Counterintuitive Properties of High Dimensional Space (2018)

I don't understand what is meant here by "dimension." Is the definition of "dimension" consistent for 3-dimensional figures and n-dimensional figures? In other words, what is the importance of the orthogonality of the axes? The axes of dimensions beyond 3-d are not orthogonal, does this change the definition of "dimension".

I cannot conceive a geometrical image of higher dimensions. Algebraically, yes, but not geometrically.

> The axes of dimensions beyond 3-d are not orthogonal, does this change the definition of "dimension".

They are orthogonal.

> I cannot conceive a geometrical image of higher dimensions.

This is normal, and essential to the point of the article. If you could visualize 10-dimensional space, it wouldn’t be so counterintuitive.

Try looking up images and videos of 4D objects projected into 3D and 2D. That might help. Hypercubes are maybe the easiest.

From Wikipedia: https://en.m.wikipedia.org/wiki/Hypercube "4 – If one moves the cube one unit length into the fourth dimension, it generates a 4-dimensional unit hypercube (a unit tesseract)."

How do we draw an orthogonal line to the three orthogonal linas that we have?

In 3 dimensional space, you can’t.

Draw a square on paper. The lines will be orthogonal. Draw a cube. The third dimension won’t be orthogonal. It can’t be. But, that doesn’t mean a third dimension doesn’t exist or can’t exist.

The same thing happens with a hypercube. The fourth dimension won’t be orthogonal on paper. It won’t be orthogonal in three dimensions (you can build a 3D projection of a hypercube). This doesn’t mean it isn’t real or can’t be real.

Whether you think of 4D space as something real we don’t have access to or something imaginary isn’t really too important. It’s just very helpful to realize it won’t have the concreteness of 3D objects in 3D space for you because you don’t have direct access to it.

This video might help. https://youtu.be/UnURElCzGc0?si=MQa2JKT_CMmM-_JL