Trillion: the new billion

To celebrate this post being the 100^th ever published on this blog, I would like to spend today’s semi-humorous discussion talking about numbers. An interesting BBC article piqued my interest on a number-related subject, and I would like to share what I read. I am pleased to report that “trillion” has surpassed “billion” as the really big number that people have a hard time grasping. One trillion, also known as a million millions, was once a number only astronomers used. But today we find ourselves using it more than ever, and not usually in positive ways: take the US national debt, for example: $11,000,000,000,000. Or the national deficit: $1,750,000,000,000. And remember when “millionaire” signified a person with unlimited riches? It turns out that there are over 10 million millionaires in the world today, making the term somewhat obsolete.

But the problem with this is that the human mind has some difficulty grasping the sheer magnitude of one trillion. “Trillion” is not a number we use in our daily lives, and until recently it was barely spoken of at all. As Kevin Connolly of the BBC puts it: “It is hardly surprising that politicians and pundits on American discussion programmes often jumble up the millions, billions and trillions - we are still in the process of adjusting to a frightening order of magnitude.” Trillion is just such an astronomical number that it is very difficult to use casually, and it makes it very hard for us to come to terms with concepts like the national debt.

Is there another way to look at a trillion to make it more understandable? Possibly: a trillion is a million millions, or a billion thousands, or 10 to the 12^th power. But this doesn’t really convey the awesome size of one trillion—here is a more practical example: Imagine you have a trillion dollars, and you decide to spend one billion dollars every day. It would still take you 1000 years to spend all of your money. This is why I stand in awe whenever the government is talking about recovery plans that require trillions of dollars—it’s easy to forget just how much one trillion is, and when we are reminded it often comes as a shock. 

What’s next? The quadrillion, apparently, which is a billion millions or 15 zeroes. I admit, I had to look up the next term, quintillion, which has 18 zeroes. But hopefully inflation (or the national debt) will not force us to use those numbers anytime soon.

More 4D mazes

Below are some more 4D mazes I made; I hope you will enjoy solving them. For the instructions, see one of my previous posts, “4D mazes.”

This first one is pretty hard: 

This one is even harder: 

I think this last one is the hardest of all, but perhaps you will find it easier: 

Anyway, I hope you have enjoyed these puzzles, which I find fascinating. Even if you do not understand the concept of the fourth spatial dimension these mazes are still interesting logic puzzles.

4D mazes

For today’s second post, I would like to talk about four-dimensional mazes. In a six-part series I wrote in January, I explained a few aspects of the fourth spatial dimension. This post is not a continuation of this series—it is more like a supplement to it.

Before I begin, though, I would like to clarify some points about mazes in general. When we see a maze drawn on paper, it is a two-dimensional maze. It utilizes the available 4 perpendicular directions that the second dimension possesses. When we see a physical maze, like those “corn mazes” out in the Midwest, this is also a 2D maze because it does not utilize all 6 directions of the third dimension. Thus, a true 3D maze would be a building where each floor contains a 2D maze, meaning that it is several 2D mazes stacked on top of each other. Likewise, a four-dimensional maze would consist of several three dimensional mazes stacked on top of each other. Surprisingly, such a maze can be drawn and solved. The maze below is one I created as an example.

This is how the maze works: Each 3x3 box is a two-dimensional slice of this maze. Each column is a three-dimensional maze, since all 3 of the boxes in each column are stacked on top of each other. Thus, the whole picture is really all three columns stacked on top of each other. But this is impossible to visualize, so it is better to explain it this way: the three boxes in each column are stacked, and all 3 boxes in each row are stacked in the same way. This means that it is possible to move from one box to the one above it, below it, or next to it.

The maze is played like this: begin at the yellow square and try to reach the green square. You can move into any adjacent square above, below, or next to you (no diagonals) that is not a wall (black square). You can also “jump” to the corresponding square in the box above you, below you, or next to you because they are on top/below the box you are in (again, no diagonals). You cannot do so if a wall is in the square you are trying to jump to. For example: on your first move you can either move to the square to the right or below or jump to the upper left corner in the box to the right or below. 

Of course, much harder mazes, such as 4 by 4 by 4 or larger ones, can be created as well. Perhaps in the future I will make a few more and put them up on this site. If you are still unclear as to how the maze works there are some articles on the internet that may be able to explain them—this idea was based off of similar puzzles I found on the internet. 

From Flatland to the fourth dimension (part 6)

Sorry, “law and order” series—this set of posts about the fourth dimension is now the longest, at 6 posts. Today I will cover the shape of the universe, wheels, and paper in four dimensions. Again, if you have not read the previous posts in this series, with the exception of #1, today’s post is going to seem even more confusing than it already is.

First, I would like to go into the shape of the universe in relation to the fourth dimension. Though there are several attempts to explain the shape of the universe, none of them have been completely proven, and the universe’s true shape is still unknown. It is possible that even if the universe is three dimensional is it “curved” four dimensionally, such as into the surface of a 4D sphere of cube. This seems difficult to visualize, but try an analogy: imagine a 2D ant living on the surface of a piece of paper. The paper can be curved, bent, or even folded but that ant would not know about it. Likewise, our universe can be curved in the same way. However, this might have some effects on exploration of the universe or our attempts to survey it. For example, our universe could be shaped like the surface of a 4D torus (doughnut). This is very hard to visualize, but think back to the analogy for a moment: consider what would happen if the 2D ant’s world were curved into the surface of a doughnut. The ant could walk in a straight line and actually come back to where he started (in several different ways—look at the shape of a torus).

Next, paper. In 3D paper is actually 3D as well, but for practical purposes it is 2D because we do not right on the “sides” of a piece of paper—not enough room. In 4D, then, paper is effectively three-dimensional. On a square 2D piece of paper, there are three paths from one corner to the others; on a 3D one there are six. Also, there are only four ways to cut a piece of paper symmetrically in half in three dimensions—in four dimensions, there are 8.

Lastly, wheels. A wheel in four dimensions is similar to a 4D torus, similar to the way a 3D tire resembles a torus. The area of a 4D tire that touches the ground, unlike a 3D one, is a plane; this means that if a 4D tire were rolling across the surface of our 3D world it would appears as if a thin rectangular prism were moving across the room. A 4D wheel as four directions to fall over, unlike a 3D tire, which only has two. Also, a four dimensional spherical tire (the 4D equivalent of a cylinder) has similar properties to a 3D cylinder—it has two directions to fall over into, but four to roll into. I realize all of this is almost impossible to visualize, but give it a try anyway.

This concludes the series on the fourth dimension for the time being. Tomorrow I would like to talk about some pressing current events that have happened over the past week.             

From Flatland to the fourth dimension (part 5)

I find it extremely amusing that this series, which is now tied with the “law and order” series for the most posts, is about math. I am by no means a math-oriented person. However, as I have said before I find this topic fascinating, and for that reason I feel it deserves this many posts. But I’m digressing. The point is: today’s post will cover the following in topics in four dimensions: water and islands. If you have not read posts 2, 3, and 4 in this series, please do so before reading this one.

First, water. In all dimensions water fills up its container, but the nature of the container is obviously dependent on the number of dimensions. In two dimensions, in order to create a water-tight container the vessel’s surface is a line and the “seal” is two points. In three dimensions, a water-tight container’s surface is a plane (like the surface of a bowl) and the “seal” is a line (imagine saran-wrap or tinfoil stretched over a bowl). In four dimensions, the container’s surface must be 3D and the seal a plane, since the water has so many directions to escape. This is difficult if not impossible to visualize, but try to picture it. Along with this is the concept of surface area: in 2D surfaces are lines, in 3D they are planes, and the 4D they are three-dimensional. This is almost certainly impossible for us to visualize, but think about it like this: imagine a warped sphere, full of curves and indents…this what a body of water in zero-g in 3D would look like. But in four dimensions this is only the surface of the water…the rest of the water is sloshing about in the other perpendicular directions. And, if the water surface is not four-dimensionally flat, we wouldn’t even be able to see all of the surface (this is really hard to visualize—try to use the 3D-2D analogy).

Along with the concept of water go the concepts of river and bridges. In two dimensions, a river moves from higher elevation to lower elevation, but it covers the entire ground surface and cannot be crossed with a bridge (unless the bridge spans the entire length of the river). In three dimensions, rivers still move from higher elevations to lower ones, but they and they zigzag depending on the terrain. A 3D river can be crossed with a bridge. In four dimensions, a river is very similar, but there are two key differences: a river in 4D would corkscrew, and there is no need for a bridge—one could simple walk around a river.

Related to rivers is the idea of lakes. In 2D, there is only one kind of lake. In 3D, though, there are two kinds: linear and round. The surface of a linear lake, as the name implies, is thinner, like a river, but a round lake is more circular. In 4D, there are three kinds of lakes, linear, flat, and globular. The surface of a linear lake is like a cylinder; the surface of a flat lake is flat, and a globular lake has a spherical surface.

Next, and somewhat related to water, is the concept of 4D islands. First, though, let’s look at an island in two and three dimensions. In 2D, the surface of an island is a line; to “search” all of it, one must simply move from one end to another. In three dimension, land surface is a plane. To search a 3D island, one must move in a 2D grid to cover all the land area. Like the surface of water in four dimensions, the surface of land is three-dimensional. To search this surface, one must move in a 3D grid along 3 axes.

The series on the fourth dimension will continue, unless for some reason I grow bored with it all of a sudden. 

From Flatland to the fourth dimension (part 4)

For the past two days I have been talking about the fourth spatial dimension and its implications. Today, the series continues with four-dimensional levitation, rotation, rolling, and truncation. If you have not read parts 2 and 3 of this series I strongly recommend doing so, and reading part 1 wouldn’t hurt either. Otherwise, today’s post is probably going to be very confusing. 

In part 3 of this series, I explained how a four-dimensional being could touch any part of a 3D object, even the inside. Today’s first topic, levitation, is related to this. In three dimensions, it is possible to attach a 2D plane, like piece of paper, to a wall with a nail. Consider hanging up a 2D of a square: the square would rest on the 3D nail, which would perpendicular to it. Now, imagine the same thing in four dimensions with a “tetranail.” To us, the cube would be suspended in the air, held by this “tetranail,” which we would see only as a hovering sphere. 

Next, I would like to discuss the rotation of 3D objects in four dimensions. Before I get to this, though, I would like to explain rotation of 2D objects in three dimensions. The key here is that an object of n-1 dimensions is being rotated in n dimensions on an n-2 axis. Consider a basic example: a square is rotated on its axis around one of its sides. This means it must be rotated out of its plane, which is different from just spinning in inside of its 2D plane. The latter means that the square is being rotated around a point; the former means it is being rotated around a line. To the “Flatland” observer, all but one line of the square disappears when the square is rotated out of the plane; when the rest of it returns, it appears to be reversed in such a way that spinning it will not turn it back to the way it was. (For example: if the letter “f” were imprinted on the inside of the square, after it is rotated the “f” would be backwards.) All this means that the four-dimensional equivalent would involve rotating a cube around one of its planar sides. The rest of it would disappear, and then re-appear on the other side of the cube. If letters were imprinted on the sides, they would be backwards. This is difficult to imagine—it would appear that it would be necessary to break the cube for it to do this, but this is not true. What we cannot see is that while the rest of the cube is “disappeared,” what it is actually doing is rotating around the remaining side on the 4D perpendicular axis.

I would also like to discuss 4D rolling rotation. Imagine a ball resting on top of a frictionless book. The book can be moved sideways in 4 directions, making the ball appear to “roll” on top of it. There are 4 perpendicular directions for it to roll. In four dimensions, though, the book would have a 3D surface, and the ball would be a glome (4D sphere). The ball now has six perpendicular dimensions to roll in—for comparison, recall that our 3D world only has 6 perpendicular directions! However, it is interesting to note that the area where the glome touches the book is still a point. (In 2D, a circle contacts with other objects at one point, and the same happens in 3D with spheres. So the same must occur in four dimensions.) Also, a 3D ball can roll in a circle, but a 4D ball has 3 possible circles to make. 

Next, I would like to talk about truncation in four dimensions. Truncation, or cutting off the corners of an object, is a simple geometrical procedure. In four dimensions, though, it has some interesting implications. In 3D, when a cube’s corner is cut off it leaves a triangle. In four dimensions, when the corner of a hybercube is cut off, it leaves…a tetrahedron. I find this incredibly mind-boggling, imagining the corner of an object to be a three dimensional shape (as if a hypercube wasn’t hard enough to visualize already!). In reality it is no different than what a hypercube’s side is normally, but I think this is an interesting way to look at it.

Tomorrow the series on the fourth dimension will continue. 

From Flatland to the fourth dimension (part 3)

Last time, I explained the concept of the fourth spatial dimension and a few applications of it. Today, I would like to discuss more of what the fourth dimension would be like. I advise against reading today’s post without reading yesterdays, and reading part 1 of this series is also advisable, though not as necessary.

Yesterday I explained how we can deduce certain aspects of the fourth dimension by analogy. This is precisely what I would like to do today: explore what it would be like to experience four dimensions by looking at what it be like to experience three after living in two.

I would first like to talk about four-dimensional beings. A four-dimensional being would obviously be in the shape of a 4D figure, and it would be able to move in all 8 perpendicular directions, at least in zero gravity (like the way we can move in all 6 directions in zero-g). This obviously hints at the structure of a 4D being—it must have at least 4 legs to be stable (humans are not stable—we would need 3 legs in a tripod). But, like human beings, it is possible to get away with less—but I doubt a 4D being could have less than 3 because there are now more directions to “fall” (fall over, that is). Also, a 4D being’s line of sight would be a cube. This is because a 2D being’s is a line (recall the explanation of how “Flatlanders” see in part 1 of this series) and human beings see 2-dimensionally. However, because we have depth perception we can distinguish 3D objects. But our field of view is still a plane—our vision is akin to a photograph; everything is reduced to a 2D image. Thus, a 4D being would see in terms of a cube; a 4D photograph would be three-dimensional. At first this sounds normal enough, but consider the implications: 2D vision grants us the ability to see all the sides and the inside of an opaque square at the same time—a 4D being would be able to see all the sides, and the inside, of an opaque cube at the same time. Therefore, from the correct vantage point, a four-dimensional being could look at a human and see every inch of his skin as well as the inside and outside of all of his organs.

Next, consider what happens when four-dimensional beings or objects interact with our three-dimensional plane. This, too, can be done by analogy. In Flatland, the three-dimensional character reaches in and touches the insides of a two-dimensional character; this is the equivalent of touching the inside of a square drawn on a piece of paper. A 4D beings would have similar powers, and would be able to reach inside of sealed 3D objects or beings.

Also, consider what happens when a 4D object is passed though our 3D plane. This, too, can be explained by using an analogy from Flatland. When a square passes through a two-dimensional plane, a 2D observer sees a line appear and then later disappear. The observer can only perceive only one “square” at time (think back to when I described a cube as a stack of paper squares). Likewise, when a sphere is passed through a 2D plane an observer would see a line appear, grow in size, and then shrink and disappear. (Again, think back to part 2.) Similarly, if a hypercube is passed through our 3D plane we would see a cube appear and then disappear, since we can only see one cube at a time. If a glome (4D sphere) was passed through our 3D plane, we would see a sphere appear and grow in size and then shrink back down to nothing again. This is a bit harder to visualize, but it is essentially the same: the glome is made up of an infinite number of spheres, but we can see only one at a time because of the way they are stacked. 

I would also like to talk about geometrical nets. A net is a shape of n-1 dimensions that can be folded into an object of n dimensions. The net of a cube, for example, is 6 squares in the shape of a cross; these can be folded into a box. Interestingly, the net of a hypercube is 8 cubes arranged into a 3D cross: 4 cubes are stacked vertically, and the other 4 are attached to the other 4 exposed sides of the cube second from the top. The nets of other 4D figures are also 3D figures. However, though we can make the nets, we cannot fold them, since they must utilize the four-dimensional 8 perpendicular directions in order to fold. Note that once folded, one cube of the net of a hypercube remains in our 3D plane; the others will appear to have simply vanished because they are outside of our plane. In an amusing short story by Robert A. Heinlein, an architect builds a building that is the net of a hypercube, and an earthquake causes the building to fold into the 4D shape. This is impossible, of course, but the concept is clever.

Tomorrow, I will cover even more aspects of the fourth dimension as extrapolated by analogy. 

From Flatland to the fourth dimension (part 2)

Before I begin, a quick word on inauguration day: All I can say is that I am proud to be here during this historic event. We should rejoice at the changing of the guard in US politics and at the fact that Obama’s victory represents a step in the right direction for race relations. I already talked about my predictions in “In 103 days…” and I hope that what I talked about will really come to pass.

Now, to business: yesterday I briefly mentioned the mathematical concept of the fourth dimension while discussing Edwin Abbot Abbot’s Flatland. Today, I would like to devote an entire post to this intriguing mathematical topic.

First, let me make one thing clear: I am discussing the fourth spatial dimension; today, the layman uses the term “fourth dimension” to describe time, in accordance with Einstein’s idea of “space-time.” But here I am using the term in reference to the hypothetical next spatial dimension, as I shall explain.

The best way to explain the fourth dimension is by progression and analogy. First, consider a line, which has one dimension. It has two perpendicular directions, north and south (also known as length). Now, “drag” the line in a direction perpendicular two the existing two directions, like rolling a pencil covered in paint across a table. The result is a square, with two dimensions. The second dimension has four perpendicular dimensions: north, south, east, and west (also known as length and width). Now, drag the square in a direction perpendicular to the existing four (imagine lifting a paper square with string attached to the corners straight up off a table). The result is a cube, with three dimensions. The third dimension (the one we live in) has six perpendicular directions: north, south, east, west, up, and down (also known as length, width, and depth). Now, this is where it gets tricky: move the cube in a direction perpendicular to all 6 directions. The result is a shape with 8 perpendicular directions and four dimensions. This fourth dimension is to us what the third is to second; imagine how a box differs from a piece of paper and you will get an idea of what the fourth dimension is to us. Can’t imagine it? Don’t worry: it’s technically impossible to fully visualize it. But don’t give up yet—using analogies, we can figure out many aspects of the fourth dimension.

By moving a square out of our three-dimensional “plane,” a figure called a hypercube is created. As I stated previously, this object is the 4D equivalent of a square of cube. I would like to discuss some of the properties of this shape by analogy. For example: a line has 2 vertices, a square has 4, and a cube has 8, so a hypercube must have 16. A line has 2 sides (which are its vertices), a square has 4, and a cube has 6, so a hypercube must have 8. But remember that a square’s sides are lines, and a cube’s sides are squares, so a hypercube’s sides must be three-dimensional cubes. Confused? Remember that the fourth dimension has 8 perpendicular directions. Think about how a cube can be created by stacking square pieces of paper on top of each other; likewise, a hypercube is essentially an infinite number of stacked cubes, placed on top of each other on the new perpendicular axis. Still having trouble? Look at the figure to the top right; this is a drawing of a hypercube. Can you see the 8 cubes? If not, try looking at the figure below to the left and try to see the 8 cubes there, then look back at the other one. See how they are the same picture only projected differently? (Some of the cubes are “slanted” because of perspective, like the squares in a drawing of a cube. As you can see there is actually more than one way to draw a hypercube—read this But I prefer the one to the right because it shows the extrusion into the fourth dimension.)

A 4D sphere, called a glome, can be explained in a similar way. A 3D sphere is a infinite number of circles of increasing-then-decreasing size stacked on top of each other. (Try to imagine making a sphere out of a stack of pieces of paper. How would you cut the paper? The answer is into circles of increasing-then-decreasing size.) Likewise, a glome is made out of spheres stacked in increasing-then-decreasing size. Like the hypercube, these sphere are stacked in the new perpendicular direction.

Geometry-minded people’s brains are probably going crazy with all the applications of this idea. Personally I am not particularly interested in geometry, but I would like to elaborate just a little on 4D geometry. For example, look at a triangle, the simplest polygon it is possible to create. In three dimensions, the closest equivalent is a tetrahedron, a polyhedron with four sides made of triangles. Thus, the 4D equivalent is a pentachoron, a 4D figure with 5 sides made of tetrahedrons (a polychoron is the word for a 4D shape, like polygon or polyhedron). Also, similar to how regular polyhedrons exist in 3D, regular polychora exist in 4D. However, geometry is not really my area and I am not entirely sure about all the properties of these figures. Perhaps another day I will explain more about them.

If the concept of the fourth dimension still isn’t making sense, I recommend perusing this website—it’s very helpful.

Tomorrow I will look into more aspects of the fourth dimension.

From Flatland to the fourth dimension (part 1)

Today I would like to talk about Edwin Abbot Abbot’s famous 1884 science-fiction novel Flatland. This book is renowned for its criticism of Victorian society as well as for its explanation of abstract geometrical theories.

The setting of Flatland is a two-dimensional world in which all of the “people” are geometric shapes moving around in a plane. The first half of the story explains the nature of Flatland and its society; the second half focuses on the main character’s encounter with a sphere.

Abbot begins with an explanation of what Flatland actually is. I will take a moment to do so as well, since it is very important to the story and to the points I am trying to make. Flatland is a two-dimensional plane, like a piece of paper. The inhabitants are two-dimensional polygons that move about the plane; they have no concept of the third dimension. A Flatlander can move in four perpendicular directions (north, south, east, west) and its field of vision is a line. No part of a Flatlander is three-dimensional; for example, a Flatlander’s “face” is not on its surface but on one of its sides.

Confused? Here’s a demonstration that will help: Put several coins on a table or desk and then bend down next to it so your eyes are level with the surface. See how the coins look like lines? This is how a two-dimensional being sees. Move the coins around a bit and try to visualize them as living beings, thinking about what they biological structure would have to be. (This is actually how Abbot explains it.)

The plot of Flatland is very simple; in the first half, Abbot explains what Flatland is (like I just did) and how its society functions. The second half is considerably more interesting; a Sphere visits the main character, A. Square, and tries to explain the idea of three dimensions to him. The Sphere first attempts to do so with words, notably with the phrase “Upward, not Northward” but A. Square simply cannot understand. To demonstrate, the Sphere pokes A. Square in his intestines, which he can reach by putting his finger into the Square insides from “above” (three dimensional above). However, A Square is still not convinced, so the Sphere knocks him out of his plane and drags him about outside of Flatland.

The duo then visit other societies that are defined by dimensionality: they first visit Lineland, in which the inhabitants are forever stuck next to the same people because they are in a line (one dimension). Next, they look at Pointland, which is inhabited by a single dimensionless being. Communication with it is impossible because the being is a solipsist; it believes everything it hears is its own thoughts, since it cannot move, see, or feel. A. Square then suggests that there could be more than three dimensions, since until recently he only knew of two. The Sphere dismisses this as nonsense, and sends A Square back to Flatland. A. Square later tries to teach others about the third dimension, but without the sphere he is incapable of doing so. Since he cannot show his peers the direction of “Upward, not Northward,” he is thrown in jail for heresy.

Flatland clearly satires Victorian society: Abbot goes into detail explaining a class system based around shape. The few sides a “person” has, the lower their rank; triangles are working class and are considered non-intelligent; polygons with many sides are priests or leaders. The king is a polygon with so many sides he is almost indistinguishable from a circle. Certain shapes can “evolve”—all shapes except triangles gain a side each generation, increasing their rank. The lower class triangles do not share this attribute; they are forced to first become equilateral triangles. However, they only progress at a rate of half a degree per generation—this obviously signifies the perpetual enslavement of the working class by the Victorian elites. Women in Flatland are straight lines, and they are forced to move in such a way that they constantly swing back and forth so they can be seen (remember that looking at a straight line in a certain way makes it look like a point, which is hard to see). This is also a parody of Victorian women, who were also “invisible” unless they made their presence known. Colors are banned in Flatland society, since lower classes could paint themselves to look like higher-class shapes.

But Flatland is better known for its explanation of dimensional theories of geometry. Abbot explains the theory of the fourth spatial dimension by analogy in the story, and certain plot events are actually methods for “studying” the fourth spatial dimension. Tomorrow, I will delve deeply into what the fourth spatial dimension actually is, some applications of it, and how it relates to Flatland.