LOGO-L> davi's star, some general ideas

[neuwirth <neuwirth@smc.univie.ac.at>]

here are some unpolished ideas about what solutions may
exist for david's star accoruing to our rules

we have to draw 18 line segments.

18=3.3.2

so possible rotational symmetries are
2, 3, 6, 9 

additionally, we have 12 outer and 6 inner line segments.


since every subfigure
produces the same number of inner and outer segments,
each one of our subfigures has to produce
twice as many outer segments as inner segments.

so 9-fold symmentry is impossible

6 fold symmetry would imply that
we draw 2 outer an 1 inner segment.

outer-outer-inner produces eitehr a triangle,
which is not permissimpel because that wen have to
do some line smore than once,
or a figure which gets back after 3 repetitions,
so it does not work.

so we have to look at 3 fold rotational symmetry.
4 outer and 2 inner line segments.
the rays of the star produced by one such figure
cannot be in an angle of 120 degrees.
then we would again get segments drawn twice.
so they have to have either 60 or 180 degrees.

180 degrees is my cstar solution

120 degrees is the sigma solution.
and my funnystar solution
(which is a variant of the sigma solution
i think i "see" that ther cannot be any other solution.
the 4 outer segments are fixed, and the only choice
is to add either 2 inner segments at one end
(my solution) or one inner segment at each end
(sigma solution)

by the same argument,
the cstar solution is the only one with opposite rays in
tha basic figures because both inne segments
are taken to connect the two opposite rays.

so the last possibility is 2 forld symmetry,
or symmery by rotation of 180 degrees.

we need 6 outer and 3 inner segments for the basic figure,
and we need to end diagonally from where we started.
only 3 neighboring rays are possible,
and there is no way doing 3 neighboring rays
and 3 inner line segments and ending up "just opposite".

please check my arguments and tell me if they are correct.

greetings

erich











--
Erich Neuwirth <neuwirth@smc.univie.ac.at>
Computer Supported Didactics Working Group, Univ. Vienna
Visit our SunSITE at http://sunsite.univie.ac.at



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