New dissections to the special classes of triangles

The even solutions of the PP-plus class (referring to squares) yield an unhingeable 5-piece class of triangles:
1² + (2n)² + (2n²)²  =  (2n²+ 1)². There are at least two possibilities for dissecting this class, and the given examples show both of them for n = 2 (left side) and only one solution for  n = 3  (right side).

n =2                                                                                                          n = 3

The first shown case can be dissected hingeable without using more pieces than required for unhingeable dissection of the class:

Hingeable triangles of ratio 1 : 4 : 8 :: 9 ; 5 pieces

In Dissections: Swinging and Twisting this kind of dissection is called "wobbly hinged", because certain pieces can not swing from one position to the other without crossing each other at least partwise.

The odd solutions of the square-sum-plus class are a hingeable 5-piececlass of triangles (compare unhingeable solution of figure 9.13, page 96 in Dissections Plane & Fancy ).

121²  =  72² + 49² +84² ;  (12n²+4n+1)²  =  (8n²)² + ((2n+1)²)² + (4n(2n+1))²

It should be mentioned that there is something like an unhingeable variant of the T-step, and it can be applied to the even solutions of the square-sum-plus class. What’s interesting is that fewer lines have to be cut out of the z-triangles as for the comparable dissections that use the common step technique (figure 9.13,page 96 in Dissections Plane & Fancy).
 

86² = 36² + 50² + 60²

The PP-double class for triangles splits into two classes if you ask for hingeable dissections (from my opinion there would be two classes for the unhinged solutions too). I you use  (n²+2)²  =  (2n)² + (2n)² +(n²-2)²  in order to describe both classes, then a "hidden" Q-step dissects the odd solutions, and a hidden T-step will do for the even solutions. Unfortunately all solutions are wobbly hinged!

27²  =  10² + 10² + 23²

66²  =  16² + 16² + 62²

The Cossali-twin class (see New dissections to the classes of squares) yields hingeable 5-piece dissections of triangles.

39²  =    10² + 14² + 35² ; (4a² + 3)² =  (4a -2)² + (4a + 2)² + (4a² - 1)²

The following class of four hingeable triangles to one is related in the same way to the class for which David Collison gave 5-piece dissections (page96 in Dissections Plane & Fancy ) as the Cossali-twin class is related to Cossali's class.

29²  =  4² + 10² +10² + 25² ;  (n² + 4)²  =  4² + (2n)² + (2n)² + (n²)²

There is a formula that can be used to describe a numerical class that splits into two differently dissected classes of four hingeable triangles to one:  (2n² + 2n + 2)²  =  (n²)² + (n²)² + (n² + 2n)² + (n² + 2n+ 2)².  Again the odd solutions use a Q-step and a T-step is applied to the even solutions.

26²  =  9² + 9² + 15² + 17²

86²  =  36² + 36² + 48² + 50²

Top of Page  Home