Pick up a piece of paper. Draw a cube (a solid one, not a thread one). Observe your results.
Congratulations. You just drew a 3-dimensional object in a 2-dimensional area. This may not seem all that important, but it is. For example, imagine a 2-dimensional world, with 2-dimensional people. You just drew a 3D world, on a 2D piece of paper. So those 2D people should be able to draw a 3D world easily too, right?
Wrong. They could, but it wouldn't be easy, for 2 reasons.
One: The 2D paper is solid for them. They would have no way to draw in the inside of the paper, because they have no third dimension from which they can draw on the paper1, so they can only draw at the edges.
Two: It would be quite hard for them to visualize what a 3D world would look like.
But, we'll tackle those problems later. First, let's translate our little thought experiment into three dimensions. In order for us to draw a 4D world, we'd need a 3D drawing board. For example, a large cube of concrete. But then we'd face the same problem as our 2D people, you can't draw on the inside!
So the concrete is a no-go. There's another possibility, stack lots and lots of paper on top of each other. This gives you a 3D cube but you still have the ability to draw on the inside. It's gonna take a while though.
Great! Now we've got our drawing board! Let's go to the other problem now. What does a 4D world look like, when you draw it in a 3D object? Here, our 2D world comes in handy again.
First of all, here's a cube with some letters for reference.
E_______ F
/ /|
/ / |
D ______ C |
| | |G
| | /
| |/
A|______|B
What do you do when you draw a cube on a piece of paper? You start off by drawing a square (ABCD). What then? You draw a line from one of the corners. The line does not make any right angles with the other lines, although in a 3D world it would make a right angle, with both other lines going from the corner actually.
For my experiment, I am drawing a line from corner D going up, and a little to the right (I'm calling that line A). Then one from corner C, and one from corner B. Then, we go from the end of line DE to the right, until we meet the next line (CF). Then we go down, untill we meet the last line (BG). After that, we go down with a slope towards B again. We don't draw a line from E downways, because it is behind CDEF. We also don't draw one from G to the left, the line is behind BCFG, nor one from A upwards with a slope, because that's hidden from view by ABCD.
So, can we pick up a general rule from this which we can use when drawing our tesseract (4D hypercube)?
Sure: Draw one plane at a time, and make sure your lines/planes do not lead into or cross into any other lines/planes.
Now it's getting complicated. Let's try to visualize us drawing a tesseract on our 3D sketchboard. We'll start off by drawing a cube fully. This is possible since we have the freedom of 3 dimensions, which means no lines or planes will interfere. Now, let's draw a line from our point E (referring to the ASCII drawing) upwards, a little to the right, and a little to the back. Now, a line to the right, and from there a line back to F. We just drew our first ever plane into the fourth dimension. We stuck to the rule, we did not cross any of our former lines or planes. Or, seeing as we've got 3 dimensions, bodies.
If you can't imagine this, there's a picture at the bottom that might help (the one with the letters), but if you can, try imagining it in your head. It's easier since you can imagine a 3D area, but I can't draw a true one on a 2D screen.
Let's continue with our drawing. Draw a plane from CD into the 4th dimension, by drawing a line from D, then to the right, then back to C. Now we're just lacking 2 lines from completing a skewed cube. Draw the 2 lines.
Just so you know, I'll call the width the X-axis, the height the Y-axis, and the depth the Z-axis. The 4th dimension will be the W-axis.
So now what we've got is a cube, with a skewed cube on top of it. these cubes are sharing one plane. The first cube is lacking any length on the W-axis, the second cube is lacking any length on the Y-axis. Let's remember our drawing of the cube on a 2D piece of paper. How did we start? We drew a square, and a skewed square on top of it. These two squares shared one line. The first square didn't have any depth, the second square didn't have any height. See the similarities? This means we're on the right way.
So, what's now? We've already got a plane CFF'C', where the accent signifies the other end of the line. For example the end of the line from C going into the 4th dimension, is C'.
Let's draw a plane BGG'B', and connect B' with C' and G' with F'. this gives us another skewed cube. This cube is lacking any length on the X-axis, which means it doesn't have any horizontal lines.
But we're not done yet! Here's the hardest part to visualize. Imagine the body which we now have, consisting of 3 cubes. We've got a concave (dented) area at the back! Why is that? Well, we've had the cube lacking the W-axis, the cube lacking the X-axis and the cube lacking the Y-axis. But we don't have a cube lacking the Z-axis yet. We can make that cube by using the concave at the back.
The cube already has 3 of its sides, namely EFGH, EFF'E' and FGG'F'. We can give it all of its corners by drawing a line from H (which is to the left of G, hidden in the ASCII drawing) into the 4th dimension. Then we draw a line from H' to E'and from H' to G', and our solid drawn tesseract is done!
Things to note:
- If you try to add any other of the lines that you would see in a thread tesseract, you'd enter the tesseract's body, which was not allowed.
- If you actually want to make a thread tesseract from this, draw one line into the 4th dimension from A, getting to point A', then one line down from D' to A', one line to the left from B' to A', and one line from H' to A'. This would instantly create (or finish, depending how you look at it) 4 new cubes (3 skewed, 1 straight), and would complete all the neccesairy lines for the thread tesseract. You can check by checking if each corner has 4 lines going from it, 1 for X, 1 for Y, 1 for Z and 1 for W.
- You can't actually see your tesseract. That's because you drew it in a solid cube. But a 4-dimensional being could see it, since they can observe the whole 3D world at once. Don't bother wearing clothes. You're just a mere 3D being. They can see right through your clothes (or rather, into your clothes). And 4D clothes are way too expensive.
- Drawing a tesseract in a 3D cube is much easier for a 4D being. They would just take their pencil, and insert it right into our concrete block from the 4th dimension. It's like the 2D people who couldn't draw on our paper. We don't have a 4th dimension from which we can draw inside the cube, the 4D being does. Mindboggling, yet logical at the same time.
- If you look tesseract up with google image search, you get lots of pictures where you see a cube inside a bigger cube. This would only be right if you draw a hypercube with perspective, which I didn't. It's a good way to visualize a hypercube though.
- A nice link to help you visualize the basics of a tesseract is this one:
http://www.traipse.com/hypercube/index.html
With some help of paint, ASCII Pic and notepad I have managed to create this 2D representation of our 3D representation of a solid tesseract.
EEE EE FFF FF
E F
EE FF
E F
EEE F
##############################################################
################################################################
#### ######
####$ #######
##### $ #########
###### $ #####$ ###
#### $ $ #####$$ ###
##### $$ $ #### $$ ###
#### $$ $ CC CC #### $$ ###
#### $ $ C #### $$ ###
DD DD #### $ $ C ##### $$ ###
D D ##### $$ $ C ##### $$ ###
D D #### $$ $ CC ##### $$ ###
DD ################################################################ $$ ###
################################################################ $$ ###
#### $ $ #### $$ ###
### $ $ ##### $$ ###
#### $$ $ ###### $$ ###
#### $ $ ####### $$ ###
#### $ $ ####### $$ ###
### $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$########$$$$ ###
#### $$ $ EEE $ #### ###$$$$ FFF ###
#### $$$ $ E $ #### ###$ $ F ###
#### $$ $ EE $ ### ### $ FF ###
#### $$$ $ E $ ### $### $ F ###
### $$ $ EEE $ ### $$$### $ F ###
#### $$ $ $ #### $$ ### $ ###
#### $$ $ $ #### $$$ ### $ ###
#####$$ $ $ #### $$ ### $ ###
####$ $ $ #### $$ ### $ ###
####$ $ $ ###$$ ### $ ###
#### $ $ ####$ ### $ ###
############################################################### CC ### $ ###
############################################################### C ### $ ###
############################################################## C ### $ ###
### DD $ $ ### C ### $ ###
### D D $ $ ### CC ### $ ###
### D D $ $ ### ### $ ###
### DD $ $ ### ### $ ### GG GG
### $ $ ### ### $ ### G
### $ $$$$$$$$$$$$$$$$$$$$$$$$###$$$$$$$$$$$$###$$$$$$$$$$$$$$$$$### G GG
### $ $$ H H HH ### ### $ #### G G
### $ $$ H H ### ### $ ##### GG
### $ $$ HHHH ### ### $ #####
### $ $$ H H ### ### $ #######
### $ $$ H H ### ### $ #######
### $ $ ### ### $ ########
### $ $$ ### ### $ ########
### $ $$ ### ### $ #### ####
### $ $ ### ### ##### ####
### $ $$ ### BBB BB### ##### ####
### $ $$ ### B B ### #### ####
### $ $$ ### BBB ####### ###
### $ $$ ### B B ##### $ ####
### $ $ ### BBB ### $ ###
### $ $ ### #### $ ####
### $ $ ### #### $ ###
### $$$ ### #### $####
### $$ ### ### ####
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### $$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$###$$$$$$$####$$$$##### GG
### $$ H H ### #### #### G
### $$ H H ### ### #### G GG
### $$ HHHH ### #### #### G G
### $$ H H ### #### ##### GG
### $$ H H ### ### ####
### $$ ### ### ####
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### $$ #########
### $$$ ########
###$$ ######
################################################################
###############################################################
A BBB
A A B B
AAAAA BBB
A A B B
A A BBB
And here's a 2D representation of a thread tesseract.
..
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/--/;;#@XX++XMMMMMHHH%%%;;;;-----------;;;;;;;%%%%%%%HHHHHMMMMHX++++++
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/;-/;;#%#+=#X+++++H=XM#MMMMMH@@;/---#+%#%%%%@@@HHMMMM#MX#/-/;;#@##X+++++H
.;--;;;%HH M#X++++++ =#MMMMMM;-/;M+++%HHHHHHMMMM#+= ;/--;;;%@ MX++++++
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1. Thanks themanwho