For the uninitiated, a tesseract is the 4-dimensional equivalent of a cube. Here’s a model you can play with.
I have somewhat of a mathematical background, so intellectually I understand what one is. Take a line - 1 dimension. A square has 4 sides, each a line. A cube has 6 faces, each a square. That’s where I lose it - I can’t envisage an object whose each “face” is a cube. It won’t fit in my brain.
Can anybody really, honestly visualize this? Or is it impossible for us 3-dimensional beings?
I’m the same as you, and I’ve heard people saying that they can do it. I can visualize two lines coming together to make a right angle. I can visualize three squares coming together to make one vertex of a cube. But I cannot, for anything, visualize four cubes coming together to make the vertex of a hypercube (or tesseract).
I can’t… I have enough trouble in three dimensions 
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I figured this isn’t enough to merit my own thread but…
I’ve gotten interested in fractals recently (the computer-generated images are so cool!) but I can’t for the life of me figure out why they say that fractals are in between dimensions. It sounds [sub]sort of[/sub] reasonable and all, but I just can’t envision something that exists somewhere in 1.something or other decimal dimensions!
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Contemplates for a moment
Then again, I’m one of those imagination-challenged people who don’t bother assigning faces to the characters whenever they read a book… :rolleyes:
Of course I can visualize a tesseract. it’s simple: all you need is a skirt and an ant.
Uhm.
AFAIKnew, a tesseract is NOT the 4-dimensional equivalent of a cube - that would be a hybercube. A tesseract is an ‘unfolded’ hypercube, such as Figure 9 in this paper.
I see that that page defines it in the way that you have done… and now I am confused. Hmmph.
There’s an interesting model here that does more for me than the one in the OP.
Thanks, I have something interesting to bother me today now 
I’ve always heard the tesseract defined as a 4-D analogue to a cube, not its 3D unfolding.
I’ve heard people claim that they can visualize it, or almost do so, but I’m skeptical. I’ve tried, and I know I can’t
What I know I can do is to note its properties and projections. There are various ways to project and unfold a tesseract in 3-space. You can also note the progressions as you go up the dimension scales and extrapolate. You translate a 0-D point along 1D to get a 1D line defined by 2 points. You translate that 2-point line perpendicular to itself to get a 2D Square bounded by 4 points and 4 lines. You translate the 2D square perpendicular to the two previous dimensions to get a 3D cube bounded by eight points, 12 lines, and 6 squares. So you translate the cube perpendicular to the previous three dimensions to get a tesseract defined by 16 points, 8 cubes, and 32 lines.
I guess I’ll toot my own geeky horn here: I can visualize a hypercube, along with one or two other very simple 4-dimensional shapes like the hypersphere and the simplex (the 4-D analogue of a tetrahedron).
Is was largely a matter of practice over many, many years. (Or in other words, my user name is not just for show.) I figured out other, simpler things before I got the hypercube straight in my head: for example, visualizing two planes (two-dimensional planes) which meet in exactly one point, or visualizing objects rotating around a plane (in 2-D things rotate around point, in 3-D they rotate around a line, in 4-D they rotate around a plane).
I ultimately would like to be able to visualize all 6 of the regular 4-D polyhedra (of which the simplex and the hypercube are the first two), but it’s not exactly a high-priority goal for obvious reasons. It also may be beyond my abilities, especially the last one, which is supposed to have 600 hyperfaces. Eek.
Sure, it’s easy. Just visualise an n-dimensional cube and set n to 4.
Nah, not really. I’ve never been able to visualise more than simple 2-D objects: I have a very abstract mind, which doesn’t work well with images. Besides, Euclidean geometry isn’t very topologically interesting.
“Speaking of ways, by the way, there is such a thing as a tesseract.”
– Mrs. Whatsit, in A Wrinkle in Time
I can sorta visualize it.
The problem with any of the tesseract representations that fit into your screen is that they’re trying to give you a picture of a 4-dimensional object in only 2 dimensions. (Imagine trying to render a 3-dimensional cube in a one-dimensional drawing, and you get a feel for the challenge.) It’s easy to draw a believable picture of a cube on a 2-dimensional piece of paper, and it shouldn’t be hard to construct a respectable model of a tesseract in 3-dimensional space.
Slap me for being over-simplistic here (because I know nothing at all about the subject), but I’ve always thought that, since the 1-dimensional projection of a 2d cube (i.e. a square) is a line, and the 2-dimensional projection of a 3d cube is a square, then surely the 3-dimensional projection of a 4d cube would be a 3d cube, rather than any of those weirded-up diagrams. Would this be rubbish?
The projection of a cube onto a plane can be a square, if you do it along one of the axes. But it can also be a diamond or a hexagon or a weird polygonal shape if you do it along other axes. In the same way, the projection of a tesseract into 3-space can be a square. But it can also assume other, weirder shapes (and remember, there are more axes to project it along in 4-space). It just goes to make life interesting.
Read Abbot’s wonderful book Flatland about the ways of dimensions. I like the part where the Sphere shows the Triangle that it can apparently change its size (its intersection, rather than Projection into 2-Space) by moving up and down along the third dimension axis (which is incomprehensible to the 2-D Triangle), and can eventually “disappear” altogether.
Not so much a projection / cross-section, but a graphical representation.
You can draw a “glass cube” on a piece of paper that clearly show all six sides, right? Sure, the perspective of at least four of those sides will be skewed, but you can still render the drawing.
Assuming that the spatial rotation was such that it would be cube-shaped, why would it look like a wire-frame? Why wouldn’t it just look like a solid cube.
But the tesseract isn’t cube-shaped. It’s tesseract-shaped. So both its projection into and its intersection with our 3-Space can be very different from a cube, just as the projection into and intersection with 2-Space of a cube can give you a diamond or a hexagon or a polygon.
Sorry, I’m not phrasing myself correctly.
The correct question is this: since an ‘edge’ in 1D is a dot, in 2D is a line, and in 3D is a plane, why would the ‘edges’ of a 4D shape intersecting with the 3rd dimension appear as lines (“wire frame”) instead of planes?
In other words, if the tesseract were rotated in the 4th dimension such that it would appear as a cube, why would it look like a wire frame rather than having solid planar edges?
I think bordelond already answered your question, jjimm. The wireframe diagrams of a hypercube are just that…diagrams. They show the vertices and the 1-D edges of the hypercube, but not the 2-D faces or the 3-D “hyperfaces”. The positions of the faces and hyperfaces can be inferred from the edges and vertices, but including the faces and hyperfaces would block our view of some of the edges and vertices.
An actual projection of a hypercube into 3-D space would indeed look like a cube, if the hypercube was positioned just right. But wireframe diagrams show more of the hypercube’s structure.
Thanks everyone, I now get it, inasmuch as I’m ever likely to get this stuff.
I managed to visualise a rotating hypercube once. It was very difficult… one of the greatest mental efforts I’ve ever done.
What would it look like if the angle of projection were different?
CougarFang: here’s what I was taught.
Take a line segment. One dimension. Some finite length. Apply some kind of fractal function (e.g. subdivide it into thirds, replace the middle third with an equilateral triangle , erase the bottom line of the triangle (this produces a “snowflake” fractal)). Repeat this infinitely many times. Your line segment now has infinite length, right? Now, it’s pretty darned odd for a line segment, with two fixed points and all, to have a length of infinity. So we have an object that exhibits some properties of 1-d objects, and other properties of 2-d objects. So we say it’s a 1.5-dimensional object.
This is probably garbled, but it makes sense in my own garbled little mind.
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My humble little mind can’t visualize a hypersphere. I read somewhere a different analogy for higher dimensions:
dimension 0: letter in a word
dimension 1: word in a line
dimension 2: line in a page
dimension 3: page in a book
dimension 4: book on a bookshelf
dimension 5: bookshelf in a library
etc…
It doesn’t translate very well to actual applications, but it does help me think about higher dimensions. Fortunately, I don’t often need to do this.