it “unfolded” in a lower dimension – in this case, back into two-dimensional space.
Now imagine a three-dimensional house folded into fourth-dimensional space. We here in the 3rd dimension can no more point toward a direction that is at a right angle to the 1st, 2nd and 3rd dimensions than a two-dimensional Flatlander could point to the 3rd dimension, but in theory a dimension outside our world does exist. From our lofty three-dimensional perch we can look “down” and peer into the two-dimensional world of Flatland, just as someone – or some thing – from the 4th dimension can gaze down into our three-dimensional world.
If your gorgeous Park Avenue penthouse were suddenly folded into fourth dimensional space with you inside it, you would find yourself trapped forever within your apartment. Every wall, floor and ceiling would be connected to another wall or floor or ceiling. And if you think of each wall, floor and ceiling as simply another surface on a cube – i.e., the room in which you are sitting and sulking – then you may find that, unless your apartment was folded into the fourth dimension with care, you could exit the sliding glass door on the west wall of your bedroom and find yourself standing on the ceiling of your living room.
Needless to say, 24 hours in this funhouse might well punish you more cruelly and unusually than anything the Supreme Court could have imagined.