Chapter 1: Calculus III, Pringles Cans, and Extradimensionality(C) 2012 Michael Belote, All Rights Reserved. Appears originally in Rise of the Time Lords: A Geek’s Guide to Christianity (ISBN: 978-1-300-02022-6)
When I first went to college to be an engineer, I thought it would be easy. Maybe this was naïve of me, but I honestly thought I was pretty awesome stuff. After all, I had breezed through high school with a 4.2 GPA on a 4.0 scale (which may seem a bit mathematically suspect, but that’s another topic for another day). I was the best in my class at AP chemistry and physics. As a junior I made a 31 on my ACT, good enough for a full ride plus some spending money, so I never even bothered to re-take the exam as a senior (when I undoubtedly would have done even better). I arrogantly thought that I could ace my way through engineering without ever missing a Razorback football game.
So as I sat down in Calculus I on the first Monday of college, I was feeling pretty good. Until our teacher decided to prove a point. The teacher had us raise our hands if we were math, engineering, or physics majors – it was everyone in the class. She then had us keep our hands raised if we were honor students in high school – again, the whole class. She had us keep our hands raised if we were Chancellor’s Scholars – perhaps two thirds of the class still kept our hands up. Then she had us keep our hands up if we had at least one year of calculus before this year – about half of the class kept their hands up…and mine went down.
You see, I had opted to skip Calculus as a senior in high school. I was a bit of a math whiz, and I was already taking AP Physics, so I figured why not take it easy? I had fulfilled my math requirements, so I might as well enjoy my senior year, right?
My first day in Calculus, I began to regret that decision. Because for the first time in my life, I was surrounded by an entire classroom of people equally as bright as I was, and half of them had the advantage of already knowing the material.
The teacher told us that the rest of us would struggle just to get by. Around midterm, she would start teaching us how to do the Calculus shortcuts; until then we had to do everything by long-form, while those who already knew the shortcuts could check their work more easily than the rest of us.
She was right, it was far more difficult than I had imagined. I fought and struggled to get a C. I had never even flirted with a B in my life. My first semester (since I had arrogantly taken 17 hours of hardcore engineering and math classes), I narrowly kept a 3.0 to avoid scholarship probation. It took everything in me to keep my scholarship in my second semester.
By the time I reached Calculus III, though, I had found my groove and was excelling above most of my peers. I found the subject fascinating. We started studying multivariate calculus. In particular, we spent a tremendous amount of time on studying the mathematics of complex curved surfaces like saddles.
Our professor (whose name, sadly, I cannot remember) was a jolly Scotsman who wore shorts and flip-flops regardless of the cold, and described everything using food metaphors. When describing motion along curved surfaces, he would always talk about his imaginary ant named Esmeralda, who would walk across the surface and describe her path. One day, while confusing us with a particularly complex path for Esmeralda, he mentioned the book Flatland.
Never having heard of the book, I followed up after class and his description intrigued me. This very short book, titled Flatland: A Romance of Many Dimensions, was written by a teacher named Edwin Abbott in the 19th century and was used to describe the concepts of multi-dimensional thinking. Flatland is a two-dimensional world populated by various shapes from geometry class. The narrator, for example, is a square; women are line segments; soldiers are triangles; hexagons are nobility, etc. The circle is the ‘perfect’ shape, and so the more one approximates a circle, the higher his social status.
Within Flatland, they only have access to two dimensions, which we would call “x” and “y.” So when they speak of something being “up,” they cannot conceive it as being “out of the page,” as we would; instead, “up” means, “toward the top of the page.” Because of this, all of their science and mathematics are limited by being trapped within these two dimensions.
For example, a Spacelander (like you or I) can see the entire “page” of Flatland all at once. When we view the narrator, we clearly see that he is a Square. Further, we can see both inside of him (into his “guts,” as it were) and outside of him (his “skin” being the four lines that make him a square), at a glance. We can also see the nature or shape of every other being in Flatland at a glance. Why? Because we can look down from our third dimension, and thus see everything in their two-dimensional world from a superior vantage point.
A Flatlander, however, cannot at a glance tell who they are looking at, because they are limited by being inside Flatland. So if someone in Flatland walks up to Mr. Square, all the observer sees is a line—whichever side of Mr. Square is facing him at the time. It is impossible for him to know whether he is viewing Mr. Square, or a Triangle at an angle, or a Line from the side. The only way to know is to approach their Flatland neighbor and feel them physically while walking around them. Then and only then can it deduce the form of the other person. (I would imagine that this makes detective work rather difficult on Flatland: “No officer, I was not molesting that woman! I just thought she was my wife, and I had to feel all over her to know for sure.”)
Yet, even a two-year old Spacelander can glance at the page and say, immediately, “That is a Square.” Some knowledge which would be unobtainable by even the wisest Flatlander is simple and obvious even to a fool or a child from Spaceland. Being three-dimensional creatures, we have a much different perspective of their world than the Flatlanders do.
The book Flatland is completely fascinating and well worth a careful read for any geek. It offers an intriguing description of our world today. With the advances of modern physics, we now believe that the universe is not simply the three-dimensional world that we can perceive: we appear to be three-dimensional beings living in a universe of at least four and possibly as many as twelve dimensions! So while we consider how laughably foolish the Flatlanders are at trying to understand our third dimension, we find ourselves in the same predicament when challenged to picture the fourth (or higher) dimensions.
Think about some of the weirdness of being confined to a lower dimension. Take out a piece of paper, and let’s call this your Flatland. (Seriously, do this, don’t just read about it. It will make more sense this way.) On your paper, draw several dozen shapes. Now let us imagine that each of these is a person.
My, what a powerful god you must seem to them! Not only did you create them with minimal effort, but you are omnipresent, for you are able to see and interact with all of them momentarily, even if they are on opposite sides of the world! You are able to erase one and redraw him in another spot instantly. You are able to see not only their outsides but also what is happening inside their skin. You can see their true nature (shape, circle, triangle, etc.) at a glance, with no “feeling” investigation.
Imagine the miracles you can do for this two-dimensional world. Go grab a Pringles can from your pantry. (If you do not have a Pringles can, shame on you. They are delicious. Grab some other cylinder and it will have to do.) Place the can on your Flatland. When they examine this gift, what do they see? Draw around the parts touching the paper and you see what they see: a circle – because this is the only part which crosses through their page. As two-dimensional beings, they can only perceive of a single flat cross-section of your Pringles can at a time.
Now, flip the can on its side. Trace the edges which touch the paper again to see what they see. They now see a rectangle—the height and width of the can.
You see, a lower-dimensional being (like a Flatlander) cannot possibly “see” all of a higher-dimensional being (like a Spacelander). All they can see is the cross section which touches their world. If we step into their world they see not a boot but a footprint.
Now let’s have some real fun. Pick out one of your shapes to be your “prophet” to Flatland; this is the shape who will try and explain about you to the other shapes. Try and think of a way to explain to the Flatlanders that these two drawings of the Pringles can are, in fact, the same shape: both a rectangle (with which they are familiar) and the circle (with which they are familiar) are actually the exact same object. They will of course think you are crazy! It is a fun thought experiment: get a friend or spouse or yourself (if you lack friends and spouses) and one of you pretend to be the Flatlander prophet. Try and explain to the prophet, using only words which they understand, that the Pringles can is both the circle and the rectangle simultaneously. It is impossible. They end up taking the whole silliness on faith, or rejecting you as a lunatic teaching fairy tales.
+ + + theology: the Trinity + + +
As we geeks begin our journey into understanding some of the fundamental theologies of Christianity, we should begin with trying to understand something about God Himself. From the very beginnings of Christian theology, there has been a battle over the nature of God. The Old Testament was as clear as it could be that there was only one God. God hammered this concept into the Israelites for centuries. Then along comes Jesus. He is the best Jew anyone has ever seen, and does amazing miracles to boot. He claims to be the Messiah, the savior prophesied in the Old Testament. In front of a few of His disciples, He transfigures miraculously; later He is put to death for blasphemy because He claims equality with God. After He dies, His body mysteriously disappears and hundreds of His disciples, from all areas of the countryside, claim to see Him again. Then He tells the disciples that He is sending them God’s Spirit, a being and person, to live within them and advise and counsel them.
So from the start, Christians have had this very strange view that God was three unique persons—God the Father (the Jewish Yahweh), God the Son (Jesus, God in the Flesh), and the Holy Spirit (present at Genesis 1:2 and later dwelling inside believers). This was really, really confusing. And it caused a lot of debate in the early church.
In fact, this is one of the most common things that the first Christian creeds were written to clarify. Century after century during the Roman Empire, Christian leaders all gathered together, debated, and issued creeds stating what Christianity believed. What these bishops came up with is something like the following definition:
1. There is only one God.
2. This God has three Persons making it up: Father, Son, and Spirit.
3. These Persons have identical natures; all are equally God.
4. All are of the exact same substance.
5. All are co-eternal and co-equal.
Quite a few people throughout history, upon seeing this list, have had reactions ranging from, “Say what?” to, “You’re insane.” Skeptics call the above logically impossible; Christians call it a mystery.
I agree that it is a mystery, but not nearly as much of one as we may think.
Let us return to our Flatland for a moment. We all remember that we Spacelanders, as three-dimensional beings, have different perspectives from our two-dimensional creations. We see things that they do not see. So when we look at our Pringles can, we can see that this shape (which we engineers would call a “three dimensional solid”) is neither a circle nor a rectangle, as the Flatlanders think of them. Rather, to use engineering terms, we would say that the can is a circular cross-section which has been extruded into the third dimension, resulting in a single, coherent solid. If you take a two-dimensional cross-section you see a circle; if you take a two-dimensional slice along the long axis, you see a rectangle.
This is demonstrable to any Spacelander, and quickly obvious to any engineer who has had to sit through an engineering graphics class and draw blueprints. In engineering graphics, we represent a three-dimensional solid on our prints using a top view, a side view, and a front view. By looking at these flat drawings and using our understanding of the third dimension, engineers can picture the Pringles can and instantly see how it is both a circle (top view) and two rectangles (front view and side view) while still being only one object.
But try explaining this to a Flatlander, after they examine the blueprint. What they will see is one circle and two rectangles (side and front). When you tell them that this is all one object, they will think you are crazy. In the end, after talking to you, they will end up summarizing your ridiculous claims with some list like this:
1. There is only one Pringles can.
2. This Pringles can has three Persons making it up: Mr. Circle, Mr. Front Rectangle, and Mr. Side Rectangle.
3. These Persons have identical natures; all are equally the Pringles can.
4. All are made out of the exact same substance.
5. All came into existence at the same time and all are equally part of the Pringles can.
6. Pringles are delicious.
Sounds familiar, doesn’t it?
We, too, are trapped in our dimensional perceptions. We are three dimensional creatures, capable of full perception of up/down, left/right, and in/out. We also are aware of a fourth dimension, time, but only vaguely: we can only experience one moment in time in our minds. So just as the Flatlanders are two dimensional beings who, at any given moment, can experience only one cross-section of the Pringles can (a “three dimensional moment,” if you will), so too are we three-dimensional beings who can only experience one cross-section of the fourth dimension (a “fourth dimensional moment”). That is why we can feel and experience “now,” but cannot feel or experience the past or future. Like the Flatlanders, we only have access to a single cross-section from the next-highest dimension.
And yet, we find from the Bible that God created this entire universe, including time. So God is something more than a four-dimensional being. It is only natural, then, that we will be just as limited in understanding God’s universe (let’s call it Timeland) as Flatlanders are limited in understanding our Spaceland.
So when Christians speak of a Trinity, think of your Pringles can, and how it is at every moment both a circle and two rectangles. You cannot remove the circular cross-section or it is no longer a Pringles can; nor can you remove the rectangular extruded length and maintain its identity as a can. It must always be both at the same time, or it ceases to be a “can,” by definition. This confuses Flatlanders because of their perspective: they cannot see anything but a two-dimensional cross-section, so it seems ridiculous to them to speak of something more than two dimensions. After all, their “nature” is two dimensions: anything more, by definition, is “anti-science” because it is unobservable and, therefore (by their definitions) supernatural.
Yet Pringles cans exist, don’t they? They are actually real in the natural world and scientifically observable. It is not that the Pringles can must be accepted by “faith”: no, it can be directly observed and proven…if you have access to the correct perspectives and dimensions. But since their perceptive abilities are limited by their access to two dimensions, the Flatlanders of course must use some form of faith combined with evidence in order to accept the very-real Pringles can.
How foolish would a Flatlander scientist seem if he pridefully refused to consider the teachings of the Pringles priests, because they referred to knowledge outside of his two dimensions of perception? Is he doing something admirable by self-imposing a handicap which limits his ability to understand reality? No! It is the upmost foolishness to say, “Since I cannot measure it, it does not exist.” But of course, let us not be too hard on the Flatlander scientist, because here in Spaceland scientists often do the same. Saying that they are only interested in “natural” things, they reject any branch of knowledge which cannot be based entirely upon things available to our three dimensional measurements. As a result, they are handicapped and guarantee that they will never see the “entire” truth. The best that they can do is develop a reasonable theory to explain the three dimensions which they can directly access, just as the best Flatlander scientists can do is explain the two dimensions which they can directly access.
Ultimately, though, the Flatlander has good evidence of circles and rectangles; they have good evidence that miracles are going on, with circles and rectangles changing into one another; and they have a word from a Spacelander, from which they derive their strange doctrines of this “Pringles can” being circle and front rectangle and side rectangle, all at once, yet at the same moment retaining the identity of a single solid object.
We are in the same situation as Christians. We are told that God is somehow to us like the can is to Flatlanders: God is Father and Son and Spirit all at once, distinct persons but one Being. We cannot comprehend this—not because it is incomprehensible, but because it is simply outside of our perspective and capacity for observation at this time. As three-dimensional beings, it is impossible for us to picture a fourth-dimensional being; much more so, a fifth (or whatever God is). Only the greatest of mathematicians have even the foggiest understanding of multiple dimensions above our own, and theirs are abstract comprehensions only, which may have no basis in reality. For someone to actually attempt to describe a five-dimensional being…well, that is just as impossible as describing the concept of a “can” to a Flatlander.
So can I describe the Trinity? Not really, no more so than anyone else. But what I hope I have shown is that just because a higher-dimensional Being is unexplained does not make Him unexplainable or untrue. Indeed, if God is a being of greater dimensions (and He is, based on the Scriptures and the logic that He must be greater than this universe He created), then we know that He could only reveal Himself to us in three-dimensional parts and concepts at a given moment. Like the Flatlanders viewing the Pringles can, we can only see one “cross-section” or one Person of God at a given moment.
And this is what we see in our Bible, as it turns out. Some people (e.g., Adam, Eve, Enoch, Moses, and the Prophets) interact directly with the Father and have experiences with Him as Lawgiver and Covenant-Maker and loving creator of mankind. Other people (the disciples, John the Baptist, Pilate, etc.) have experience with the Son, seeing a flawless Jew who does mighty miracles and dies to reconcile us to God. Yet others (the Apostles, Paul, church members today) interact with the Holy Spirit inside of them, guiding them through divine inspiration. Because God is greater than our dimensions, our three-dimensional world cannot hold all of Him any more than you can make a Pringles can fit completely inside Flatland: it is a logical impossibility, for you cannot take a higher-dimensional object completely into a lower-dimensional reality without destroying what it is.
The best that we can do for understanding the Trinity is the same exercise that the Flatlanders must go through: we study each of the individual Persons (Father, Son, Spirit) to the best of our abilities; and we fuzzily understand that these three beings connect together in some fundamental way.
In some way that our three-dimensional minds cannot possibly ever picture, these three unique Persons are at the same time one single Being, one God, indivisible yet still individuals.