Is there a difference between the Tesseract and a 4D Euclidean cube?

No, there is no difference but the name.

tesseract = hypercube = 4-cube = 4D cube = 8-cell regular polytope = measure polytope of 4D

The idea behind visualizing a tesseract is by analogy with portraying a 3D solid (polyhedron) in a 2D medium, like a printed page.

When you draw a cube on paper, you are drawing a 2D projection of it. Well, there are many ways to do that -- many angles to view it from.

Similarly, you can make a projection of a (4D) hypercube onto 3D-space in many different ways.

3-cube: If you look straight down on the front face of a cube, then that square forms the outermost part of your drawing, the back face is a little smaller and inside the front face, and each of the 4 corners of one is connected to the corresponding corner of the other.

4-cube: That is the cube-drawing that's analogous to your first hypercube-drawing -- one cube inside a slightly larger one, with each of the 8 corresponding corner-pairs connected.

3-cube: You can also move a little to the left, and see the back face appear to overlap the left side of the front face on your drawing. Those 4 connecting lines will then slant left and up, from the bottom 2 corners; and left and down, from the top 2 corners.

4-cube: Analogous to this, would be a large "front" cube with a slightly smaller "back" cube emerging through its left side, with the 8 corresponding corner-pairs connected, once again.

3-cube: Then if you instead move a little left and up from the front of the cube, the back face appears to overlap the upper left corner of the front face, and the 4 corner-pair connecting lines become nearly parallel.

4-cube: The analogous hypercube drawing has the slightly smaller "back" cube overlapping the upper left back corner of large "front" cube, with the 8 corresponding corner-pairs connected by lines that are nearly parallel.

All these aspects for portraying the hypercube, and others, are possible with certain 4D-viewing tools. Of course, you expect 4D figures (polytopes) to be richer diagrammatically, than 3D figures (polyhedra), because of that extra dimension, but there's an even bigger increase in "richness" in the portrayals because of the rotation groups in the two different dimensions.

In 2D, all rotations are completely specified by a single angle. The 2D rotation group, SO(2), is 1-dimensional.

In 3D, it takes 3 angles to completely specify all rotations; SO(3) is thus 3-dimensional.

In 4D, it takes 6 angles to completely specify all rotations; SO(4) is thus 6-dimensional.

These numbers can be seen by understanding that an elemental rotation, in a space of any number of dimensions, is defined by an angle and a plane in which the rotation takes place.

In n dimensions, there are n coordinate axes, and so, ½n(n-1) different ways to choose a pair of axes to make a coordinate plane.

Each possible coordinate plane is orthogonal (perpendicular) to each of the others, and rotations in orthogonal coordinate planes are independent, when it comes to composing them to make a general rotation in n-space.

So if you happen to run across one of the aforementioned 4D-viewing tools, you should find that there are 6 separate rotation controls in it.

Unless things have changed, a tesseract is another name for a 4-cube. "4D Euclidean cube" sounds like an even longer way to say the same thing.

They are the same thing. Since you cannot actually visualize a four-dimensional object, the images you have seen are various representations of what a 3D cross section, shadow, or 3D net of a tesseract would look like.

tesseract and 4d cube are the same thing.

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You have never seen a tesseract.