My differential equations professor told us this during the first class session and it has reverberated ever since. Mathematics beautifully describes the world around us. Of course we can also describe the world through other means such as expressing the colors and odors of a flower or the pleasant sound of birds singing in the mornings through words but sometimes there's just no words to describe what we see. It's like the old saying 'a picture is worth a thousand words'. However, mathematics takes this much further, to the point were we cannot even completely visualize the concept yet it has a direct influence on us. There exist concepts in mathematics for which words in English or in any other language cannot capture their essence. The only way to express such concepts and ideas is through mathematics. One of the first times I came across such a concept was in my calculus 3 class. The professor was demonstrating how to apply multiple integrals and their relationship to dimensions when he told us the following story (based on the book Flatland by Edwin A. Abbott).
Imagine you live in 1D world. There is only 1 dimension in your world so you can only move in one of two directions, forward or backward. One day when you're strolling about you hear a voice call out to you, 'Hey, how you doing?' You hear this voice very clearly but you cannot see who is talking. Perplexed, you ask 'Who are you?'
The response, 'I'm a circle! I exist in a 2 dimensional world!'
'2 dimensions?!? There's no such thing as 2 dimensions! You can only move in two directions.'
'Just because you cannot imagine what 2 dimensions look like doesn't mean it doesn't exist. Let me show you who I am. I'll pass by through your dimension so you can see me.'
In front of you a point appears. The point turns into 2 points right next to each other. Then the 2 points start distancing from each other until all of a sudden they stop and start approaching each other. Eventually the 2 points merge into 1 point and the point disappears. Afterwards the circle exclaims 'I just passed through your dimension. Did you see me?'
Now imagine you live in a 2 dimensional world so you can move in 4 directions, forward, backward, left, and right. Out of the blue a voice exclaims 'Hey there, I'm a sphere!'
You respond 'What is a sphere?'
'I'm like a circle but in 3 dimensions. Let me show you.'
Out in front of you a point appears. The point then turns into a tiny circle. The circle starts growing larger and larger. Suddenly the circle starts shrinking until it turns back into a point and disappears.
Now back to our 3 dimensional world. From our perspective we can see that the points that appeared in the 1 dimensional world are 1 dimensional slices of the circle. The circles that appeared in the 2 dimensional world were 2 dimensional slices of a sphere. Now take this a step further into our dimension. How would a 4 dimensional sphere look like? Obviously we can't imagine the direction of a 4th dimension. We only know 6 directions (forward, backward, left, right, up and down). However, we can see 3 dimensional slices of the 4D sphere passing through our dimension. How would that look like? Let's imagine such object passes through our dimension. In the previous 2 cases the object started as a point then turned into the highest dimensional representation possible in that world. In our case such representation would be a sphere. Thus as the object passes through our world we would first see a point. The point would expand into a tiny sphere. That sphere would keep on growing and growing until suddenly it reversed direction and began shrinking back into a point. Eventually the point would disappear. Obviously such visualization does not show us the entire 4 dimensional sphere but the math behind it can still provide us information about such object.
Going back to high school geometry and algebra 2, the equation of a circle is `x^2+y^2=r^2` where x and y are the center's coordinates and r is the radius. Extending this to 3 dimensions the equation would be `x^2+y^2+z^2=r^2` where x, y, and z determine the center's coordinate in 3 dimensions. How about 4 dimensions? Just add another variable, `x^2+y^2+z^2+w^2=r^2`. This time x, y, z, and w determine the 4 dimensional coordinate of the center and still tells us the distance to the edge of the object in all 4 dimensions. What about a 4 dimensional cube? Let's start in 1 dimension which would simply be a line segment. How many endpoints does a line segment have? Only 2. Going to the second dimension how many endpoints does a square have? 4 endpoints, each one being a corner of the square. In 3 dimensions, a cube has 8 endpoints. Comparing each one side by side:
| Dimensions | Endpoints/Corners |
As can be seen from above the number of corners in a square/cube of any dimension can be represented by the expression `2^n` were n is the number of dimensions. Thus a 4 dimensional cube would have `2^4=16` corners. Can we depict them pictorially? We can try our best but we can never fully depict a 4 dimensional cube in 3 dimensions. Information will be lost when creating any physical representation of higher dimensional objects. The expression from above though provides us with information about such cube which we could only find out visually otherwise. All of this seems like a nice thought experiment but how can it be true? It's just a thought experiment after all, higher dimensions are simply figments of our imagination, aren't they? Absolutely not. Einstein's general theory of relativity depends on higher dimensions.
General relativity states gravity is the curvature of space-time (time is an additional temporal dimension directly connected to our 3 spatial dimensions). How can space-time be curved? Let's bring this down to 2 dimensions with a sheet of paper. Draw a triangle on a flat sheet of paper. The sum of the internal angles will be 180 degrees. Now fold the paper in such a way that it makes an arch. The angles still add up to 180 degrees but the sides are not straight anymore. Drawing another triangle with straight sides and adding up the angles will return a value larger than 180 degrees. From the viewpoint of an ant on the sheet of paper it can't directly tell if the paper is curved but it can measure the effects of any curvature on the paper (such as adding up angles of polygons). The curvature can only happen if there exists a higher dimension. Thus in the 3 dimensional case the curvature occurs in the 4th dimension. We can't directly see such curvature from our perspective but we can measure its effects, which turns out be the gravitational force between objects.
Trying to fully visualize higher dimensions pictorially is outright impossible since we have no concept of additional directions apart from our 3 dimensions. Despite this, mathematics provides us a way of retrieving information from higher dimensions. Dimensions are just one example of how mathematics can provide information about the unimaginable. Carl Sagan provides a brilliant illustration of the same story in the video below.