perhaps *this* will refresh your memory
Posted by vlorbik on November 23, 2010
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Posted by vlorbik on November 23, 2010
This entry was posted on November 23, 2010 at 7:40 PM and is filed under Graphics, Lectures Without Words, Zines. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.
vlorbik said
the seven points in the center represent
the seven “directions” from 000 on a cube.
[to wit {001, 010, 011, 100, 101, 110, 111}
(binary 1 to 7; one calls 001, 010, and 100
“unit vectors” (in the x, y, and z direction
respectively… or not… but such things
are good to know…).]
the bigger circles around (& in one case,
amongst) the seven-in-the-center “plane”…
are associated with the *lines* of
P = P_2 ({Z}_2).
These lines are then treated as the *points*…
think of the circles… of the “dual” space P^*
(“pee-star”). then there’s a
structure preserving map
(called, i suppose, “star”) from P to P^*:
a mapping P—-}P^* that not only takes
points of P (dots, say) to points of P^*
(circles), but simultaneously takes the
lines of P to the lines of P^*.
the appropriate arithmetic is modulo 2;
in the two bottom lines i’ve given examples
of “why” the points 111 and 100 do and do not
(respectively) lie on the line [110]:
the “dot product”
[xyz]abc := xa +yb + zc (mod 2)
produces 0 (it *is* on the line)
or 1 (it’s *not*).
the mod-2 addition table is given;
just remember everything is in
“remainders on division by two”.
(think “even” and “odd”, rather,
i suppose, for most people…)
this drawing was something of a breakthrough
for me. it was soon followed by
https://vlorblog.wordpress.com/2010/11/23/now-heres-something-youll-really-like-that-trick-never-works/
cut & paste | the livingston review said
[…] trying again dammit P_2(Z_5)11/23/10 perhaps *this* will refresh your memory P_2(Z_2)*11/23/10 now here’s something you’ll *really* like more […]
Vlorbik On Math Ed | the livingston review said
[…] 11/23/10 trying again dammit P_2(Z_5) 11/23/10 perhaps *this* will refresh your memory P_2(Z_2)* 11/23/10 now here’s something you’ll *really* like more […]