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New dissections to the special relationships of squares

Unhingeable 6-piece dissection of squares with  x² + y² + z²  =  w² ;  w + x  =  y + z ;   x < y < z . This is an improved dissection that covers the entire possible range of sizes (compare 7-piece dissection of figure 4.22, page 37 in Dissections Plane & Fancy). I have simplified the relationship too (compare 4.5, page 35). For Cossali's class, the P-slide can be converted to a step.


x² + y² + z²  =  w² ;  w + x  =  y + z

Extended solution (y > 8x) for unhingeable 6-piece dissections of squares with  x² + y² + z²  =  w² ;  x + y  =  w (compare figure 4.20, page 36 in Dissections Plane & Fancy). For  y : x  =  2n²  (half the PP-plus class) the P-slide can be converted to a step.


x² + y² + z²  =  w² ; x + y  =  w ;  y > 8x

Frederickson gave a hingeable 7-piece dissection for squares with  x + y  =  w  that uses a T-slide and works if
y < 2x. The range of sizes for hingeable 7-piece dissections can be extended up to  y < 2,5x  by using a T-strip as in the following solution.


x² + y² + z²  =  w² ; x + y  =  w ;  2x < y < 2,5x

Since the general dissection of hingeable three squares to one affords 9 pieces, I shall show an 8-piece dissection for  x + y  =  w  that works up to the size of  y < 8x.


x² + y² + z²  =  w² ;  x + y  =  w ;  y < 8x

I have found a hingeable 8-piece dissection for any size of y that is larger than  (3 + 2Ö2)x. It combines two T-slides: one is applied after taking the z-square out of the w-square. The remaining area is transformed to a rectangle of length w. Then take the x-square out of the opposite corner so that the remaining area can be defined as two attached rectangles of length w and length y. Now the second T-slide reduces the length of the larger rectangle from  w  to  y, and a piece from the first T-slid "reunits" with a neighbored piece of the second T-slide, because both pieces share the same anchor point ("split" the yellow colored, five-sided piece).

You may have noticed that there is a very small red colored and triangle shaped piece in the dissection shown bellow. If I had chosen a larger ratio between  y  and x, than that piece would have been relatively larger ( but alas, - the size problem with the bitmap!).


x² + y² + z²  =  w² ;  x + y  =  w ;  y > (3 + 2Ö2)x   or   w > 2z

On page 35 in Dissections Plane & Fancy Frederickson gave a limit of  4x < 3w  for the relationship
y  =  x(w - x) .  He gave no lower limit for  x, but if you change the orthogonal orientation of the P-slide (Figure 4.23, page 37), then the extension would be  5x > w. From there on, and for any smaller size of  x, the following 7-piece dissection will do:


x² + y² + z²  =  w² ;  y  =  x(w - x) ;  5x < w

The upper bound for the dissection of figure 4.23 on page 37 in Dissections Plane & Fancy is  4x < 3w. The following solution will produce 7-piece dissections for any instances with  5x > 4w, and of course there is a similar solution for  0.75w < x < 0.8w , in which the z-square stays in it's (upper right) corner and y-square is dissected to a  x ´(w - x)-rectangle with the help of a P-slide.


x² + y² + z²  =  w² ;  y  =  x(w - x) ;  5x > 4w

I have found a hingeable 8-piece dissection for  y = x(w - x)  that works whenever  2x < w . It uses Thabit's dissection of two squares to one twice, - but of course only for this special relationship a minimum amount of pieces are achieved. Cutting another pair of squares out of one of the already dissected squares would lead to additional crossings for any other cases. The resulting dissections are unhingeable if  2x > w . There is an alternative for dissecting all sizes with  5x < w , and it leads to a hingeable class (go to New dissections to the classes of squares)


x² + y² + z²  =  w² ;  y  =  x(w - x) ;  2x < w

The dissection of figure 4.21 on page 36 in Dissections Plane & Fancy does not work for the entire range of sizes as given on page 35 for the special relationship  z  =  2(w - y) . The limit for the shown dissection would be y > 3z (or 2x > 3z and not x > z).

I have found a 7-piece dissection for all sizes of the relationship that extend from  x < 2z  up to  z < 2x  (or y > z).  Of course the P-slide used for the x-square has a different orientation within the w-square for all solutions with
x < z  than for those with x > z
 


x² + y² + z²  = w²   ;    z  =   2(w - y)    ;    x < 2z < 4x

I should mention that there is also a similiar dissection for all sizes with  x > 2z  where the x-square is taken
out of the w-square and the remaining gap within the area of the y-square is filled with the help of a P-slide,
- but the dissection shown on page 36 in Dissections Plane & Fancy has a greater extension of sizes.

If  y < z, then almost the same method can be used in order to achieve unhingeable 7-piece dissections as well as hingeable 8-piece dissections:


x² + y² + z²  =  w² ;  z  =  2(w - y) ;  y < z ;  unhingeable: 16y > 13z ;  hingeable: 9y > 7z

From  16y < 13z  on down to any smaller size for  y, it is suitable to position the  x-square and the  y-square into opposite corners. This time I decided to show an 8-piece dissection that uses a T-slide for the hingeable solutions, because it has the same range of sizes as the unhingeable 7-piece solution that uses a P-slide. Of course, a Q-slide would have had a greater range of sizes, - but that isn't necessary since the hingeable dissection from above extends the upper bound for y in the dissection bellow anyway. For  x = 0  (with 4y = 3z) the method shown bellow leads to a hingeable 4-piece dissection of a famous Pythagoreen triple (shown by Frederickson in his new book).


x² + y² + z²  =  w² ;  z  =  2(w - y) ;  16y < 13z

For the special relationship of three squares to one with  y = z , Frederickson gave two different dissections that cover all instances with  y < 2x  (page 36 in Dissections Plane & Fancy). I have found a third dissection that handles all instances with  y > 2x, so that it is now possible to produce unhingeable 7-piece dissections for all possible sizes of the relationship.


x² + y² + z²  =  w² ;  y  =  z ;  y > 2x

In order to make the dissections of three squares to one with  y = z hingeable, Frederickson used a simple 8-piece dissection that cuts the two squares of equal size into four pieces and fits them into the smaller square from Perigals dissection of two squares to one. I decided to show that it is possible to leave the two squares of equal size uncut, because for certain cases the hingeable 8-piece dissections with that property can save a piece by converting a T-slide to a T-step, - thus leading to hingeable classes (see there).

In the dissection shown bellow there are again two T-slides that share an anchore point, - thus saving one piece. Alternatively the double T-slide could have a different orientation within the  w-square, - which leads to another hingeable class, but reduces the range of sizes.


x² + y² + z²  =  w² ;  y  =  z ; x > Ö2y

In his new book, Frederickson gave a hingeable 9-piece dissection for the special relationship  y  =  z . It again uses Perigals dissection of two squares to one twice and covers all instances with  x < Ö2y . I do not know how to handle the entire range of sizes from Frederickson's dissection with less than 9 pieces, - but I have found a solution that produces hingeable 8-piece dissections whenever  y > 2x . Again the dissections take advantage of a double T-slide, - but this time it is used to transform the area of the  z-square that is left over after taking the x-square and the  y-square out of the  w-square.

I most apologize for having to draw a very special case due to the size-problems with the bitmap. The two small pieces (yellow and red) that fill a rectangle shaped area within the  z-square (second square bottom left) are only exceptionally of the same length. As you may have noticed, a T-step would have been possible for this special case, - thus saving one piece (but there are no classes that derive from the same method).


x² + y² + z²  =  w² ;  y  =  z ; y > 2x
 
 

I end this expositions to the relationships of three squares to one by introducing a new special relationship:
x + (1 - Ö2)y + z  =  w   with   x < y < z . Though I have found an unhingeable 7-piece dissection that covers all possible sizes, I know of no hingeable 8-piece dissection. Of course there are no rational classes to this special relationship, and probably no classes with hingeable 7-piece dissections either. In order to understand the mathematics of the following dissection method, simplify the sizes by using   w  =  z + 1 (there are no rational solutions anyway!), - then a stripe with 1 ´ w  plus another stripe with  1 ´ z  add up to the area of the  x- square and the  y-square. Since the  x-square is positioned in the upper right corner next to a rectangle-shaped piece of length  y  that has the same height as the  x-square, we can assume that the remaining left part of the  1 ´ w  stripe has an area that adds up to the  x ´ y-rectangle in order to fill in the missing part of the  y-square when being reshaped. Then the remaining part of the 1 ´ z-stripe (bottom right) must have an area that equals the gap in the  z-square (left over after taking out the x-square and the x ´ y-rectangle). Thus the  y-square and the  z-square both have a piece of the desired length that is attached to a piece of a different length, but of the desired area in order to be reshaped with a P-slide.

By using the simplified relationship  x + (1 - Ö2)y  =  1 ,  there should be no problems for calculating the sizes of the squares or deducting suitable formulas. The following graphic shows the case when  z = 6 , which leads to  y  =  (sqrt.(49+24Ö2) - 1) / 2Ö2 .

--------->    in preparation    <----------
 

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