Where we learn that some soothing patterns can kick into high-gear in spacetime.
Do you know about spacetime?
I didn’t till recently. And I’m not usually into things like ‘space exploration’ – it gets me as fussy and uncomfortable as the phrase ‘deep-sea diving’. I’m a researcher, not an explorer, and I don’t have any interest physically going to places which other human beings haven’t already checked it all out. I also most definitely do not want to creep up on some other being’s homestead.
But then I came upon this thing called spacetime while researching symmetry stuff and I was like, huh, are you telling me there’s some sort of symmetry in space? Well, I wanna see that!
Though, baby steps. First, I bet it’d be helpful if I shared some definitions of symmetry. I came across some delightful ones, actually; ones that inject a lil subjectiveness in the form of feelings, a state of being, even quality of life in reaction to the existence of symmetry in your immediate locus {😉 to all you KLM* subscribers}.
Symmetry
“Ten Patterns That Explain The Universe” by Brian Clegg
(noun)
Measured in a like way.
(pg. 202)
Mathematician Hermann Weyl
A thing is symmetrical if there is something you can do to it so that after you finish doing it, it looks the same as before.
1885 – 1955
Symmetry is comfort food for the eyes.
Frank A. Farris
MoMath online event – April 19, 2023
Here’s the thing for me when I read all of these definitions. I nodded my head vigorously for each of them. That first one, Yup {nod}. Oh and…Yup {nod}. Oh, yup yup, totally {nod nod}. I did do a little head tilt on that first definition. It gave me a new lens to look through at symmetry that I hadn’t consciously thought of before – that whole measuring thing.
I’d always thought of symmetry as only about the (pleasing) pattern aspect of it, not what made those patterns, errr, pattern-y and consistent. It really does all go back to precise measurement, not just of an object itself whether it’s in 2D or 3D space, but of its actual distance from others. In real everyday life. And in the bio lab, too.
Before my rendezvous with biology, I thought that the power of symmetry was exclusive to physics, but I was wrong. I learned that viruses were endowed with varying amounts of symmetry. Like Lego building blocks, the individual proteins self-assemble into an icosahedral structure characteristic of a virus. According to my friend Brandon Ogbunu, an MIT biophysicist, in biological systems symmetry can persist from the molecular to the organism itself to maximize its evolutionary fitness. A straightforward example is the bilateral symmetry of our legs – they have to be the same length if we want to run and hunt successfully in the jungle. The various symmetries of virus function, for instance, to provide mechanical stability and the efficiency to bind to a host cell. …Symmetry seemed like a key link between particle physics and the function of life.
“The Jazz of Physics: the secret link between music and the structure of the universe”
by Stephon Alexander (pg. 120-121)
Ok, so I guess that means the whole measuring thing is always occurring, even when we encounter our everyday symmetry, yeah?
Oh, what’s this?
Mirror symmetry… involves taking the distance of each point in half of an image from an imagined mirror and reversing it in the other direction. Each point on both halves of the mirror-symmetric image is the same distance as its equivalent point, measured from the line of symmetry where the reflection takes place.
“Ten Patterns That Explain The Universe” by Brian Clegg
(pg. 203)
The most significant property of mirror symmetry is lateral inversion – that’s when the left side appears to be the right side. You know, it’s like on the plethora of video calls we’ve all been on these past few years. This mirror symmetry – which reflection of yourself do you choose in your settings, and why? I for one don’t think about it unless I have on clothing that has some messaging on it or my background includes something that others on the call could be read (and that I want them to read).
When lateral inversion is happening in mirror symmetry (also known as reflection symmetry) it’s not so easy to detect cuz…well, since the left and the right side are visually symmetrically the same therefore, measurement-wise, everything on each side is the same. Any differences are indiscernible.
When I talked about transformation before, the action of a reflection transformation is the rotation (or I suppose it could be a flip) of an original image by 180° and in that essay I referred to as a night/day transformation.
But I hadn’t considered what it might mean when you’re considering symmetrical imagery. Since the spatial orientation in a reflection is changed, and even though you can’t visually see it because both the left and the right half are reflections of each other, something has still happened. You may not see it with yer eyeballs, but…there’s been some sort of change, right? Your perception is what has changed.
Invariance also plays a role, a rather big role, in symmetry’s existence in real life. As it was defined in that essay, an invariance is a quantity or an expression that is constant throughout a certain range of conditions. Symmetry solidly meets up with invariance at some point, and one of those points is when you get to conservation laws.
Now now – I am gonna barely peek into physics right now because it is not my thing but this incredibly brief sojourn into it – it’s just a few paragraphs – is worth it, because you will get to meet one of the greatest mathematicians ever, a mathematician who Einstein thought was friggin awesome (that’s not quite a direct quote but he did say that their work was beyond rad [also not a direct quote]), a mathematician who just happens to be a woman, a woman who was a mathematician at the dawn of the 20th century.
I’m happy to intro you to Dr. Emmy Noether.
Conservation laws are these fundamental principles of physics.
There’s one on energy, and a couple about momentum, and another about electric charges.
And true to the very definition of what being fundamental is, these conservation laws about those things are absolute and they are what is the basis for things like, you know, the theory of relativity.
Dr. Noether did her Ph.D thesis on invariants and what followed is that a lot of her work was about studying hidden symmetries.
The relation between conservation laws and invariants is subtle. With an invariant, you take a system and do something like turn the axes or move the origin or move the hands of your clock and show that certain numbers, the invariants themselves, don’t change. A conservation law, on the other hand, describes a quantity that stays the same over time. The total amount of energy – or charge in the universe is a conserved energy.
“The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality”
by Dave Goldberg (pg. 112)
And then:
The pioneering German mathematician Emmy Noether would prove mathematically that the conservation laws, physical laws that are responsible for much of the everyday behavior of the world around us, were the result of symmetry in nature. Symmetry is far more than a visually attractive layout: it is the pattern that shapes the heart of reality.
“Ten Patterns That Explain The Universe” by Brian Clegg
(pg. 200)
Daaannnng, the ‘heart of reality’. For a few moments, after I read that, I started thinking on how symmetry and fractals might play together. You know, like the ferns on the homepage of this very site. But before I could get too intertwined in that examination, I reminded myself – and now you – that though the fern leaves on either side of a branch could absolutely be symmetrical, the scaling action of a fractal was something else entirely.
Symmetry assisted in giving that comforting pattern to many a fractal. Symmetry in nature. Patterns of attraction and that calming sense brought on by familiarity.
Architects knew the importance of symmetry long ago. And it isn’t just that it’s eye-pleasing, but it also contributes to safety. Bridges are symmetrical – they gotta bear weight evenly.
And how about inside the home? Symmetry is pretty much everywhere, with everything from how you decide to hang 2 or more pieces on a wall to the wallpaper itself. There’s a whole bunch of ways (especially when you consider all the possible rotation and mirror transformations) for symmetry to show up in wallpaper design.
Same goes for tiling. Have you heard of Penrose Tiling?
As I go on and on about the comfort of patterns, I must share that I break this mode when I am hanging anything on a wall. Blame it on some finely attuned energy detector within me, but as anyone who has seen how I hang things on a wall, it is very rarely symmetrical with anything else on that same wall. Everything that is hung is level – I’m not completely out-of-control, please – but I leave …mmm, let’s call it seemingly unexplainable spacing between some items and others I smoosh on up against its next closest mate that’s hanging there. Because, vibrations of energy. It works for me and my partner says it works for him💖, so, takeaway conclusion is: Do You. Always and Forever. In All Your Spaces.
Mathematician Roger Penrose and his father, psychiatrist(!?!) Lionel Penrose, developed this aperiodic tiling, and there are two ways to do a 5-fold symmetry about a single point – which creates what’s known as either a star or a sun configuration.
Ah, the expansive delight of symmetry 🙂
I haven’t forgotten about Symmetry! In! Space!
All this comfort and calmness we get from patterns of symmetry in our homes, in our cities, on our planet, well, apparently, its effect not only stretches into the entire universe – as with Dr. Noether’s conservation law theorem when it comes to Relativity and all of its forms – but it also goes even bigger. So big that it breaks out of our 3D world and into the 4th dimension. Yeah, that’s what spacetime means.
Those wacky invariances, those quantities that remain constant no matter if the conditions that they exist in get a bit altered, they are the counterpart – and the basis for – the existence of spacetime symmetries, where time and space are the same in all places and the conservation laws are the same in all directions.
It’s a cube within a cube. It’s a hypercube. It’s a tesseract. And none of us can really picture it exactly, cuz we live in 3-dimensions.
Honestly, I don’t quite get it. But it feels pretty cool? I mean, imagine if you were a 2D comic strip character attempting to grasp the concept of 3D. Same same – our 3D selves think of 4D like… what? Why? No.
(gif from Math Images)
Okay, let’s step back from the 4D. Look around you right now, see if you see some symmetrical things. Maybe some symmetry you hadn’t even realized before. Many folks faces are pretty symmetrical. It’s great knowing that this wonderful world of symmetry is everywhere and it really serves as something that is grounding – in all the possible (that I know of) dimensions that can exist, just humming along with its very reliable little buddy, invariance.
If you didn’t arrive at this essay via my newsletter, I shared some cool symmetry imagery there that may quiet down anything that I unintentionally worked up with all that this 4D talk.
Anyhoo. SPACETIME!
Thanks math, you’re the best.
“Humans are bilateral animals, that is we are generally symmetric with respect to the sagittal plane. The need for directed movement results in bilateral symmetry of muscles and skeletons though internal organs are often asymmetric.” [Cleveland P. Hickman, Jr.; Larry S. Roberts; Susan L. Keen; Allan Larson; David J. Eisenhour. Animal Diversity. 5th ed. McGraw-Hill Higher Education 2009.]
aaahh, so many goodies in that quote – all parenthetical commentary from moi 😄: ‘…the sagittal plane(!)’ and ‘the need(!) for directed movement…’ and ‘internal organs are often asymmetric’ (often but not always??? 🤨😜) Thx for sharing & citing this, Gerald!