Philosopher Bertrand Russell once remarked that “Mathematics, rightly viewed, possesses not only truth, but supreme beauty.” But for those of us who haven’t studied math on an advanced level, it can be difficult to even guess how fractions and algebra hold deep meaning for the universe. Fortunately, some brave souls are bridging the gap between mathematical complexity and the world of meaning. As Edwin Abbott’s classic Flatland linked geometry and spirituality, some mathematicians have used the idea of “higher dimensions” to help us think about God.
Dr. Kim Jongerius earned her PhD from Colorado State University. She has maintained her interest in language throughout her career, arguing that mathematics is itself a language, and has written rules and insights connecting the mathematical and linguistic abilities of C.S Lewis, exploring connections between the languages of mathematics and faith by co-authoring chapters on dimension, infinity and proof in mathematics through the eyes of faith, and developing a cross-disciplinary honors course titled “Pattern and Structure: Mathematics and Literature.” She is a professor of mathematics at Northwestern College, a Christian liberal arts college, and she has been involved in her church as a small group leader deacon and member of the leadership team. Kim has also long been active in the Association of Christians in the Mathematical Sciences, including terms as ACMS Board Secretary.
This excerpt from her talk shows just one dimension of her findings between religion and mathematics – you can see the full presentation here.
(This post is part of Sinai and Synapses’ project Scientists in Synagogues, a grass-roots program to offer Jews opportunities to explore the most interesting and pressing questions surrounding Judaism and science. “Does God Live in Another Dimension? (How We Assume Shapes What We Conclude)”, a talk given at Temple Mount Sinai on April 25, 2021, is the second in the series Higher Meanings: Connecting Religion and Mathematics. The next event in this series, “How Religion in Medieval Times Shaped the Jewish, Muslim and Catholic Study of Mathematics,” with Dr. Victor Katz, will be on June 10).
As we can see, even in ancient times, folks had a hard time reconciling the concept of a God who is with us with a God who is beyond our understanding, who is all-powerful, who is all-knowing, who is infinite. How do we picture an infinite God being with us? Well, we have to think about what our assumptions are. And I’m going to get into dimension to convince you – like Larry did last time – [that] infinity is more complicated than you think it is. (If you didn’t see Larry’s talk, you need to look it up because he really did a great job with this). So God, who is infinite, is more complicated than we think He is as well. Dimension is similar, right? Dimension is not as simple as we think it is. So we’re used to thinking about it in certain ways, and it doesn’t always fall out exactly that way. So I’m going to talk you through some of the basics of dimension. And then we’re going to get to this connection with the nature of God as we go.
All right, so what is “dimension” anyway? What do I mean by that? Well, there are a couple different definitions people throw out around. One is the number of coordinates needed to specify a point in the object. So if the object is a single point, and you’re at that point, I don’t need anything to tell me where you are, I know you’re at that point. So this is zero-dimensional. If I have a line segment, though, now it gets more complicated, right. I don’t know where you are on the line unless you tell me some information. So if I give it a coordinate system, I tell you, “All right, that’s going to be part of the real number line, we’ll just go from 0 to 1.” Now, if you tell me I’m at 0.75, hey, I can figure that out. That’s maybe about here, right. So I know where you are on the line. If you tell me I’m at 0.259, I’m going to go “Well, okay, that’s a little over ¼, so I’m somewhere in there.” I know pretty much where you are on the line. You give me any value and I can I can tell which point it is on the line. So this is one dimensional, right.
What if I get what I normally think of as a two-dimensional object, like a rectangle and its interior? Well, let’s do the same thing. So I’m going to put a 0-to-1 axis horizontally, and a 0-to-1 axis vertically. And now if you tell me, for example, that you are at (½, ½), then I’m going to figure you went one half-unit over and one half-unit up, and so you’re right in the middle of this thing. If you’re at (1, 1), you went one unit over and one unit up – one unit over, so you’d be right up here in this corner. Two coordinates tell me where you are in the object. This is two-dimensional.
Similarly, for a three-dimensional block, I need three coordinates to tell where I am there. I’m going to freak you out a little by putting the origin, the place where they’re all zero, up here. And the reason for that is that if I put it down here, then this one is – well, I can’t see it. So you can see in the picture, right. So at the point ½, ¼, 0, I would be going one half-unit down, one quarter-unit to the right, and then not moving, because my third coordinate was 0. So I’d stay here – if the third point was 1, I would go back to the back of the block.
All right, well, that’s one way to think about dimension. And mathematicians like this way, especially algebraics like me, because I can easily think of higher dimensions from here. If I think of dimension as the number of coordinates that I need, then what’s the fourth dimension? Well, I’m just going to add a fourth coordinate to my points. And what’s the fifth dimension? Well, I’m going to add a fifth coordinate to my points. So I can have infinite dimensional space, I’m just going to keep adding coordinates – as many as I want.
Some people like a little more concrete feel to their geometry, so maybe we want to think of dimension, instead, as the number of mutually perpendicular directions of possible travel within the object. So back to the point, if you’re there, well, you are not going anywhere, right. So there’s zero directions of travel; it’s zero-dimensional. If you’re on the line, well, you can go to the right or you can go to the left. That’s two directions, but they’re not perpendicular to each other, right – they do not form, right angles. So there’s really only one direction of travel. If you’re over here in the solid, well, I could choose to go one way, and I could go at right angles to that. Now I could have gone at right angles the other way, but that wouldn’t be perpendicular to both of these, right. So there’s only two that are perpendicular to each other. I can’t go anywhere else and be perpendicular to both of those things.
But if I’m in a solid box, I can go one way, I can go at right angles to that, and then I can go perpendicular to that. Now, that’s a little hard to see here,but I’ve got that going back into the box. We often think about it in math and science as a right-hand rule. So I’ve got three axes here, these two are perpendicular to each other, these two are – if I move my fingers right – perpendicular to each other, and these two are perpendicular to each other. I’m not as flexible as I used to be. Anyway, that’s sort of my axis system that can move around with me.
Okay, well, this is more insightful, right. Thinking about directions of travel, I can picture that in my mind. Coordinates are a little bit abstract. The problem is, because this isn’t as abstract, it’s harder for me to move up into higher dimensions, and it’s hard for me – I can understand it conceptually that I now need a fourth direction of travel that’s perpendicular to all three of those, but I cannot picture it, because I’m sort of trapped in this three-dimensional world in terms of how I can picture things. So that’s why I like to go with the coordinate method, because when I have coordinates, then I can go up as high as I want, as many dimensions as I want.
Okay, but as I said, things are trickier than that. So with coordinates, we can think of any whole-number dimension, but in fact there are things that are in-between dimensions. So let’s look at that Koch Snowflake a minute. I’m going to start with an equilateral triangle, and we can assume for the moment that all the sides are a length of 1, okay. so we’re thinking about this side as being length = 1 that’s length = 1, that’s length =1. And it’s just a triangle – not the part inside, okay, just that boundary.
Now here’s what I’m going to do, I want you to watch this side as I do it on each of these sides. I’m going to replace the side with this – I’m going to take out the middle third, and I’m going to put equal lengths to that down like this as a kind of a triangle attached to it. So my path will now go over a third, down a third, back up a third, and then over the final third. So this length of one thing is now going to become a length of ⁴⁄3, and that’s going to be important for what we talk about. So watch that, and over here you’ll see what I talked about in terms of taking out the middle third and replacing it. But I want you to see what happens here okay see that’s what happened on all three sides, that I took out the middle third and made that part of it longer. Hey! Integration of faith and learning, look at that. You know I’m kidding, that doesn’t really count. But it is kind of neat when little connections like this come up, like, “Oh, the Star of David in my construction of the Koch Snowflake.”
Okay, now I’m going to keep this process going. So focusing on this side now, and you’ll see I’m going to take out the middle third and replace it with a little two-sided triangular path, right, so here we go. I do it on all of them at once, right. So this is the iterative process that I mentioned earlier. Now I’m going to do it again on each of those little tiny sides, I’m going to take out the middle third and bump it out. There we go.
And this is about as far as I can draw it so that we can still see anything going on. You can tell it’s already starting to look kind of fuzzy. We just keep doing this, on into infinity, and we call the end result the Koch Snowflake. So it’s the boundary of this figure.
And as I said, some weird things go on here. Like, its enclosing a finite area because it can all be trapped in a circle, but remember, we talked about how each line was replaced. So when I started with 1, what replaced it was now ⁴⁄3. So the thing that originally had perimeter 1 or 3, that triangle, now would have ⁴⁄3 + ⁴⁄3 + ⁴⁄3 – that’s ¹²/₃. So now that has perimeter 4. So the curve is longer. And that happens every time I multiply by ⁴⁄3, so it gets longer and longer and longer and goes to infinity. As I iterate this again and again and again, I get an infinite curve. So I have infinite length enclosing a finite boundary, which feels a little weird.
Well what about the dimension of this thing? So normally a curve I would kind of think of as one-dimensional, even if it’s bending, it’s not a straight line, you can kind of tell me one point and you would be able to tell me where you are on that line. The Koch Snowflake is different; this has what we call fractal dimension, and it’s about 1.2619.
So now I know what you’re wondering, “How did you get that number? And please don’t tell me if I don’t like math because it might freak me out too much.” So I try not to do that, but I am going to try to tell you where this comes from. First of all, the pieces – I’m not going to go into the computational details, so don’t worry about that – the pieces of it are – this is what makes it difficult. First of all, fractions – which if you don’t like math, and that started in elementary school, fractions is probably what did it to you, so I’m sorry about that. Also logarithms – if you made it through fractions and you stopped liking math in high school, this might be what did you in. So again, sorry about that. And limits, which if you didn’t get as far as calculus you probably never had to do, but they also involve fractions and sometimes logarithms. So we’re stuck with that.
I’m not going to go into the details of that, but I actually do want to talk to you a little bit about logarithms, because they’re not as bad as you think. Really, a logarithm is just an exponent. So I think of seeing an expression like log₁₀(100) is asking me a question, “What number would I have to put on 10 as an exponent to get 100?” The logarithm is just an exponent. And then I think, “It’s not so hard to see, right, if I square 10, I get 100, 10 * 10 is 100, so 2 is the exponent that I would need.” So the log₁₀ of 100 is 2. What about the log₂(8)? Well, what number do I need to put on 8 as an exponent, on a 2 as an exponent, to get 8? Well, 2² is 4, 2⁴ is 8. So 3 is the number I’d have to put on 2 to get 8. So log₂(8) is 3. I won’t torture you with more of those, but I just want to again kind of reiterate that a logarithm is an exponent, it’s what you put on the base to get the other number. Now, what is log₁₀(49)? I’ll torture you just a little bit more. Well 10¹ is 10, 10² is 100, 49 is somewhere in between there, so the log₁₀(49) is going to be a number between 1 and 2. So I don’t take 10 to the first power, I don’t square it, I do something in between. And that’s what we’re getting at with fractal dimension.
But often we get at it by just counting boxes, so that we can kind of see, “Are the number of boxes we need to cover this thing one-dimensional, two- dimensional, three dimensional, or what? What’s the exponent?” So I’m going to quickly go through this, but I think I can make it understandable. So I’m going to call a ¼ box a segment of length – ¼. Okay, so this is a segment of length 1, and I’m going to think of making a number line with segments of length ¼. So these are one dimensional boxes, okay. And if the number line, if the segment is sitting exactly in there, it intersects four of those boxes. And maybe I shifted it a little bit, now it intersects 5, right – 1, 2, 3, 4, 5.
But it’s not going to intersect more than five of them, no matter how much I move it around. So we say that the maximum number of these one-dimensional 1/n boxes is about n. And we’re going to keep making n, which is 4, from my example, get bigger and bigger and bigger. So once it’s one over a million, each of those little things is one over a million in length – whether you’re covering a million of those or a million + 1, probably doesn’t make a really big difference.
What if we take that same thing we think of as one dimension, and we try to put it in a two-dimensional grid? Well, we could put it there so it hits exactly four of those boxes. So my side, ¼ squares – I need four of them to contain this thing – I could shift it a little, so now it needs 5. I could angle it a little. See, if I count there, 1, 2, 3, 4, 5, 6, it needed. So that went up a little bit. But if you think about it, no matter how I put this thing in here, absolute worst case scenario is that it would go down n and over n. Of course it won’t make it that many, right, because it’s still only length 1. So I can’t go length 1 down and length 1 over. There won’t be enough for the diagonal segment that I put in there. But that’s kind of worst case scenario, right, so it’s as, again, as n shoots off really high – like once I have 1,000,000 by 1,000,000-side boxes that I’m covering this with, at most I’ve got 2n of them. So this is still n to the 1st power, right. It’s that exponent that we’re going for.
Suppose we put that same segment in a 3D grid, okay. So I only built one layer of this — so like here, the dark square would be the front of the cube, and the light square back here would be the back of the cube. And then here, this side, the edges — can you kind of picture those? So this is like one layer of cubic subdivided space. Well again, I’m not going to try to put it there, so I don’t think we can really see it. But if you put this thing in here, it could go down as far as 4 cubes. These’s your ¼, in their sides, it could go over 4, it could go back 4. It’s certainly not going to do all of all those, because it runs out of length. But basically, worst case scenario, you get n + n + n of those boxes that you’re hitting. So that’s 3n again. Notice the exponent is 1, and when n is super, super huge, the 3 isn’t going to make that much difference either.
All, right, what if we try with something we think of as two-dimensional? So a square and its interior. I’m going to try to use these same ¼ boxes, right, these one-dimensional boxes, segments of length, ¼. Well, there’s no way I can put enough of them in there to cover the square. It’s just not going to happen. So that’s because the square is two-dimensional, not one-dimensional, right. It won’t fit in a one by in a one-dimensional grid. But if we get a two-dimensional grid, now it fits. And it could cover exactly 16 of those little squares, or could cover a little bit more if we move it, shift it around. Like this one is – worst-case scenario, you’re covering like 2n² of […] but notice a square coming in there. If I put it in the cubic grid, now it could go down 4, could go over 4, it could go back 4. Worst case, you’re covering 3n², you’re not going to pick up a whole cubic number of these things. So again, that exponent is the important thing.
All right, done with that. But let’s think about this applied to something like the Koch snowflake. So here’s the idea, I start with a big grid and I count how many boxes – 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30 – how many boxes? I’ve lost track – I think 14, have some of the snowflake in it, right. So this one counts, actually, because it’s got a little, but this doesn’t count, that doesn’t count. And so I count those up, and then I do it with a smaller grid. So I’m letting my n – the denominator of the side lengths of these boxes – shrink down. So n goes up, and then the side goes down. And I have one of these over here of course, too, and down there. And I count again the boxes that I hit, and then I put in a finer grid. And now I count the boxes that are hit. So we do this finer and finer and finer until we start to converge on an exponent in the number of boxes that we get, and that’s where we came up with 1.2619 for the fractal dimension of the Koch Snowflake.
So let me put this in the context of other things that you’re more familiar with. If we were doing this idea with the state of Wyoming, this is one of the simplest state boundaries there is, right. Three sides of it are just totally straight lines, and the fourth one has a little bit of river action going on. Like, I think this is the Yellowstone River. And so maybe something a little odd goes on there. But for the most part, that’s not going to affect things very much. And so basically Wyoming has a one-dimensional boundary.
Well, what if we look at South Africa? What’s the dimension of South Africa’s boundary? It’s got some smooth parts but it’s got a lot more kind of craggy parts. So there’s more interesting stuff going on here. So someone blew up the map and put on these grids on it, and got the grid smaller and smaller and smaller, and figured out that it’s about 1.02 dimensional. So pretty close to one-dimensional, but not exactly. What about Britain? Wow, that’s a craggy map. There’s all kinds of inlets and just stuff, all the promontories, different things that are going to affect the dimension. So again, this was computed as about 1.25-dimensional. So it’s not two-dimensional, the edge, but it’s more than one dimensional somehow.
Okay, what about the surface of a head of broccoli? Now, surface – this is counting each of these little, you know, leaf things as part of the surface. So I’m not talking about the whole clump, but just the the boundary part of it, not the insides. And yeah, that’s 2.66-dimensional. So that’s getting pretty close to three- dimensional.
What about the human brain? So again, just the surface, not the whole thing with the surface, the outer coating. But the outer coating covers all those little nooks and crannies, right, they’re kind of crammed in against each other. So it turns out this is 2.79-dimensional. Now, this is kind of fascinating. So it made me think about neurodegenerative diseases, what do they do to the fractal dimension of your brain. Our brains are pretty efficient – in fact, it’s said that we don’t use anywhere near all of them. Well I looked this up – I have not read this article, but just the abstracted article, it’s really fascinating. And they noted, among other things, that – and I can’t remember now which one is which – one hemisphere of our brain increases, its fractal dimension increases, until adolescence, and then it starts to decrease. The other side increases until adulthood and then it starts to decrease. The fractal dimension of gray matter in the brain decreases in Alzheimer’s patients, but increases in MS patients. The fractal dimension of the white matter in the brain decreases in both ALS and MS patients. So this could, at some point, be a diagnostic tool for helping doctors understand what they’re dealing with in patients. Again, smaller fractal dimension, less efficiency.
How about the surface of the human lung? So here’s an image – you know the artist can’t draw all the little air sacs and things with as much care as they could, but anyway, because of the limitations of the picture – but think about all those little air sacs in there and how they are pulling oxygen out of the air and putting it into the blood. Well, that surface it turns out to be – wow – almost three-dimensional. And that’s why they’re so efficient at their work. But again, you get a lung disease, and it’s going to decrease that fractal dimension.
Okay, what about higher dimensions? Well, as I’ve said, we can add on extra coordinates. That works. Physicists – some physicists, anyway – speculate that there might actually be up to 11 dimensions of the world that we live in. Some are wound so tightly up that we can’t perceive them. This is actually part of String Theory, in fact. The 11th dimension is proposed by superstring theorists. This is not an area of mathematics that I really know anything about, but I’m going to take them at their word for that. We have long talked not only in science, but in popular culture, about the space-time continuum. Might our sense of time really be our perception of a fourth dimension? In fact, time as a fourth dimension is one of the 11 that physicists talk about.
Might God be somehow outside time, so that God’s experience of time is not the same with ours? Well, this is not inconsistent with the Bible. Isaiah 57:15 says that God is eternal, Psalm 90:2 says that “God is from everlasting to everlasting.” Even C.S. Lewis got into this, and suggested that “All times are eternally present to God.” That perhaps it’s true that “God forever” – here’s how Lewis put it: “It sees you in the nursery pulling the wings off a fly.” I’ve not been able to forget that since I read it. He goes on and suggests other things too. Now I think, in my childhood, I never pulled the wings off a fly, but I certainly did other horrible things that I don’t like to think about God seeing all the time. I kind of like to think He’s focused in on the positive aspects of my life. But this makes me think about forgiveness in a different way, and the importance of forgiveness, and the fact that God’s ability to forgive is so different from my own.