A tesseract is a "four-dimensional" object that is analogous to a three-dimensional cube in many aspects. Here, I will explain how to draw a two-dimensional representation of a tesseract, as well as explain exactly what it is and what is meant by four-dimensional.
Of course, this can either be done on the computer or with actual drawing tools; either way, the steps will be the same.
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Step 1: Step 1
The first step is to simply draw an octagon. Since I made the images for this 'Ible in Microsoft Paint (which has no octagon tool), I overlaid two squares over each other.
Step 2: Step 2
Step two is to extend the sides of the octagon so that it forms an eight-pointed star (octagram).
Step 3: Step 3
Add squares to each horizontal and vertical side of the star.
Step 4: Step 4
The last step is to draw diagonal lines between certain corners of the squares you drew in the last step, as shown.
Step 5: Understanding the Tesseract
Now you have your finished product. Looks pretty cool, doesn't it? But how exactly does this two-dimensional image represent a four-dimensional figure?
It goes like this: take a pencil and paper and draw a single point on the paper. The point is a zero-dimensional entity, meaning it has no physical definition - no length, width, or height.
Now make a second point and connect it with the first point with a straight line. The line is a one-dimensional object; it only has one physical characteristic, which is length.
Draw a second line (one that's as far away from the first as its own length) at a right angle to the first and connect their vertices (corners or ends), and you get a two-dimensional square with length and width.
If you could repeat this process by drawing the same square perpendicular to the first square and connecting their vertices with each other, you would get a cube with length, width and height - the three dimensions of our physical world. Of course, it is impossible to draw a 3-D object on a 2-D paper, so we'll settle for an imperfect "projection" of a cube, drawn by placing the second square at a 45-degree angle to the first and connecting them with lines the same length as their sides.
The tesseract you just drew is, essentially, a continuation of this process. In a manner of speaking, it is an image of what would happen if you were to draw a second cube perpendicular to the first and connect their vertices. This is shown in the second picture; the first cube is blue, the second is red, the purple is where they overlap.
Like I said before, our physical world has three dimensions. These three are perpendicular (at right angles) to each other. The mysterious fourth dimension would be perpendicular to all three of these dimensions at once! But don't even try to imagine it; because we live in a 3-D world, it would be impossible for us to imagine such a direction, since technically it can't actually exist. You might as well try to make a square imagine a cube!
Here's an alternative way of thinking about it. If you were to take the six 2-D squares from the third image and fold them in a 3-D way, you would make a cube. In the same way, if you were to take the eight 3-D cubes in the fourth image and fold them in a 4-D way, you'd make a tesseract.
Since we can't imagine how a tesseract would actually look in all of its 4-D glory, all we can do is create a 2-D projection of it, the way we did for the cube. Some information is lost in the transition, but it's better than nothing.
Thanks for checking this out. If you let me know if my explanation was coherent, I'd really appreciate it. Please subscribe, rate, and comment, and have a good day :)