The Universe in a Cube of Cheese

By

My eighteen-year-old son is a math whiz. He’s the kind of kid who learns Calculus 2 from some website, then lands a perfect score on the AP test without ever taking the class. Meanwhile, I’m still struggling with five times seven.

Was it really fifteen years ago when I bought a gallon bucket of plastic counting bears to teach him addition? Now he’s dragging me to the kitchen table at midnight and patiently drawing out diagrams on paper napkins, unpacking the glories of the numerical universe one step at a time.

He’s a born teacher, massaging higher math into the vernacular until my fog lifts, waiting for that moment when I gasp because I finally understand. All at once I see what he means and why it matters (Hallelujah!). I see why this concept is beautiful, why he wanted me to see it; then just as fast all the light passes away, and I’m back in the dark.

I don’t know math well enough to hold on to that kind of truth, but for a half second it was mine, and the flash of insight changes me, even as it fades. It wakes me up and makes me ache to get it back again. It lingers like a perfect melody that you heard in a dream but you can’t remember well enough to sing once you wake up.

Sehnsucht is a German word that comes close to what I mean, a yearning for something beyond our reach. C. S. Lewis referred to this desire as “the inconsolable longing in the heart for we know not what.” It’s more forward-looking than nostalgia, more solid and more hopeful than sentimentality. When Sehnsucht strikes me, it feels like I am rising up through the thick, choking waters of the earth into a cleaner air.

Philosophically, we can trace Sehnsucht back to George MacDonald, and back again to the German romantic philosopher Novalis. However, I want to save all that for another essay at another time — for now I’d like to just name the thing and move on to the heart of what I mean to say about it.

This week JD has been teaching me about the possibility of a fourth physical dimension. We have all been taught about the three obvious dimensions: length, width, and depth. You may have also heard that the fourth dimension is time. However, JD’s been digging around in theories about a fourth physical dimension — a realm like depth, but beyond depth.

I’ve sensed this possibility intuitively since I was a child, but I didn’t have good images for it until reading George MacDonald’s Lilith, and I didn’t think it was scientifically feasible until watching NOVA’s The Elegant Universe, a documentary about string theory.

I also felt shadows of a fourth dimension while studying the discipline of philosophy. The transition from old, historical rationalism (think logical proofs) through David Hume’s more modern empiricism (think scientific method) left me hungry. In fact, Hume knew his own gaps. He knew it was impossible to bridge the expanse between what is real and what we perceive to be real (AKA: the mind-body gap), which means that it’s very difficult to prove anything at all.

It’s not practical to linger in that futility very long. Those who take it too seriously seem to land in a sulking existentialism, but that’s not where it has to stop. Instead of pouting that we can’t nail everything down, it’s also possible to stay properly small and properly curious in a mysterious universe. As the old song goes, ‘tis a gift to be simple, ’tis a gift to be free, ’tis a gift to come down where we ought to be. Is it so terrible to recognize that our five little senses and our three pounds of brain might not have the ability to master all that is?

There’s something healthy about admitting the possibility that “Holy, Holy, Holy” isn’t redundant, but only the straight line of the edge of a cube? A multi-dimensional… trans-dimensional God … is likely to distort when we attempt to hammer him into a post-Enlightenment flatland.

Yesterday I listened to a mathematician describe the blasted, necessary intersection of a Klein bottle. A Klein bottle is an edgeless object that can exist in four-dimensional space, but when we try to form it in a three-dimensional world, we have to insert a seam to even hint at what the thing would be like in a larger reality. As I listened to this man describe how a three-dimensional model could never truly suffice, I heard in his frustration the frustration of a theologian trying to unpack God to a materialist.

Another four-dimensional object has been named the hypercube, a 4D version of a square. Generous researchers have tried to create computer simulations of the hyper cube, and there are several videos online that allow us to watch them rotate. When I see the rotation, I understand four-dimensionality for a moment, then lose it again. It’s so difficult to think beyond our own experience.

However, the stretch to comprehend this has still been good for me, and I’m going to share with you some of the shadows of possible truth that I’ve collected while stretching. Some of you are more mathy than I am, and you can write and correct me where I am in error, but we must start somewhere, so I’ll take a stab at it.

Actually, let’s make it a story. But instead of trying to tell this story in three and four dimensions, we’ll reel it back to two and three.

Once upon a time, there was a two-dimensional square who was given consciousness. Our poor square has only ever existed in a flat cognition. He has heard about three-dimensions, but to him this is the stuff of sci-fi conventions and religious cults. Most of his fellow squares have grown out of such nonsense entirely and have declared themselves atheists. They are perfectly content to spend their lives studying the edges and planes, and in fact, they feel rather bright to be doing it.

But our square is a bit of a romantic. That’s not his fault, he’s just survived a nasty breakup with a triangle, which has led him to wandering the flat streets, staying up too late, meeting with shady polygons in flat pubs — the sort of polygons who drink flat whiskey and talk about fairies, and fawns, and one particular mythical entity called a “cube of cheese.” You can judge him if you want, but the company does him good, even if he can’t take these dreamers too seriously.

As it so happens, you stumble into the pub on a night when this motley crew is discussing three-dimensionality and you decide to help.  In fact, you have a couple of cubes of cheese in your pocket. How lucky.

You rotate one cube of cheese for the square to see, but as you do, you realize that he can only understand one side of the cheese at a time. You go over and over it, but he is clearly confused, and at one point you think he might start to cry. “This is why she left me,” he mumbles. She said I just didn’t “get” things.

“Women,” a rectangle groans.

A new idea comes to you. You take a knife and start slicing your 3-D cheese into little thin slices, like images from an MRI. You even slice the cheese diagonally, trying to help the square understand what depth means. The square says, “Oh! A cube is a million flat planes!’”

No, no. He’s closer but still missing it.

You begin to organize these slices of cheese like one of those flip books kids use to mimic motion. You show him the slices one at a time, more and more quickly, trying to help him see that they come together to make something whole. As these slices of cube pass over the square’s vision, he suddenly catches a glimmer of depth. All of these individual planes could merge into a single unity. A cube!He can almost see it. Almost. No… it’s gone.

Still, in that second, he felt what he had no real ability to perceive or validate. He caught the trajectory, an impression that woke up an ache in his wee little heart for a reality beyond him.

I am telling you this story because I think we are but squares in a universe of cheese cubes.

I am not a great scholar of science or philosophy, but I have studied enough to begin to feel the limits of the human mind. I can tell that there is a very real possibility that we we will not be able measure or reason our way through all that is. We can snicker at the idea of angels and gods, but the wisest among us are only nonagons.

Tim Keller, in his book The Reason for God, cites an old Indian story about five blind men and an elephant. The men are asked to describe an elephant by feel, so the man at the trunk says the elephant is only a sort of tube, while the man at the tail says it is a rope. The man at the ears says it is like a rug, and the man at the side believes it is a wall. On and on, and we finally understand that only someone with sight can perceive the whole of what an elephant is at once. It takes a consciousness that transcends our own limitations to put all the parts in true perspective.

It is not that we can know nothing, for God has given us senses, logic, and reason as gifts of imago Dei. Yet when an ache rises, when the universe hints to us that there is more still, and by a flicker, by a flutter, by a sudden flash of want and awe we are moved to Sehnsucht, perhaps these flat rocks are crying out to us. Perhaps this is the beginning of wisdom.

 

Rebecca Reynolds teaches Classical Rhetoric and Philosophy of Faith in eastern Tennessee, and is a contributor to the Story Warren website. She’s the author and illustrator of the pediatric series From the Medical Files of Dr. Phineas C. Bones and collaborated with Ron Block as the lyricist for his critcally-acclaimed album, Walking Song. She lives in Kingsport, Tennessee, with her husband and three children.


19 Comments

  1. Rachel

    Thank you for translating this for us less “mathy” folks. 🙂 There is a beauty in the understanding we do not have and can never fully attain… but rather the point is to always strive “further up and further in.” There is no end of His divine mystery, and I am glad of the calling to joyously pursue that with you.

  2. Laure Hittle

    i can accept that the universe is made of cheese. It explains the moon, for starters.

    i’m not very mathy, either. By G-d’s grace and a friend’s unreasonable patience, i mastered algebra just a few years ago—and it’s gone again. But i had a few of those extra-dimensional moments as well, before the veil fell back into place. i felt it again while watching cheese slices flip just now.

    When i was ten or so i used to lie in bed and, behind my closed eyes, see red and blue balls morphing. They were big and tiny all at the same time, and as they grew bigger they were also growing littler. i’m sure that doesn’t make sense. i don’t know how else to explain it. How can something be both big and little? How can it grow and shrink simultaneously? My mind couldn’t comprehend it, and it scared me. i tried my best to make them stop, and eventually they did. Years and years later, i wondered if maybe what i was seeing was a glimpse of eternity, of infinity.

  3. Rebecca D. Martin

    “…it’s also possible to stay properly small and properly curious in a mysterious universe. As the old song goes, ‘tis a gift to be simple, ’tis a gift to be free, ’tis a gift to come down where we ought to be.” Wonderful. I look forward to that “another essay at another time” on Sehnsucht.

  4. Mark Proctor

    Rebecca, thank you for sharing this. I had a math wizard friend tell me once that reason does not come before faith. Even a mathematician, if he is honest, must eventually come to this conclusion or go insane trying to avoid it. Many have; many will. Even mathematical knowledge, supposedly the most strictly logical and rational, rests ultimately on faith- faith in the axioms which form its basis. I am thankful for Sehnsucht which draws us to faith…both are a gift from the Father.

  5. Lisa

    I am basically allergic to math, but I loved this! I have been reading through Ezekiel, and this reminded me of his “wheels within wheels”… glimpses of something strange and impossible to explain yet utterly familiar even so.

  6. Ann

    This is beautiful. I remember driving home late at night after an organic chemistry lab hyped up on Mountain Dew, snickers and trying to understand the mysteries of life. I stopped at a red light which triggered the light to change. As I watched another car stop so I could go, I suddenly realized that they only had to stop because I was there and I had a brief glimpse of all the ripples that could potentially play out because I had stopped, the light had turned and they were now three minutes later than they would have been otherwise. I thought my head would explode. So, I very sensibly went home to bed!

  7. Jody Lee Collins

    Rebecca, this was lovely! ~ when I went back to school at the age of 36 to get my teaching credential I too found I was ‘allergic to math’ (thank you, Howard Gardner).
    But alas, I had to take Algebra even if it killed me (which it nearly did–tears in class and the whole bit.) My final paper for this professor was about fractals–self-similar repeating patterns.
    I was astonished at the beauty that had been ‘discovered’ by mathematicians to explain my world of pinecones, cloud edgings and sunflower centers. Mandelbrot had reduced the wonder to a formula, but to me it was art and magic, wonder everywhere.
    Why couldn’t people see it? I have no answer to that, but I do knowhHigher math proves there is a God, I am convinced.
    (I’ve been teaching now for over 20 years- in Elementary School :-).

  8. JamesDWitmer

    Never before have I read an article about math that moved me to worship. Thank you – again – Becca, for sharing your gift.

    Such a blessing to be so small, in the hands of a God so great in every dimension. (Ephesians 3:14-19

  9. Jud

    I’m curious: did you come up with this idea on your own, or have you read “Flatland: A Romance of Many Dimensions” ?

    It’d be pretty amazing if it was merely coincidence.

  10. Pete Peterson

    @pete

    Good question, Jud. I do own that book, but I don’t think I’ve ever read it. In fact, I’d forgotten about it until I was either finished or mostly finished with this post. First I tried to write the pub scene with cubes discussing a hypercube first, but that was too difficult. So, I rewrote it all in 2D and 3D, and it was a lot easier to follow. Does Flatland have a scene similar to this? I guess that book might be somewhere in my subconscious if I’ve heard someone talking about it in the past, but sources I was actively thinking about when I was writing were George MacDonald’s _Lilith_, the Dufflepuds in Narnia, Lewis’ _Space Trilogy_, and L’Engle’s _Wrinkle in Time._Oh, and I thought about _Winter’s Tale_ some, too.. those sorts of.narrative expressions of metaphysical concepts.

    Thanks for your kind words, everybody. Love the Rabbit Room. You’re a good audience.

  11. Jud

    The entire book is “a scene similar to this.”

    A few years ago I challenged myself to read 52 books in a year. I finished at 64, and my favorite of the whole set was Flatland. I highly recommend it.

  12. Peter B

    Seriously? I just assumed she was pulling from the book; the lack of attribution seemed to be an implication that everyone would recognize the reference.

  13. Pete Peterson

    @pete

    Oh, neat. I’ll have to go check it out. I think I have this same book you’re talking about, though it seems like mine is a picture book, so maybe I’m thinking of the wrong thing. Who is the author of the one you love, Jud?

    The similarity doesn’t surprise me, though, if it’s there. It’s a natural thought once you start getting into dimensionality at all. Anthropomorphizing lines and shapes is funny. One of my favorite students was sitting in a math class once when he was asked what a certain angle of a triangle was. He answered the”anus.” He got into a bit of trouble over that, but I’ve laughed for three or four years over it, wondering about the digestive system of polygons. Could you identify triangle scat if you happened upon it in a forest?

  14. John Covil

    I was thinking of this geometric relationship again, though in a less profound way than multidimensional it’s, when reading Hebrews this morning:

    Hebrews 1:7-8

    Of the angels he says,
    “He makes his angels winds,
    and his ministers a flame of fire.”
    But of the Son he says,
    “Your throne, O God, is forever and ever,
    the scepter of uprightness is the scepter of your kingdom.

    Angels have an eternality of a sort, as do human souls. It’s eternal in one direction. But God’s eternality is of an entirely different quality. We are rays, created at a single point, extending on forever. But God (and his begotten) is a line; the only true line, extending from eternity to eternity.

  15. Laura Peterson

    I absolutely got the “A Wrinkle in Time” vibe from this, Rebecca.

    When Mrs. Whatsit is explaining the tesseract: “I got it! For just a moment I got it! I can’t possibly explain it now, but there for a second, I saw it!”

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