Hingeable Greek crosses of ratio 1 :Ö8 :: 3 ; 8 pieces, - 7 pieces if unhingeable (attache one of the yellow squares to the red piece of the 1-cross)
New dissections of Greek crosses and Latin crosses
1. Greek crosses
Hingeable Greek crosses of ratio 1
:Ö8
::
3 ; 8 pieces, - 7 pieces if unhingeable (attache
one of the yellow squares to the red piece of the 1-cross)
Hinged pieces for Greek
crosses of ratio 1 :Ö8
::
3
On page 46 in Hinged Dissections: Swinging and Twisting, Greg Frederickson shows a hingeable 8-piece solution of Greek crosses with ratio 1 : 3 :: Ö10. He does not mention me as having sent him an unhingable 7-piece solution to the same case, befor he got the idear to make it hingeable by using only one piece more (I had not cared much about hingeable dissections to that time), - so I decided to show that there is a (slightly) more simple structured and somewhat better proportioned hingeable solution existing.
Hingeable Greek crosses of ratio 1
: 3 :: Ö10
; 8 pieces, - 7 pieces if unhingeable (attach a blue quadrilateral to the
top red piece)
Hinged pieces for Greek crosses of
ratio 1 : 3 :: Ö10
There are many 9-piece solutions for hingeable Greek crosses of ratio 1 : 2 :: Ö5 , but there seems to be only one 7-piece dissection if the crosses are not required to be hingeable.
Unhingeable Greek crosses of ratio
1 : 2 :: Ö5 ;
7 pieces
As for the hingeable crosses, I am still suspiciouse if there could be an 8-piece solution existing, because I have found a few partially hinged dissections of Greek crosses with ratio 1 : 2 :: Ö5 , where only one piece remained unhinged. Just to give anyone who is interested some examples of the wide varietys of 9-piece hingeable solutions; — here are some hingeable 9-piece dissections that you may try to find:
— A symmetrical solution where the
1-cross is dissected with three pieces.
— Any solution that requires four
pieces for the 2-cross.
— Another symmetrical solution with
an uncut cross than the one shown on page 46 in Hinged Dissections: Swinging
and Twisting (or any solution with that property if you haven't read that
book yet).
I had first found an unhingeable 9-piece solution with rotational symmetrie for the following Greek crosses, but than discovered that there are a few unsymmetrical 8-piece solutions existing. Interesting enough, - I had not found them befor looking for hingeable solutions, because my unhingeable symmetric solution was not suitable to be made hingeable.
Uningeable Greek crosses of ratio
Ö2
: Ö3
:: Ö5
;
9 pieces
There seems to be only one 8-piece solution for Greek crosses of ratio 1 : Ö2 : Ö2 :: Ö5 , but there are many hingeable 9-piece dissections of the same crosses existing. Can you find a hingeable solution with only one piece more as required for the dissection shown below and where the larger, pale colored piece of the little cross stayes uncut?
Unhingeable Greek crosses of ratio
1 : Ö2
: Ö2
:: Ö5
; 8 pieces
Hingeable Greek crosses of ratio 1
: 2 : Ö5
:: Ö10
; 10 pieces
Hingeable Greek crosses of ratio 1:
2 : Ö20
:: 5 ; 11 pieces, - 10 pieces if unhingeable (then only one square of length
2 is required for the 2-cross)
Special 11-piece class of unhingeable
Greek crosses with ratio 1 : n :: sqrt.(n² + 1) ; shown example is
of ratio
1 : 5 :: Ö26
The cases for n = 1; 2; 3 and 4 can
be dissected with fewer pieces, and here is an unhingeable solution for
n = 4:
Unhingeable Greek crosses of ratio
1 : 4 :: Ö17;
10 pieces
2. Latin crosses
Hingeable Latin crosses of ratio 1
: Ö
8 :: 3 ; 10 pieces, - 8 pieces if unhingeable
(attach one of the squares to the large piece of the small cross and let
two equal triangles build a triangle twice their size)
Hingeable Latin crosses of ratio 1
: 1 : Ö2
:: 2 ; 11 pieces, - 10 pieces if unhingeable
(attache one of the squares to the yellow piece of the 1-cross). The hinge
for the two connected red pieces can be frozen (quadrilateral and small
triangle).