Dimensional Venturing Part 4 – To infini-D and beyond!

apple plumNow that you’ve read my previous posts and have the 4-D thing working well, you’re ready to go for a few more dimensions.  Consider the apple that struck Isaac Newton’s head.  The event occurred in 1665, in England at 52°55´N by 0°38´E, roughly three feet above ground level.  The apple, variety “Flower of Kent,” weighed about 8 ounces and was probably somewhat past fully ripened.  Got that picture in your head?  You’re doing great.

Now visualize the apple taking thirty seconds to move twenty feet diagonally upward, northward and eastward as it morphs to an underripe 4-ounce Damson plum.

The change you just imagined followed an eight-dimensional path: three dimensions of space, one of time, one of weight, one for degree of ripeness, and two category dimensions, species and variety.

Length in a given direction is only one kind of dimension, as Sir Isaac’s example demonstrates.  A mathematician would say that a dimension is a set of values that can be traversed independently of any other set of values. A dimension can be confined to a limited range (360 degrees in a circle) or be infinite like … well, “infinitely far away.”  A dimension might be continuous (think how loudness can vary smoothly from sleeping-baby hush to stadium ROAR and beyond) or be in discrete steps like the click-stops on a digital controller.  The physicists are arguing now whether, at the smallest of scales, space itself is continuous or discrete.

colors_post
Photo by Becky Ziemer

Color vision’s a good example of dimensions in action.  For most of us, our eyes have three types of cone cells, respectively optimized for red, green and blue light.  We see a specific color as some mixture of the three and that’s how the screen you’re looking at now can fake 16 million colors using just three kinds of color-emitting elements (phosphor dots in old-style TVs, LEDs in most devices these days).

Where did that 16 million number come from?  The signal-processing math is seriously techie, but at the bottom the technology uses 256 intensity levels of red, 256 levels of green and 256 levels of blue — each is a discrete dimension with a limited range.  Together they define a 256x256x256-point cube.  Any point in that cube represents a unique mix of primary colors.  One of the colors in the little girl’s hat, for instance, is at the intersection of 249/256 red, 71/256 green, and 48/256 blue.  The arithmetic tells us there are 16,777,216 points (possible mixed colors) in that cube.

Well, actually, there’s one more dimension to color vision because our eyes also have rod cells that simply sense light or darkness.  Neither brown nor grey are in the spectrum that cones care about.  A good printer uses four separate inks to produce browns and greys as mixtures of three dimensions of red-green-blue plus one of black.

So color is 3-dimensional, mostly.  But that’s just the start of color vision because most of us have millions of cone cells in each eye.  A mathematician would say that any scene you look at has that number of dimensions, because the intensity registered by one cone can vary in its range independently of all the other cones.

Ain’t it wonderful that you’re perfectly OK with living in a multi-million-dimensional world?

Next week – a word from the other side

~~ Rich Olcott

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Dimensional venturing, Part 2 – Twirling in 4-space

Last week we introduced the tesseract, which is to a cube what a cube is to a square — an extension into one more dimension.  That’s why it’s also called a hypercube.  The first tesseract diagrams I ever saw were so confusing — they looked like lots of overlapped squares tied together with lines that didn’t make much sense.  I wondered, “Wouldn’t it be easier to understand a tesseract if I could see it rotating?”

Years later computers and I had both moved ahead to where I could generate the pictures you see in this post.  What I learned while doing that was that 4-D figures have two equators.  In four dimensions, it’s possible for something to rotate in two perpendicular directions at the same time.  Read on and please don’t mind my doggerel — it doesn’t bite.

line2c The LINE is just a single stroke,
a path from here to there.
Stretch it out beside itself
and you will have a SQUARE.
Where’s its face when it turns around?
Gone, ’cause its back’s not there.
square2c
cube2c The CUBE’s a square
made thick, you see.
Length, breadth and depth
comprise a full 3-D.
Add yet a thickness more,
crosswise all to X, Y, Z.
A TESSERACT on a corner spins
but an XY-slice is all we see.
tess2cxy
tess2czw But the axis, too, can rotate through
a path that’s drawn invisibly.
Four faces grow and shrink in place —
it’s hard to do that physically.
This tesseract is tumbling ’bout
two equators perpendicular.
Were I in such a state, I vow,
I’d be giddy, even sickular.
tess4cxyzw

In the 4-D views, when one of the tesseract’s cubical faces appears to disappear into an adjacent face, what’s actually happening is that the face is sliding past the other face along that fourth dimension (which I called W because why not?)

You’re looking at a two-dimensional picture of the three-dimensional projection of a four-dimensional object as it moves in 5-space (X, Y, Z, W, and time — if it didn’t move in time then it couldn’t be spinning).

Next week — Herr Klein’s bottle, or rather flask, or rather surface.

~~ Rich Olcott

Dimensional venturing, Part 1 – What’s 4-D?

Whenever a science reporter uses the phrase “string theory,” it’s invariably accompanied by a sentence about tiny strings vibrating in 10 or 11 dimensions. Huh? How can you have more than three? And what does it really mean to say that that comix villain comes from the 4th dimension?  Actually, we live in many dimensions, though it’s not easy to visualize them all at once. Let’s get some practice.

Right now, you’re reading along a line, a one-dimensional path from left to right. Imagine a point drawing a straight line about a foot in front of you. Let that line just hang out there in the air, glowing a gentle green color, with one “edge” (the line itself) and two “corners” (its ends).

As you read down the page, you traverse a series of lines laid out next to each other in the two-dimensional plane of the page. Imagine your green line moving upward, leaving a plane of yellow sparkles behind it. Stop when you’ve got a sparkly yellow square in front of you showing its one face, four edges (one green, three yellow) and four corners (two green, two yellow). Let’s put some red paint on one of those yellow edges.Cube

Stack up enough printed pages and you’re got a 3-dimensional book. Imagine that nice yellow square moving away from you until you’ve got a friendly cube hanging out in the air. Our original line, the green edge, has produced a green face going into the distance. The red edge has built a pink face. All together, the cube has 8 corners, 12 edges and 6 faces. OK, now make your cube disappear.

But we’re not done yet. Time is a dimension. Consider that cube. Before you dreamed it up – nothing. Then suddenly a cube. Then nothing again. During the interval the cube was floating in front of you, the green line was tracing out a green face in time. The pink face was drawing a pink cube. The whole cube, from when it started to exist until it went away, traced out a four-dimensional figure called a tesseract, also called a 4-cube or hypercube. The tesseract was bounded by a cube at the beginning, six cubes while it existed (one from each face of the initial cube), and a cube at the end of its time, for a total of eight.

Just for grins, count up the faces, edges and corners for yourself.

But wait, there’s more. The tesseract doesn’t just sit there, it can spin. Being four-dimensional, it can spin in a surprising way. We’ll get to that next week.

~~ Rich Olcott