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.
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.
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.
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.
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.
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.
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.
When the Huygens probe flashed us those images of lakes on Saturn’s moon Titan, people chattered that maybe there’s life in those hydrocarbon “waters.”
Huygens descending on Titan (image courtesy NASA/JPL-Caltech)
If there is life there, we might not even recognize it as such. Not just because of the frigid temperatures and other reasons laid out in en.wikipedia.org/wiki/Life_on_Titan, but for a couple of reasons having to do with the physical chemistry of solvents.
Water is a polar solvent, good at dissolving salts and other substances in which centers of positive and negative charge are in different parts of the molecule. Conversely, water molecules interact so strongly with each other that interspersed hydrocarbons and other non-polar molecules are forced away and out of solution. Hence the existence of oil slicks … and cell membranes.
Every kind of life on Earth, or at least everything that a biologist would be willing to call life, is composed of units whose integrity depends upon hydrocarbon moieties (molecules that contain significant amounts of hydrocarbon structure) being forced together in escape from a polar environment.
For bacteria and multi-cellular life forms, the boundary between interior and exterior is the cell membrane (see diagram), two layers of two-tailed molecules laid tail-to-tail with their non polar (black) hydrocarbon chains sandwiched between negative (red) polar groups that face out of and into the polar (red and blue) cell. Furthermore, our cellular life also depends upon a whole collection of two-layer membranes that isolate different metabolic functions within the cell — respiration over here, protein construction over there, and so forth.
By some definitions we have smaller kinds of life, too: viruses and phages. Many viruses (e.g., herpes) have a non-polar fatty coating. Others make do with proteins to hide their DNA. However, biochemistry tells us that these structural proteins are almost certainly held together in large part by patches of non-polar regions with precisely matching shapes.
However, Titan’s surface is dominated by a hydrocarbon solvent, a liquid mix of methane and ethane, that behaves very differently from water. Critically, hydrocarbon solvents do not dissolve water and other polar materials. The amino acids from which we build our proteins, the nucleic acids from which we build our RNA and DNA, even the carbohydrate groups from which plants build sugars and cellulose — all are essentially insoluble in hydrocarbons.
If lightning or some other process were to generate some nucleic acids in a Titan lake (as in the Miller-Urey experiment, see en.wikipedia.org/wiki/Miller_experiment), those molecules would immediately aggregate and fall as a sludgy solid onto the lake floor. There’d be no opportunity for those small molecules to interact with each other, much less find some amino acids to tie together to produce a protein. Life as we know it could not begin.
Well, how about a non-polar kind of life? The properties of hydrocarbon solvents permit two possibilities, both of which are very strange from an Earth-life perspective.
The easier one to visualize turns that double-layered cell membrane inside out. A Titanic cell membrane could be a sandwich with a layer of polar stuff between two non-polar layers. Given that structure, the cell’s internal non-polar metabolic processes could operate in isolation from the non-polar outside, just as our cell membranes isolate our watery internal cellular metabolism from our watery outside. A reversed cell membrane on a Titanic cell would wrap around some very interesting biochemistry.
But things could be even stranger. All hydrocarbons can intermix in all proportions with all hydrocarbons. That’s why petroleum crude is such a complex mix, and why different crudes break out differently at the refinery. Any non-polar molecule can slide in between any other hydrocarbon molecules with very little effort. On Titan, then, non-polar materials can diffuse freely throughout those ethane seas.
Moreover, liquid hydrocarbons have very low surface tension compared to water. At the surface of a pool or droplet of water, those H2O molecules cling to each other so tightly that another object must exert force to get between them and into the bulk liquid. The threshold force, measured by the surface tension, is so high for water that pond skaters and similar bugs can live their lives on a pond rather than in it. In contrast, surface tension for a liquid hydrocarbon is only one-third that of water.
What’s important here is that surface tension is the force that works to minimize the surface to-volume ratio of a blob of liquid. The form with the smallest possible ratio is a sphere. Sure enough, small droplets of water are spherical. Hydrocarbon fluids, with their much lower surface tension, tend to accept looser forms. Water on a tarry surface beads up; oil on a wet surface spreads out. Scientists think that water’s powerful sphere forming propensity was crucial in creating proto-cells during Earth-life’s early stages.
Suppose that Titan’s hydrocarbon life doesn’t depend on nearly-spherical cells. Maybe Titan life has no cell boundary as we know it. Titan’s “biology” could be one titanic (in both senses) “cell” with different metabolic processes isolated by geography rather than by membranes the way Earth life does it.
Maybe the lakes closest to Titan’s equator generate long-chain hydrocarbons. Maybe another set of lakes links those molecules to form complex benzenes and graphenes that catalyze reactions in still other lakes.
Maybe Titan’s rivers and streams carry information the way our nerve cells do.
Hi, there. Ever seen a Kline bottle before? Well here’s one…
It’s a sample of the attitude you’ll find in the items I plan to post here. You’ll find articles on everything from Life in Titan to Infinite Monkey Business and beyond. Come on back in a week or so, after I’ve learned more about WordPress.
Hard Science Ain't Hard 