Pug, dun be a Pug! It's obvious that what we see in the avatar is a 2D representation of a 3D representation of a 4D cube
In that post of mine that you quoted I was not talking about the avatar, just about the very tesseract.
And you surely agree that a tesseract is a 4D cube -- or a hypercube, if you prefer ('4D hypercube' is a bit of a redundance, isn't it?).
Actually, at the age of 13 I drew a 2D projection of a tesseract and submitted it to my tech design teacher.
She did not understand, she thought I was trying to make joke of her -- and no, not because of so bad drawing skills of mine.
She threw my drawing away and she assigned me a sound 2 (out of 10).
That was my time to realize that adults and authorities may be incredibly dumb...