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Ok, for a 5 foot square space on a your usual grid the area contained in each is 5 square feet. That means that the length of the sides of this square have to be √5 feet long (2.2360679775... feet).
The movement speeds list your movement in square feet rather than just regular feet. A normal human can move up to 30 square feet in a single movement action. If they are walking in a continuous path of 5 foot squares without any diagonals, (those come into play later) then they travel at most 6*2.2306=13.4164... feet
Twice that on a double move for about 26.8328 feet. Four times that on an unencumbered, unarmored run for 53.6656.
These are all in a period of a single round which is 6 seconds.
Converting these three from feet/6 seconds to miles per hour gives
1.5246 mph single move
3.0492 mph double move
6.0984 mph run
These are for Bob McAverage.
The wizard Pythagaryl determines through years of study and intimate knowledge of sacred geometry that the superior way to move is diagonally. By walking the diagonal of a 5 foot square, they cover a distance of √10 per square (3.1623 feet per second).
Doing the same calculations before and presuming a normal rate of human movement
18.9737 ft/6 second single move, 2.1561 mph
37.9473 ft/6 second double move, 4.3132 mph
75.8947 ft/6 second run , 8.6244 mph
Pythagoryl moves √2 times faster moving diagonally compared to Bob McAverage speeds.
So clearly serpents, crustaceans, and other manner of sideways scurrying critters have been styling on bipeds for millennia. Why don't people move that way?
Ok, for a 5 foot square space on a your usual grid the area contained in each is 5 square feet. That means that the length of the sides of this square have to be √5 feet long (2.2360679775... feet).
Edges are 5 feet long. A single square contains 25 square feet. Each diagonal is approximately 7.07 feet. With every other square counting as 2 for the purposes of movement, you actually travel slower moving diagonally than you do moving in a straight line.
4 squares of diagonal movement therefor counts as 6 squares of movement. Diagonals are 7.07 feet, and you traveled 4 of them = 28.28 feet. Traveling the same 6 squares of movement in a straight line would be 30 feet in the same span of time.
Running, a human gets 24 squares of movement in a straight line, which is 16 squares diagonally. There, you cover 113.12 feet compared to the 120 feet you would of covered had you run straight instead of diagonally.
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As has been pointed out, a 5' square is 5' on each edge. Pythagoras tells us that the diagonal of a 5' square is a bit over 7' long (close enough that we'll just call it 7' for the sake of simplicity).
With a 30' movement, a character can move 6 full squares forward/backward/left/right. Moving 6 squares diagonally would actually be ~42'. But don't forget that we have to do that funky double-counting of every-other square when moving diagonally, so 1, 2-3, 4, 5-6, such that moving only on a diagonal gets you 4 total squares, or ~28'.
People don't move like that because the game's highly abstracted movement system makes it inefficient. Also, math.
-Skeld
Edit: Ha! Jeraa ninja edited their post while I was writing this up. Mine is sorta redundant now. :)
Had this been 4e D&D, you would have a point. 4e D&D did away with the "every other diagonal square counts as 2 squares" thing. It also had square fireballs.
So clearly serpents, crustaceans, and other manner of sideways scurrying critters have been styling on bipeds for millennia.
But I use a hex grid. At a side length of 1'4.644", I am not sure how to count the strait line.
/cevah