Thread: "Parity aspects in skew MagicTile"

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NEW parity aspect in skew MagicTile!

The even Duoprismes are also interesting and different.

2nd theorem of Baumann, "PitDeeDom"

Unlike in odd cases here in the even case the / edges are separated in
two orbits. Dito the \ edges!

I encountered a bad parity situation where I had exactly one edge swap
left in each of these 4 orbits.

If I do 4 twists in most compact constellation (corners of a small
square with horizontal and vertical sides), I hit 4 orbits with 12
diamond face elements exactly twice. This can be undone by a 3-cycle.
And I get one edge swap in each of the 4 edge orbits (plus edge swap
pairs).

This repairs the parity.



Recapitulation of parity aspects in skew MagicTile



theorem

name
=81@
restore parity with

twist
number

puzzle

Astrelin

PitDvoRom

odd

turn whole by 60=81=8B

0

{4,6|3} 30 v020 runcinated

1st Baumann

PitDeoBom

odd

turn whole by 90=81=8B

0

{6,4|3} 20 e010 bitruncated

Schumacher

PitDeoDom

odd

big X

14

{4,4|7} 49 e 1.41 duoprisme

2nd Baumann

PitDeeDom

even

small square

4

{4,4|6} 36 e 1.41 duoprisme



Remarks

* In the smaller PitDeoDoms (9 and 25) the big X needs only 6 and 10
twists
* The 4 edge orbits in PitDeeDom have checkerboard pattern
* In the smaller PitDeeDom (16) I was lucky enough to not encounter
the parity problem

















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NEW parity aspect in skew MagicTile!

The even Duoprismes are also interesting and different.

2nd theorem of Baumann, "PitDeeDom"

Unlike in odd cases here in the even case the / edges are separated in t=
wo orbits. Dito the \ edges!

I encountered a bad parity situation where I had exactly one edge swap l=
eft in each of these 4 orbits.

If I do 4 twists in most compact constellation (corners of a small squar=
e with horizontal and vertical sides), I hit 4 orbits with 12 diamond face =
elements exactly twice. This can be undone by a 3-cycle. And I get one edge=
swap in each of the 4 edge orbits (plus edge swap pairs).

This repairs the parity.

Recapitulation of parity aspects in skew MagicTile

Remarks

• 
  
  In the smaller PitDeoDoms (9 and 25) the big X needs on=
  ly 6 and 10 twists


• 
  
  The 4 edge orbits in PitDeeDom have checkerboard patter=
  n


• 
  
  In the smaller PitDeeDom (16) I was lucky enough to not=
  encounter the parity problem

Congrats for finding another parity!

OK my name becomes Schumacher in this table. But I'm happy with it. The rea=
son I use schuma for online forums etc is because I'm a big fan of Michael =
Schumacher.

Nan

--- In 4D_Cubing@yahoogroups.com, "Eduard" wrote:
>
>=20
> NEW parity aspect in skew MagicTile!
>=20
> The even Duoprismes are also interesting and different.
>=20
> 2nd theorem of Baumann, "PitDeeDom"
>=20
> Unlike in odd cases here in the even case the / edges are separated in
> two orbits. Dito the \ edges!
>=20
> I encountered a bad parity situation where I had exactly one edge swap
> left in each of these 4 orbits.
>=20
> If I do 4 twists in most compact constellation (corners of a small
> square with horizontal and vertical sides), I hit 4 orbits with 12
> diamond face elements exactly twice. This can be undone by a 3-cycle.
> And I get one edge swap in each of the 4 edge orbits (plus edge swap
> pairs).
>=20
> This repairs the parity.
>=20
>=20
>=20
> Recapitulation of parity aspects in skew MagicTile
>=20
>=20
>=20
> theorem
>=20
> name
> =81@
> restore parity with
>=20
> twist
> number
>=20
> puzzle
>=20
> Astrelin
>=20
> PitDvoRom
>=20
> odd
>=20
> turn whole by 60=81=8B
>=20
> 0
>=20
> {4,6|3} 30 v020 runcinated
>=20
> 1st Baumann
>=20
> PitDeoBom
>=20
> odd
>=20
> turn whole by 90=81=8B
>=20
> 0
>=20
> {6,4|3} 20 e010 bitruncated
>=20
> Schumacher
>=20
> PitDeoDom
>=20
> odd
>=20
> big X
>=20
> 14
>=20
> {4,4|7} 49 e 1.41 duoprisme
>=20
> 2nd Baumann
>=20
> PitDeeDom
>=20
> even
>=20
> small square
>=20
> 4
>=20
> {4,4|6} 36 e 1.41 duoprisme
>=20
>=20
>=20
> Remarks
>=20
> * In the smaller PitDeoDoms (9 and 25) the big X needs only 6 and 10
> twists
> * The 4 edge orbits in PitDeeDom have checkerboard pattern
> * In the smaller PitDeeDom (16) I was lucky enough to not encounter
> the parity problem
>

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I add a picture for PitDeeDom:

PitDeeDom Illustration
370073/view?picmode=3D&mode=3Dtn&order=3Dordinal&start=3D1&count=3D20&dir=
=3Dasc>


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I add a picture for PitDeeDom:

90/pic/547370073/view?picmode=3D&mode=3Dtn&order=3Dordinal&star=
t=3D1&count=3D20&dir=3Dasc">PitDeeDom Illustration