Geometric Models – Plane Dissections

To illustrate the Pythagorean Theorem, makers of geometric models have long made models with pieces that could be arranged either as a square with a side equal to the length of the hypotenuse of a right triangle or as two squares, with sides equal to the length of the two other sides of the triangle. In 1928, R. H. Wood, a student of high school teacher A. Harry Wheeler in Worcester, Massachusetts, made such a model.

Any two polygons of equal area can be divided into a finite number of polygonal pieces that can be arranged to form either polygon. This result was well known from the mid-1800s. A few model makers, such as Wheeler, took great delight in developing specific models of dissected polygons and figuring out different ways to arrange the pieces. Surviving notes from the early 1930s indicate that Wheeler designed models of relatively complicated plane dissections for his own pleasure. Then, mindful of the popularity of jigsaw puzzles in the Depression years, he made and encouraged his students to make dissections of simpler forms. Some of these models were hinged at vertices.

Wheeler classified his dissections according to the number of pieces used. The arrangement of records below follows this scheme. In some instances, pieces that fit together were assigned separate numbers. The separate records have been kept, with text indicating what fits together. Records on documentation relating to these models are at the end of the group. Throughout, clicking on the title of an object brings up further images and description.

The three pieces of this hinged dissection are all divided in half, with half red and half black.
Description
The three pieces of this hinged dissection are all divided in half, with half red and half black. They may be arranged to form a regular hexagon with an inscribed black equilateral triangle or a regular hexagon with an inscribed red equilateral triangle.
This is Wheeler's model 3HA. A drawing (indicating two colors) dated March 23, 1932 is in 1979.3002.55 (folder 1). The model name is not indicated there.
On his list of dissections, under regular dissections, Wheeler calls this "Hexagon Inside Out," and denotes it by 3HA See 1979.3002.43 (folder 2).
Location
Currently not on view
date made
ca 1932
c. 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.788
accession number
304723
catalog number
304723.788
The three pieces of this hinged dissection have white, black and red plastic tops attached to wooden bases.
Description
The three pieces of this hinged dissection have white, black and red plastic tops attached to wooden bases. They may be arranged as a quadrilateral in two ways and as a triangle in two ways.
This is model 3J in Wheeler's list of decompositions of polygons, listed with the triangles.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.809
catalog number
304723.809
accession number
304723
The three pieces of this hinged dissection may be arranged as a rectangle or as an isoceles triangle.Sheets entitled Regular Hexagons in 1979.3002.43, folder 2 discuss this model, where it is shown in a list of dissected rectangles. Wheeler apparently denoted it by 3O or 3Alpha.
Description
The three pieces of this hinged dissection may be arranged as a rectangle or as an isoceles triangle.
Sheets entitled Regular Hexagons in 1979.3002.43, folder 2 discuss this model, where it is shown in a list of dissected rectangles. Wheeler apparently denoted it by 3O or 3Alpha. There is a drawing in 1979.3002.43, folder 1, dated January 29, 1932 and labeled 3alpha.
Location
Currently not on view
date made
ca 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.781
accession number
304723
catalog number
304723.781
The hinged dissection has three hinged pieces (two of the triangles and the small square are glued together). They may be arranged as a large square or as two rectangles and a smaller square.
Description
The hinged dissection has three hinged pieces (two of the triangles and the small square are glued together). They may be arranged as a large square or as two rectangles and a smaller square. If the side of the large square is c and the sides of the triangles are a and b (a longer than b), then the area of the small square is (a-b)2. The area of each of the four triangles is 1/2ab. Hence c2 = (a-b)2 + 4(1/2)ab. Hence c2 = a2 + b2, as the Pythagorean theorem would indicate. The model resembles Wheeler's 6SI (square into gnommon), but has only five pieces and is quite different from MA.304723.763, which is labeled as an example of that model.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.767
catalog number
304723.767
accession number
304723
The four pieces of this hinged model are made of plastic glued to wood. The pieces can be arranged as a triangle, a right triangle, or a rectangle. The green plastic is cut, but it covers only one pieceThis is 4A on Wheeler's list of decompositions of polygons.
Description
The four pieces of this hinged model are made of plastic glued to wood. The pieces can be arranged as a triangle, a right triangle, or a rectangle. The green plastic is cut, but it covers only one piece
This is 4A on Wheeler's list of decompositions of polygons. It is a dissection that transforms a triangle into a right triangle.
A pattern for this model found in 1979.3002.55 is dated 1932.
Location
Currently not on view
date made
ca 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.810
accession number
304723
catalog number
304723.810
The four pieces of this hinged dissection can be arranged as a triangle and as an isosceles triangle.
Description
The four pieces of this hinged dissection can be arranged as a triangle and as an isosceles triangle. The pieces are made of yellow, purple, blue and red plastic, glued to a wooden base.
This appears to be model 4B in the list of triangles in Wheeler's list of decompositions of polygons.
Drawings for this dissection, dated February 2, 1932, are in 1979.3002.43 (folder 2). Other drawings, dated February 4, 1932, are in 1979.3002.55, folder 1
Location
Currently not on view
date made
ca 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.750
accession number
304723
catalog number
304723.750
The four plastic pieces fit in metal container with cloth-lined lid. The pieces may be arranged as a square or an equilateral triangle. Compare MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823.
Description
The four plastic pieces fit in metal container with cloth-lined lid. The pieces may be arranged as a square or an equilateral triangle. Compare MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823. This small version of the model has no hinges.
This is similar but not identical to Wheeler's Model 4ST. Two drawings for that puzzle, both dated January 17, 1932 and April 26, 1932, are in 1979.3002.43, folder 1. The drawings refer to the Mathematical Gazette - no volume or page is given. A drawing in 1979.3002.53 dated January 5, 1932 refers to this model as 4C.
Location
Currently not on view
date made
ca 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.819
accession number
304723
catalog number
304723.819
The four pieces of this model may be arranged as a square or and equilateral triangle. The container is a reused aspirin tin. A drawing in 1979.3002.53 dated January 5, 193? refers to this model as 4C. This small version of the model does not have hinges.
Description
The four pieces of this model may be arranged as a square or and equilateral triangle. The container is a reused aspirin tin. A drawing in 1979.3002.53 dated January 5, 193? refers to this model as 4C. This small version of the model does not have hinges. It is stored in a metal case marked: ASPIRIN.
Compare MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823.
Location
Currently not on view
date made
ca 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.823
accession number
304723
catalog number
304723.823
This crude, unpainted, incomplete wooden model consists of three pieces that can be arranged to form a five-sided figure that, with an additional triangle, would form a square.
Description
This crude, unpainted, incomplete wooden model consists of three pieces that can be arranged to form a five-sided figure that, with an additional triangle, would form a square. There is a pencil line across part of the front of the model, and masking tape at various intersections of the pieces on the back. With the additional triangle, the pieces could be arranged to form am equilateral triangle.
For a complete examples of this model, see MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823.
Location
Currently not on view
ID Number
1985.0112.222
catalog number
1985.0112.222
accession number
1985.0112
The four pieces of this hinged dissection may be arranged as a square or as an equilateral triangle.Compare MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823.
Description
The four pieces of this hinged dissection may be arranged as a square or as an equilateral triangle.
Compare MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823. There are two drawings for a related puzzle, 4ST, both dated January 17,1932 and April 26, 1932, in 1979.3002.43, folder 1. The drawings refers to the Mathematical Gazette. A drawing in 1979.3002.53 dated January 5, 1932 refers to this model as 4C.
This is Wheeler's Model 4ST.
Location
Currently not on view
date made
ca 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.764
accession number
304723
catalog number
304723.764
These four hinged pieces may be arranged a an equilateral triangle or a square.Compare MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823.This is similar but not identical to Wheeler's Model 4ST.
Description
These four hinged pieces may be arranged a an equilateral triangle or a square.
Compare MA.304723.764, MA.304723.819, MA.304723.766, and MA.304723.823.
This is similar but not identical to Wheeler's Model 4ST. Two drawings for that puzzle, both dated January 17, 1932 and April 26, 1932, are in 1979.3002.43, folder 1. The drawings refers to the Mathematical Gazette - no volume or page is given. A drawing in 1979.3002.53 dated January 5, 193? refers to this model as 4C.
References:
Fredrickson, Greg N., Dissections Plane and Fancy, pp. 136-137 attributes publication of this dissection to Dudeney (1902).
H. E. Dudeney, The Canterbury Puzzles, pp. 49-50, 178-180. Dudeney specifically points out on p.180 that the puzzle may be made with the pieces hinged. The first edition of the book appeared in 1907, the second in 1919.
Location
Currently not on view
date made
ca 1932.
maker
Wheeler, Albert Harry
ID Number
MA.304723.766
accession number
304723
catalog number
304723.766
The four pieces of this hinged dissection have plastic tops glued to wooden bases.
Description
The four pieces of this hinged dissection have plastic tops glued to wooden bases. They can be arranged to form a triangle two ways.
This is model 4D in Wheeler's triangles in his list of decompositions of polygons.
A drawing of this model, dated January 4, 1932, is in 1979.3002.55, folder 1. It refers to Macaulay, Math. Gaz, 1915-1916, p. 382. Another drawing, marked D and also dated Janury 4, 1932 also shows this dissection. It is in 1979.3002.55, folder 2. There is a drawing in 1979.3002.50, dated November 21, 1931, that refers to a hinged model using this dissection - this model is larger than that drawing.
Location
Currently not on view
date made
ca 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.808
accession number
304723
catalog number
304723.808
This partially hinged dissection has three pieces. It can be arranged as a near-parallelogram or as a triangle in two ways. A protrusion from one edge and a hole in another edge do not meet.These are three of the four pieces in Wheeler's model 4H.
Description
This partially hinged dissection has three pieces. It can be arranged as a near-parallelogram or as a triangle in two ways. A protrusion from one edge and a hole in another edge do not meet.
These are three of the four pieces in Wheeler's model 4H. A drawing for the model, dated January 6, 1932, is in 1979.3002.55, folder 2. It indicates that the missing piece is a triangle which can be added, with the pieces arranged to form a triangle in three ways.
Location
Currently not on view
date made
ca 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.753
accession number
304723
catalog number
304723.753
The four pieces of this hinged dissection may be arranged as a regular hexagon in two ways. An inscribed rectangle in red goes to one in green.
Description
The four pieces of this hinged dissection may be arranged as a regular hexagon in two ways. An inscribed rectangle in red goes to one in green. The wooden core of the pieces is covered with plastic on the top and on the bottom.
In his list of dissections, under "Regular Hexagons," Wheler describes this model as "Hexagon with a Hexagon Inside out" and denotes it as 4HG.
A drawing of model 4HG (without the inscribed rectangle) in 1979.3002.55, folder 1, is dated April 21, 1932.
Location
Currently not on view
date made
ca 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.795
accession number
304723
catalog number
304723.795
The four pieces of this hinged dissection have green and black sections and may be arranged as a regular hexagon with a black rectangle inscribed or as 30-60-90 degree triangle with a green rectangle inscribed. Compare to 304723.794. This is Wheeler's model 4HH.
Description
The four pieces of this hinged dissection have green and black sections and may be arranged as a regular hexagon with a black rectangle inscribed or as 30-60-90 degree triangle with a green rectangle inscribed. Compare to 304723.794. This is Wheeler's model 4HH. There is a drawing in 1979.3002.55 (folder1), although it is not marked 4HH and the inscribed rectangle is not shown. A second drawing in this folder is denoted 4HH and dated April 21, 1933.
In Wheeler's list of "Regular Hexagons" this is denoted as 4HH. See 1979.3002.43, folder 2.
Location
Currently not on view
date made
ca 1933
maker
Wheeler, Albert Harry
ID Number
MA.304723.789
accession number
304723
catalog number
304723.789
The four pieces of this hinged dissection can be arrnaged to form a hexagon or a triangle. Decorative plastic pieces make it possible to show a red rectangle inscribed in the hexagon or a green rectangle inscribed in the triangle.Compare to 304723.789.
Description
The four pieces of this hinged dissection can be arrnaged to form a hexagon or a triangle. Decorative plastic pieces make it possible to show a red rectangle inscribed in the hexagon or a green rectangle inscribed in the triangle.
Compare to 304723.789. This is Wheeler's model 4HH.
In Wheeler's list of "Regular Hexagons" this is denoted as 4HH. See 1979.3002.43, folder 2. A drawing is in 1979.3002.55 (folder1), although it is not marked 4HH and the inscribed rectangle is not shown. A second drawing in this folder is denoted 4HH and dated April 21, 1933.
Location
Currently not on view
date made
ca 1933
maker
Wheeler, Albert Harry
ID Number
MA.304723.794
accession number
304723
catalog number
304723.794
This model has three triangular pieces, with black plastic on top and wood on the bottom.
Description
This model has three triangular pieces, with black plastic on top and wood on the bottom. They can be arranged to form an equilateral triangle or hexagon with a hole in the shape of an equilateral triangle.
The object is shown (with hole) as model 4HI in the list of "regular hexagons" in 1979.3002.43, folder 2. A drawing in 1979.3002.55 (folder 1) shows the model but does not give its name. A second drawing, dated April 21, 1931, is in the same folder and has the name designated.
Location
Currently not on view
date made
ca 1931-1933
maker
Wheeler, Albert Harry
ID Number
MA.304723.806
accession number
304723
catalog number
304723.806
The three pieces of this hinged dissection can be arranged to form an equilateral triangle or a into a hexagon with a hole in the shape of an equilateral triangle.With a second model of an equilateral triangle, this is Wheeler's model 4HI, a hexagon transformable into two equilat
Description
The three pieces of this hinged dissection can be arranged to form an equilateral triangle or a into a hexagon with a hole in the shape of an equilateral triangle.
With a second model of an equilateral triangle, this is Wheeler's model 4HI, a hexagon transformable into two equilateral triangles or a rhombus. This is in Wheeler's list of dissections, 1979.3002.43, folder 2. There is a drawing in 1979.3002.55 (folder 1), but the name is not designated on the drawing. There is a second drawing with the name designated in the same folder, dated April 21, 1931.
Location
Currently not on view
date made
ca 1931-33
ca 1931 -1933
maker
Wheeler, Albert Harry
ID Number
MA.304723.807
accession number
304723
catalog number
304723.807
Thr four pieces of this hinged dissection can be arranged as a regular hexagon or as a rhombus.A drawing dated April 16, 1932 in 1979.3002.55, folder 1 shows this model.
Description
Thr four pieces of this hinged dissection can be arranged as a regular hexagon or as a rhombus.
A drawing dated April 16, 1932 in 1979.3002.55, folder 1 shows this model. In his list of dissections, under "Regular Hexagons," Wheeler describes this model as "Hexagon into a Rhombus." He denotes it by 4HO.
Another related drawing is in 1979.3002.43, folder 1.
Location
Currently not on view
date made
ca 1932
c. 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.790
accession number
304723
catalog number
304723.790
The four pieces of this hinged dissection have plastic attached to both sides and a wooden central core.
Description
The four pieces of this hinged dissection have plastic attached to both sides and a wooden central core. The pieces form an equilateral triangle with an inscribed rectangle which transforms into a parallelogram with an inscribed square.
This is a special case of Wheeler's model 4K (trapezoid into a triangle) in his list of decompositions of polygons.
Compare MA.304723.814.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.813
accession number
304723
catalog number
304723.813
The four pieces of this hinged dissection have plastic on both sides and a wooden core.
Description
The four pieces of this hinged dissection have plastic on both sides and a wooden core. They can be arranged as an equilateral triangle or a rhombus .The pieces on one side have smaller plastic parts on them so that the equilateral triangle and the rhombus both appear to have an inscribed rectangle
A pattern for this model found in 1979.3002.55, folder 1, is dated March 10, 1932. See also 1979.3002.43, folder 2.
Compare MA.304723.813 and MA.304723.814.
Location
Currently not on view
date made
ca 1932
ca 1932 03 10
c. 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.815
accession number
304723
catalog number
304723.815
The four pieces of this hinged dissection have plastic on both sides and a wooden core.
Description
The four pieces of this hinged dissection have plastic on both sides and a wooden core. They can be arranged as an equilateral triangle or a square.The pieces on one side have smaller plastic parts on them so that the equilateral triangle appears to have an inscribed rectangle.
This is a special case of Wheeler's model 4K (trapezoid into a triangle) in his list of decompositions of polygons.
Compare MA.304723.813.
Location
Currently not on view
maker
Wheeler, Albert Harry
ID Number
MA.304723.814
accession number
304723
catalog number
304723.814
This hinged dissection has four pieces that may be arranged as a rectangle or a triangle.This is model 4P among the triangles in Wheeler's list of decompositions of polygons. A drawing dated January 9, 1932 , is in 1979.3002.55, folder 2.Currently not on view
Description
This hinged dissection has four pieces that may be arranged as a rectangle or a triangle.
This is model 4P among the triangles in Wheeler's list of decompositions of polygons. A drawing dated January 9, 1932 , is in 1979.3002.55, folder 2.
Location
Currently not on view
date made
ca 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.782
accession number
304723
catalog number
304723.782
This is Wheeler's model 4Q. It shows an inscribed quadrilateral in four pieces. Each piece is a section of a circle. It is divided into segments of two colors of plastic, so that there is a yellow triangle with a vertex at the center of the circle and a lune in another color.
Description
This is Wheeler's model 4Q. It shows an inscribed quadrilateral in four pieces. Each piece is a section of a circle. It is divided into segments of two colors of plastic, so that there is a yellow triangle with a vertex at the center of the circle and a lune in another color. The model is shown in drawing in 1979.3002.55, folder 1, and dated April 9, 1932 . The four parts can be permuted six ways to form six different inscribed quadrilaterals all inscriptible in the same circle.
Location
Currently not on view
date made
ca 1932
maker
Wheeler, Albert Harry
ID Number
MA.304723.759
accession number
304723
catalog number
304723.759

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