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I just uploaded the latest MagicTile, which includes a number of new
colorings and slicings, bringing the total puzzle count to 1020. I haven't
updated the wiki with slots for recent additions, but I did generate full
puzzle lists
e
and after this change, so you can diff those two files to see what was
added.
Some puzzles are using new code that allows defining colorings via group
relations of a regular map. I've appended the names of these with their
designation from this page
I'd like to point out one of the new regular maps and relate it to
something interesting about hyperbolic 2-manifolds.
The dual {4,7} 42-color and {7,4} 24-color share the same symmetry group as
the Klein quartic (168 orientation-preserving symmetries), but these
surfaces are *genus 10* rather than genus 3. The Gauss-Bonnet theorem
tells us The Gauss-Bonnet theorem tells us the area of a hyperbolic
2-manifold is a function of its Euler characteristic, =CF=87
A =3D -2*=CF=80*=CF=87
So high genus surfaces (with more negative Euler characteristic) have
larger areas.
I wonder if solving a simple slicing of the latter (F0:0:1 or E1:0:0, say)
would feel different than the same slicing of the KQ. My guess is that
even though it has the same number of colors and a larger area, it may
somehow feel more cramped (in a similar way to how the Rubik's cube feels
cramped compared to Megaminx). I'll have to try.
A quick aside: I like the organization of the spherical and elliptical
puzzles in the MagicTile tree, but the hyperbolic folder feels like a
mess. I'm thinking about organizing by genus, then maybe
orientable/non-orientable/orbifold under that. This would scatter the same
Schl=C3=A4fli symbols throughout the tree though, so I'm not sure. If folk=
s
have opinions on what would be best, I'd appreciate them.
Happy 2017 everyone!
Roice
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ber of new colorings and slicings, bringing the total puzzle count to 1020.=
=C2=A0 I haven't updated the wiki with slots for recent additions, but =
I did generate=C2=A0Cubing/files/MagicTile/">full puzzle lists=C2=A0before and after this c=
hange, so you can diff those two files to see what was added.
>
oup relations of a regular map.=C2=A0 I've appended the names of these =
with their designation from der/OrientableRegularMaps101.txt">this page.=C2=A0 I'd like to poin=
t out one of the new regular maps and relate it to something interesting ab=
out hyperbolic 2-manifolds.
-color and {7,4} 24-color share the same symmetry group as the Klein quarti=
c (168 orientation-preserving symmetries), but these surfaces are genus =
10=C2=A0rather than genus 3.=C2=A0 The Gauss-Bonnet theorem tells us
nnet theorem tells us the=C2=A0area of a hyperbolic 2-manifold is a functio=
n of its Euler characteristic, =CF=87
=87
racteristic) have larger areas.=C2=A0=C2=A0
er if solving a simple slicing of the latter (F0:0:1 or E1:0:0, say) would =
feel different than the same slicing of the KQ.=C2=A0 My guess is that even=
though it has the same number of colors and a larger area, it may somehow =
feel more cramped (in a similar way to how the Rubik's cube feels cramp=
ed compared to Megaminx).=C2=A0 I'll have to try.
div>A quick aside: =C2=A0I like the organization of the spherical and ellip=
tical puzzles in the MagicTile tree, but the hyperbolic folder feels like a=
mess.=C2=A0 I'm thinking about organizing by genus, then maybe orienta=
ble/non-orientable/orbifold under that.=C2=A0 This would scatter the same S=
chl=C3=A4fli symbols throughout the tree though, so I'm not sure.=C2=A0=
If folks have opinions on what would be best, I'd appreciate them.v>
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