Well... 2 ^1 is a line of length 2. To draw 2^2 you draw two lines of length 2 at right angles to each other and fill in the square. For 2^3 you draw three lines of length 2 at right angles to each other and fill in the cube. For 2^4 we just continue! The problem is drawing 4 lines of length two at right angles to each other and filling in the solid figure - we live in a 3d spatial world so run out of dimensions to draw this "hypercube"!
To finish off below (run out of characters) - you can project a 4d cube into a 3d space much as we project a 3d cube onto a 2d screen above - do a search for this under 4d hypercube!
The (2d!) pictures of hypercubes seem to show it as a sort of square within a square, with 'pyramid stumps' going from the outside square to the inner one. Are these pyramid stumps actually other squares, that we just can't draw accurately in 2 dimensions? Because that would give 16 squares overall, which is 2^4?
Not quite! Just as the surfaces of a 3d cube are 2d squares then the surfaces of a 4d hypercube are 3d cubes (there are 8 of them)! What you see in the pictures is a 2d projection of a 4d figure and it's hard to visualise! Have a look at "Tesseract" on Wiki. There's also a great page on math.union.edu website but I can't post it - email me and I'll give you the link!
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toolworks 8 months ago
So... what does 2 ^4 look like graphically?
gasmarkfive 1 year ago
@gasmarkfive
Well... 2 ^1 is a line of length 2. To draw 2^2 you draw two lines of length 2 at right angles to each other and fill in the square. For 2^3 you draw three lines of length 2 at right angles to each other and fill in the cube. For 2^4 we just continue! The problem is drawing 4 lines of length two at right angles to each other and filling in the solid figure - we live in a 3d spatial world so run out of dimensions to draw this "hypercube"!
grallator9 1 year ago
To finish off below (run out of characters) - you can project a 4d cube into a 3d space much as we project a 3d cube onto a 2d screen above - do a search for this under 4d hypercube!
grallator9 1 year ago
@grallator9 Thanks for that!
The (2d!) pictures of hypercubes seem to show it as a sort of square within a square, with 'pyramid stumps' going from the outside square to the inner one. Are these pyramid stumps actually other squares, that we just can't draw accurately in 2 dimensions? Because that would give 16 squares overall, which is 2^4?
I love maths! :P
gasmarkfive 1 year ago
Not quite! Just as the surfaces of a 3d cube are 2d squares then the surfaces of a 4d hypercube are 3d cubes (there are 8 of them)! What you see in the pictures is a 2d projection of a 4d figure and it's hard to visualise! Have a look at "Tesseract" on Wiki. There's also a great page on math.union.edu website but I can't post it - email me and I'll give you the link!
grallator9 1 year ago
@gasmarkfive go to a world with 4 dimensions and u can find out =]
Yu2Kal 1 year ago
Great - I understood it. Can you do one for quadratic equations now?
depman09 2 years ago
I'll see what I can do...
grallator9 2 years ago