Sometimes I like to spend an hour looking at something I barely understand. The inside of this guy’s mind has got to be so interesting, but it’s been shaped by geometry rather than words, so it’s very hard for him to express it. The geometry shaping it is also quite less limited than the square space we hit baseballs in, so it’s hard to draw as well.
I can offer some help on grokking what he’s saying, but there’s simply no way to absorb this stuff quickly. That said, I wouldn’t mind being able to imagine the platonic forms inside Bill Thurston’s head.
GLOSSARY-LIKE DISCUSSION
Topology. You want to understand why identifying the left and right side of a wide rectangle (declaring left = right, so that when you leave the left side of the Mario screen you appear again on the right) is the same as cutting a long strip of paper and taping the two ends together.
(There’s a slight variation on that game that results in a famously weird space—the one-sided, single-edged Möbius strip).
Quotient spaces. You want to understand what it means to quotient a space. I can give a few examples. ℝ/ℤ would be the unit real interval [0,1) — kind of a microcosm of the real numbers themselves. The Western chromatic musical scale quotients by 13 notes to make the octaves. It’s not that A4 is “equal” to A8, but it shares the same structural relationship as do E4 and E8.
An orbifold is a manifold that’s been quotiented. Like if you took the plane and made an equivalence class of the vertical [0,1)’s with all the [1,2)’s and [2,3)’s and etc., you would be looking at an infinitely wide strip with all the verticality “wrapped up” in [0,1) — not gone, just wrapped up into one microcosm. You could also think about Groundhog Day (the analogy doesn’t work precisely). He’s living through the same span of time over and over because it’s been quotiented along the time dimension (the result of the division is a length of one day)
Oh … equivalence classes are another thing you have to know about. I haven’t written about them yet. WVO Quine came up with a sensible definition of “what is 2” using the concept. And as Terence Tao wrote, when one uses a “noise-tolerant” definition — like if a lot of different ways of saying something can be taken to mean the same thing — that’s another example of an equivalence class.
Back to the music theory for a second—there are multiple ways you could set up equivalence classes.
octaves — it’s not as if all “G♯” notes sound the same — but when we talk about octaves it’s usually with reference to the same-sounding-ness of twice-as-fast frequencies
across instruments — the overtone series of a tuba playing C3 differs from the overtone series of a double-bass playing C3. Nor does blowing on your double-bass or plucking your tuba produce the note. But we equivalence-class these differences away and write their parts in mostly the same notation. (Not exactly the same since you never see the word pizzicato on a tuba score, etc.)
inversion — I can do a major A triad as AEG, GAE, EGA … it’s a combinatorics thing; 3! ways. No, they don’t all sound the same, but when I use the word “triad” I am equivalence-classing over the kind of sameness that they do have.
enharmonics — Sure, D♯♯ and F♭ sound the same — but conceptually they’re very different, and the notes around D♯♯ will be different than the notes around F♭.
slight errors — players of the cello or the voice know that pitch is a continuous variable—however we might reasonably call 398 Hz = 400 Hz = A4.
transposition — Certainly composers choose the key of D (or, if they’re Stephen Sondheim, F♯) for a reason — but if a song isn’t within your vocal range you can always subtract or add a certain fixed pitch (in notes-space, not in Hz-space!) from every note and the piece will sound “the same” — not exactly the same, but it will recognisably be a pub song — I mean, the US national anthem
If I say “Hand me that glass”, I don’t mean to reference the glass at a particular orientation, rotation, or place in the room—I mean to equivalence-class ∀ such configurations of the glass—they all mean “that glass”. And if I say “Hand me a glass” — “Which glass? This glass?” — “Any glass!” then I’m equivalence-classing ∀ glasses within a certain distance from you.
Hyperbolic geometry. In square space, four right angles ∟ add up to the whole shebang 360°. But in the logical abstract it needn’t be that way. What if “space” consisted of 3 right angles ∟, or 12? Something to think about.
Oh — and what if it took one number of azimuthal ∢ right-angles to make the whole pie round, and took a different number of planar right-angles to make that whole pie round? Yeah, that would be weird too.
Watch the 20-minute movie Not Knot, where they explain that links—knots made of several (closed/looped/circular) ropes rather than just one rope—biject uniquely to the complement of some hyperbolic geometrical space.
Since hyperbolic geometric spaces had already been explored a bit before the 1980’s, now everyone had a fun tool to unite concepts and ad-lib toward new ones. The new bijection opened up the gates to some easy logical shortcuts. I drew a picture of the way this kind of logic goes in talking about a clever way someone thought of to generate random normals with little computation. But this is in general how mathematicians solve impossible-sounding problems. I use a little bit of logic in domain X, as long as it’s easy there. Then I use this equivalence that somebody figured out to port the stuff into domain Y. Then I do that’s easy in domain Y. Then I either go back to my original domain or maybe I use some more equivalences to do easy stuff in domain ℤ, ℚ, Linear, and so on—always only using “obvious” logic in the particular domain, and letting the equivalences keep me right as I convert the problem across domains. The “link”-to-hyperbolic-complement-space was one such. Other examples include Fourier-to-regular domain, polynomials-to-sequences, equivalences-across-NP-complete-problems, graphs-to-matrices, matrices-to-characters, Lie-groups-to-matrices, …..
Oops — just used another common maths word without defining it. Bijections are one-to-one mappings from the source domain onto the entire whole of the target domain. For example a strictly monotonic function from ℝ→ℝ uniquely assigns members ∈ℝ to other members ∈ℝ — in such a way that no value is reused and every value is used. A strictly monotone function injects the source into && surjects the source onto the target—which means it can be inverted. (By contrast, a non-monotonic, up-and-down-looking function, re-uses values, so going in reverse you couldn’t tell which usage the 3 had come from.) If ∃ a bijection between X and Y, then ∃ a correspondence between X and Y. When mathematicians are trying to speak casually, they will often say something like “You can’t comb a hedgehog” or “You can turn any 3-manifold into a 3-sphere”. “You can do” is their way of saying ∃ a bijecting function that relates the two: ƒ(X)=Y. If ∄ a bijection, then it’s impossible to put X and Y into correspondence — there’s no earthly or heavenly way in which these two things could be made to look alike. For example, maps must fail to correctly show the globe because ∄ a bijection between a globe and a plane. (They also fail because of distortions; that would be asking for a conformal, area-preserving bijection instead of merely a bijection.) They also show how spaces-with-stuff-removed can biject to completely unexpected things. A punctured plane is equivalent to the surface of a cylinder, for instance. (?!?!) The punctured surface of a ball is equivalent to a (not-punctured) plane, for instance. (‽‽) Hey, I don’t make this stuff up, I’m just reporting the facts.
I guess in this talk he is showing different pictures of the associated geometry of various links.
Look up Hopf fibrations, one-point compactification, nilgeometry, solvegeometry, Lie groups (they’re groups, but continuous rather than discrete), Hopf circles, …. on Wikipedia. Be forewarned: this may turn into a months-long reading project.
Complements.Not Knot talks confusingly on this topic (“it’s not empty space, it’s space that’s not even there” … I think that way of talking only makes sense to mathematicians).
As I said in (2), spaces-with-stuff-removed can be homeomorphic to something completely unexpected. If you remove a point from the plane you introduce cylindricity around that point. Kind of unexpected that poking a hole in a square space makes a circular space, but that’s logic for you—always pointing out that illogical-sounding things are in fact inescapably true.
The symbology for complements looks too similar to the symbology for quotients. Sorry, not my decision. ℝℚ = the irrationals ℚ∁. ℂℚ∁ = the curliness of √−1without the ridiculously, insanely thick thickness of the continuum. A manageable space in which not all sequences converge. ℝTranscendentals = Algebraics. Another eminently reasonable number system that does everything you’d want without the messy insanity.
ℚ = all fractions, minus zero. This is a punctured thing. ℝ² = the punctured plane. ℝ³ = the cubic solid we seem to live in (Newton’s rigid rods) minus a point in the center of the universe. I don’t know if ℝ³ bijects to a looped thing like ℝ².
The worldSnoopy. Logically it’s equivalent to the punctured cubic thing I just described. Kind of boring, I thought removing Snoopy would be more devastating.
BTW, you can also adjoin things, like ℚ adjoin i = ℂℚ∁ mentioned above. I like this one if you can’t tell. ℝ adjoin ∞ is the one-point compactification of the line (as long as ∞ is defined to be ± ∞ so you can get there from the left or right)
Symmetries. The peace sign has a 3-way symmetry. Mirror images are 2-way symmetries. You could draw a flower with a 5-fold symmetry or a 12-fold symmetry and so on. The concept itself isn’t confusing, but the way Thurston and Not Knot talk fluidly, assuming without making explicit the implications of identification, quotienting by symmetry, topological gluing, point/line removal, and complementation together, is overwhelming.
The word ‘space’ has acquired several meanings, which is what you would expect of such a sexy, primitive, metaphorically rich, eminently repurposeable concept.
Outer space, of course, is where cosmonauts, Hubble telescopes, television satellites, and aliens reside. It’s ℝ³, or something like that.
Grammatical spaces keep words apart. The space bar got a little more exercise than the backspace key while I was writing this list.
Non-printable area (space) is also free from ink or electronic text in newspapers: ad space. Would you like to buy one?
Space on my hard drive to store an exact digital replica of all my vinyl? This kind of space also applies to human memory capacity, computer RAM, and other electronic pulsings which seem rather more time-based than spatial & static.
Businessmen refer to competitive neighbourhoods: the online payments space; the self-help books category; the $99-and-under motel space; and so on.
Space as distinct from time. Although cosmologists will tell you that spacetime is a pseudo-Riemannian manifold which looks locally like ℝ⁴, a geographer or ecologist will tell you that locally space looks like ℝ² (since we live solely on the surface of the Earth). I believe the ℝ² view is also taken by programmers who geotag things (flickr photos, twitter tweets, 4square updates): second basement = 85th floor and canopy = rainforest floor as far as that’s concerned.
Both perspectives are valid. They’re just different ways of modelling “the world” with tuples. Is it surprising that cold, rigid, soulless mathematics allows for different, contradictory viewpoints? Time is like space in the grand scheme of things, but for life on Earth time-averages and space-averages are very different.
Parameter space. The first graphs one learns in school plot input x versus output ƒ(x). But another kind of plot — like a solid liquid gas diagram — plots input a versus input b, with the area coloured or labelled by output ƒ(x). (In the case of matter’s phases, the codomain of ƒ is the set {solid, liquid, gas, plasma} rather than the familiar ℝ.)
When I push this lever, what happens? What about when I push that one?
Phase space. Paths, orbits, and trajectories taken through other spaces. Like the string of (x₍ᵤ₎,y₍ᵤ₎,z₍ᵤ₎)-coordinates that a water rocket takes across the lawn. Or the path of temperature(temp₍ᵤ₎) during a year in Bloomington.
Or the trajectory of the dynamical system(your feelings₍ᵤ₎, your partner’s feelings₍ᵤ₎) representing your marriage.
Roger Penrose uses the example of the configuration space of a belt to explain that phases can happen on non-trivial manifolds. (A belt can take on as many configurations as a string, plus it can be twisted into a Moebius band, but if it’s twisted twice that’s the same as twisted zero times.)
[Sorry, I don’t have a Unicode character for subscript t, so I used u to represent the time-indexing of path variables. Maybe that’s better anyway, because time isn’t the only possible index.] ₜ
Personal space. I forgot personal space. Excuse me; pardon me.
Abstract spaces. These are best understood as ordered tuples, i.e. “Things plus the relationships and desired interpretation of those things.” The space—more like “the entire logical universe I’m going to be talking about here”—is supposed to contain EVERYTHING you need, in order to work with any of the parts. So for example to use a division sign ÷, the space must include numbers like ⅓ and ⅝. (Or you could just do without the ÷ sign. You can make a ring that’s not a division ring; look it up.)
A Banach space is made up of vectors (things that can be added together), is complete (there are enough things that infinite limit sequences make sense), with a notion of distance (norm), but not necessarily angle. Also two things can be 0 distance away from each other without being the same thing. (That’s unlike points in Euclidean space: (2,5,2) is the only thing 0 away from (2,5,2)).
A group is complete in the sense that everything you need to do the operation is included. (But not complete in the way that Banach space is complete with respect to sequences converging. Geez, this terminology is overloaded with meanings!)
A vector space is complete in the same way that a group is. In the abstract sense. Again, a vector is “anything that can be added together”. The vectors’ space completely brings together all the possible sums of any combination of summands.
For example, in a 2-space, if you had (1,0) and (0,1) in the space, you would need (1,1) so that the vector space could be complete. (You would also need other stuff.)
And if the vector space had a and b, it would need to contain a+b — whatever that is taken to mean — as well as a+b+b+(a+b)+a and so on. In jargon, “closed under addition”.
A topological space (confusingly, sometimes called “a topology”) is made up of things, bundled together with the necessary overlap, intersection, union, superset, subset concepts so that “connectedness” makes sense.
A Hilbert space has everything a Banach space does, plus the notion of “angle”. (Defining an inner product is as good as defining an angle, because you can infer angle from inner multiplication.) ℂ⁷ is a hilbert space, but the pair ({0, 1, 2}, + mod 2) is not.
Euclidean space is a flat, rigid, stick-straight, all-joins-square Hilbert space.
To recap that: vector space ⊰ Banach space ⊰ Hilbert space, where the ⊰ symbol means “is less structured than”.
Topological spaces can be even more unstructured than a vector space. Wikipedia explains all of the T0 ⊰ T1 ⊰ T2 ⊰ T2.5 ⊰ T3 ⊰ T3.5 ⊰ T4 ⊰ T5 ⊰ T6 progression which was thoroughly explored during the 20th century. (Those spaces differ in how separated “neighbours” are taken to be.)
I don’t mean to imply that these spaces can only be thought of as tuples: ({things}, operations). There are categorical ways to understand them which may be better. But don’t look at me; ask the ncatlab!
Lastly, sometimes ‘space’ just means a collection of related things, without necessarily specifying, like above, the tools and viewpoints that we take to their relationships.
Branes, D-branes, M-theory, K-theory … news articles about theoretical physics often mention “manifolds”. Manifolds are also good tools for theoretical psychology and economics. Thinking about manifolds is guaranteed to make you sexy and interesting.
Fortunately, these fancy surfaces are already familiar to anyone who has played the original Star Fox—Super NES version.
In Star Fox, all of the interactive shapes are built up from polygons. Manifolds are built up the same way! You don’t have to use polygons per se, just stick flats together and you build up any surface you want, in the mathematical limit.
The point of doing it this way, is that you can use all the power of linear algebra and calculus on each of those flats, or “charts”. Then as long as you’re clear on how to transition from chart to chart (from polygon to polygon), you know the whole surface—to precise mathematical detail.
Regarding curvature: the charts don’t need the Euclidean metric. As long as distance is measured in a consistent way, the manifold is all good. So you could use hyperbolic, elliptical, or quasimetric distance. Just a few options.
Manifolds are relevant because according to general relativity, spacetime itself is curved. For example, a black hole or star or planet bends the “rigid rods” that Newton & Descartes supposed make up the fabric of space.
In fact, the same “curved-space” idea describes racism. Psychological experiments demonstrate that people are able to distinguish fine detail among their own ethnic group, whereas those outside the group are quickly & coarsely categorized as “other”.
This means a hyperbolic or other “negatively curved” metric, where the distance from 0 to 1 is less than the distance from 100 to 101. Imagine longitude & latitude lines tightly packed together around “0”, one’s own perspective — and spread out where the “others” stand. (I forget if this paradigm changes when kids are raised in multiracial environments.)
Experiments verify that people see “other races” like this. I think it applies also to any “othering” or “alienation” — in the postmodern / continental sense of those words.
The manifold concept extends rectilinear reasoning familiar from grade-school math into the more exciting, less restrictive world of the squibbulous, the bubbulous, and the flipflopflegabbulous.
Pictures of the 3-sphere, or should I say the 4-ball? It’s a 4-dimensional circle.
Even though these drawings of it look completely sweet, I have a hard time parsing them logically. They’re stereographic projections of the hypersphere. All they’re trying to show is the shell of {4-D points that sum to 1}. That’s lists of length 4, containing numbers, whose items add up to 100%. Some members of the shell are
The hypersphere is just made up of 4-lists like that.
The 3-sphere was the object of the Poincaré Conjecture (which is no longer a conjecture). Deformations of this shell — this set of lists — are the only simply-connected 3-manifolds. Any other 3-manifold which doesn’t look holey or disjoint must be just some version of the hypersphere.
Here’s an inside-out thought: The air around us is a 3-manifold with 3-holes where solid objects are, and the 2-boundary is the ground. Or if you think of all the sky, it’s a spherical 3-shell (with one 3-hole, the Earth) floating in empty space.
I wish I could draw what I’m thinking of. Something like this.
From a child’s-eye view, “the air” is the complement of everything that’s “actually there” (solid or liquid things).
You could correct that child by bringing up outer space (another 2-boundary on the air), or the fact that air is made of particles as well.
But wouldn’t you rather be sitting in the middle of a field imagining yourself, the trees, the grass, the clouds, the birds, and the wood chips being cut out by a Photoshop lasso?
In the loop quantum gravity approach, space-time is quantized by a procedure that encodes it in a discretized structure, consisting of spin networks and spin foams.
A spin network consists of an oriented embedded graph in a 3-dimensional manifold with edges labelled by SU(2) representations and edges labelled by intertwiners between the representations attached to incoming and outgoing vertices. These representations relate to gravity in terms of holonomies of connections, and the formulation of Einstein’s equations in terms of vierbein, or tetrads, and dual co-tetrads.
Thus, to a spin networks, or the 1-skeleton of a triangulation by tetrahedra, one assigns operators of quantized area and volume, coming from counting intersection points of a surface, or 3-dimensional regions, with the edges or vertices of the spin network with a multiplicity given in terms of the spin representation attached to the edges and the intertwiners attached to the vertices.
Loop quantum gravity is one of a few general frameworks that may eventually form the basis of how physicists think of the smallest scales of time and distance.
(These frameworks are sometimes called Theories of Everything but they’re really just thoughts-on-the-way-to-theories of brief-and-tiny matter.)
GLOSSARY
SU(2) is the group of 2×2 unitary matrices with determinant 1. They have the form SU(2) is just like the unit quaternions, which represent rotations in 3-D. Here are the pieces that make up SU(2) . , ,
3-dimensional manifold — any shape that can be made with dough. Including that if you stretch the dough, the outside and the inside stretch.
oriented graph — things like this:
embedded oriented graph — oriented graphs sculpted in 3-D
edge — arrows in the above pictures
spin network — a graph like above and each circle has value +½ or −½
representation — every group can be represented as a matrix
holonomy — how to move things in parallel within the dough (curved 3-manifold)
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Human Mathematics 