Using the Regular and Semi-regular Solids
Hallenberg, Harvey R
Silicate rocks, rocks containing primarily silicon and oxygen, dominate the Earth’s crust, even where limestone has lithified on top of them. The primary “unit” of all silicate minerals is the crystal lattice tetrahedron, formed with a silicon atom at its center and four oxygen atoms equidistant at the four “points,” or vertices.
Ancient Greek natural philosophers, more than 2,000 years ago, speculated about the fundamental composition of the living and nonliving things in the world around them. Naturally, they pondered the composition of the rocks and the mountains. When Plato (427? – 347? BCE) described the five regular solids in his dialogue Timaeus, the natural philosophers and mathematicians of his time came to believe that solid things in the world were composed of “earth” in these five forms. The multiplicity of solid things could be imagined to contain different proportions of “earth” as tetrahedra, hexahedra (cubes), octahedra, dodecahedra, and icosahedra. What a marvelous intuition!
However, as with many intuitions, the judgment was only partially correct. The silicate rocks, including quartz and granite, are truly composed of tetrahedral lattice structures of atoms. Salt (sodium chloride) is truly composed of hexahedral lattice structures of atoms. And diamonds, when found in nature, are octahedral. However, almost all other crystalline solids do not conform to the Greek philosophers’ generalization. Crystal lattice structures abound, but not in the five “regular” forms first described by Plato.
From 1913 through 1939, the British firm of Philip & Tacey, Ltd., sold pedagogical materials approved by Maria Montessori for use in her Children’s Houses and in Montessori “departments” for older children, in English public and private schools. One of the Philip & Tacey catalogs was reproduced in a facsimile edition with other Montessori materials catalogs in 1993 by Grazia Honegger Fresco. On page 136 of this facsimile edition are pictured models of the five regular solids! Could these models have been a standard part of the Montessori math apparatus for older children in the 1920s and 1930s?
Claude Claremont’s residential Maria Montessori Training College in Hampstead (Wade, 1991), a northern suburb of London, hosted Maria Montessori every two years during the 1920s and 1930s. Montessori gave the final lectures of each two-year course; she also examined the students and signed their diplomas. Were the regular solids Claude Claremont’s addition to the Philip & Tacey catalog? It is perhaps indicative of Claude Claremont’s influence on the Philip & Tacey company that a large photograph of the building at Rosslyn Hill, used for the residential training college, appears in the catalog, along with a very complimentary written tribute to Claude Claremont’s efforts to extend Maria Montessori’s educational reform in England.
The question remains, “Who inserted the illustrations of the five regular solids in the Philip & Tacey catalog?” There is no mention of the five regular solids in Montessori’s seminal work on geometry, Psico Geometria, which was first published in 1934 (Casa Editorial Araluce, Barcelona).
Whether Maria Montessori herself, or Claude Claremont, first proposed the inclusion of the models of the five regular solids into the “advanced Montessori method,” the Philip & Tacey catalog indicates Montessori’s approval of this inclusion. She was very aware of everything sold under her personal name and imprimatur.
If we assume that Maria Montessori saw the value of presenting models of the five regular solids to elementary-aged children, the larger question must be, “When and why were they abandoned?” In fact, no current supplier of Montessori’s pedagogical materials includes these models in its catalog. Various prisms, pyramids, the cone, the ellipsoid, the ovoid, and the sphere are to be found in the standard lists of geometric solids in these catalogs, but only the cube (hexahedron) survives to represent the Platonic solids.
I would rather not spend any more time speculating about when and why the other four Platonic solids disappeared from the “elementary material.” I would much rather spend my time making a case for returning models of these solids to our classroom shelves.
Whenever possible, we should put new ideas or new artifacts into an historical context for elementary students. One of the emerging “sensitivities” of elementary students is a sense of chronology, a sense of time. The youngest elementary students adore dinosaurs, the denizens of the prehistoric “Age of Reptiles.” Slightly older elementary students become fixated on the kings and queens of ancient times and, often, distant places. Maria Montessori recommended that we place everything possible in its prehistorical or historical context, to better touch the elementary child’s inner world.
So, let us put Plato into his historical context. Plato was a great teacher and thinker who lived in the city-state of Athens (in what is now Greece) during a time of war and conquest. He became a student of Socrates, a great teacher and philosopher, in 407 BCE. The Athenian army was destroyed in 414 BCE, attempting to capture the city of Syracuse in Sicily. This defeat led to a coup d’etat in Athens, in which some of the military generals were executed and political power was transferred to a people’s assembly. However, military adventures were not abandoned. The off-and-on war with the city-state of Sparta flared up again. The Athenian capture of Byzantium in 408 BCE did nothing to stop hostilities with Sparta. Sparta’s peace offer of 406 BCE was rejected by the Athenians.
In the midst of war and political turmoil, Plato learned to question “authority” from Socrates. Socrates taught Plato to think for himself and not just parrot the judgments of those around them.
Plato believed that the material world was composed of earth, water, air, and fire. (We would identify solids, liquids, gases, and the process of combustion with these “elements.” Our distinctions are called physical states.) Plato also identified a fifth element he calledprana, a substance that held the other four elements together (Lawlor, 1982).
Plato, in his dialogue Timaeus, created a cosmology using a metaphor of plane and solid geometry. Plato identified each “element” in his cosmology with one of the five regular solids. Fire was associated with the tetrahedron, earth with the hexahedron (cube), air with the octahedron, and water with the icosahedron. Prana, the “glue” that bound the four primary “elements” together in different proportions, was associated with the dodecahedron (Eawlor, 1982).
As the natural world around him seldom gave Plato direct evidence of the five regular solids, he had to invent a realm of “pure” number and form that existed on a higher plane of reality. Objects in the natural world were merely projections of these higher numbers and forms. It is interesting to speculate that Plato may have intuited the existence of molecules and crystal lattice structures composed of the chemical elements in precise numerical arrangements that do, as it seems to us now, compose the stuff of the universe. (Modern-day scientists recognize 91 natural chemical elements in the universe, not just 4 or 5.)
Plato described the characteristics of the five regular solids in his dialogue Timaeus, and is usually given credit for first identifying them. However, as most historians know, much of the knowledge of the ancient Mediterranean world was consumed in flames when the Library of Alexandria was burned-twice. Plato may not have been the first philosopher to describe the regular solids on a roll of papyrus or parchment. We will never know for certain.
Could the Neolithic farmers of the island of Britain have known about the five regular solids 1,000 years before Plato was born? There are a number of spherical stones in the Ashmolean Museum at Oxford in England that appear to have been carved with grooves that resemble the edges of the regular solids, somewhat rounded. Other stones have nodal points between the grooves that can be connected to differentiate the regular solids. Could these carved stone balls represent the knowledge of the regular solids 10 centuries before Plato? The evidence is not conclusive. If you like to carve grooves in spherical stone balls, inevitably some of these balls may acquire the characteristics of the regular solids (Lawlor, 1982).
If you wish to create models of the five regular solids yourself, you can follow the directions in the next paragraphs (also see the illustrations pictured at the top of the page). However, directing students to produce the models from poster board has far greater educational value. In my experience, boys are attracted to this activity more than girls. I find this fact to be surprising as girls, generally speaking, seem more interested in making artifacts out of cut-and-folded paper and cardboard. Boys are often characterized by teachers as being manually “challenged” by comparison with girls of the same age. The activity of creating models of these regular solids seems to overcome the inhibitions and the manual dexterity “problems” of elementary-aged boys as young as 8. I have witnessed great growth of self-esteem in boys who persevered to create models of all five regular solids, for more than 35 years!
The tetrahedron has 4 faces that are equilateral triangles. The hexahedron (cube) has 6 faces that are squares. The octahedron has 8 faces that are equilateral triangles. The dodecahedron has 12 faces that are regular pentagons. And the icosahedron has 20 faces that are equilateral triangles. all five of these regular solids can be modeled rather easily with Montessori’s Metal Insets. A “net” or network for each model can be traced with the Metal Insets on pieces of light-colored poster board. The perimeter of each network can be extended slightly with “tabs” shaped like flattened trapezoids. These tabs will make it possible to glue the models together, after all of the edges have been scored and folded. (Scoring is done with the point of a pair of scissors along the edge of a steel ruler. Care must be taken not to score the edges too deeply or the network will come apart. Always fold away from a scoring. Practice makes perfect!)
Montessori created her Metal Insets using the decimeter (10-centimeter) measure whenever she could. Years ago, I created plastic templates for the regular polygons with 10-centimeter edges. The pentagon with a 10-centimeter edge makes a rather large and impressive dodecahedron, but it also uses up more poster board. If you choose to use a regular pentagon with a 10-centimeter edge, you will probably not find sheets of poster board large enough to draw the complete network for the dodecahedron. This difficulty can actually lead to a creative circumstance. Suggest tracing the larger regular pentagons separately on pieces of poster board of different colors, or use remnants of colored poster board. Remember that if the pentagon template is traced separately, tabs will need to be drawn on all five sides of each pentagon. The regular solid models become works of art if they are made from different colors of poster board.
Tabs may be glued together inside a model or outside a model. Gluing the tabs inside a model makes the finished product look more like a solid, but gluing tabs together inside the models is harder than gluing them outside. Over the years, many of my students have used cellophane tape on the outside of their models to hold them together while the glue dries on the inside. If drawing and scoring the tabs is too difficult for your students to manage, simply taping the faces together is an adequate solution. Expect to purchase many rolls of cellophane tape. (Another problem with using white glue to assemble these models is the inevitable spilling of glue on the outer faces of the models. Spilled glue immediately discolors poster board. When using white glue, advise the students to use small amounts. Small amounts of glue dry faster and create less mess.)
Most recently, students in my upper elementary class at the Claremont Montessori School in Boca Raton, FL, have preferred using staples to attach tabs of their geometric solids. Small staplers seem to work best if the tabs are stapled inside the models.
Once the models are fabricated, the students can be shown how to calculate the surface area of each model. Finding the surface area of the hexahedron (cube) is the easiest and should be chosen first. As the edge of each square face is 10 centimeters, the surface area of each square face will be calculated at 100 square centimeters. Simply multiplying 100 square centimeters by 6 (the number of faces) will give the result as 600 square centimeters.
The tetrahedron, octahedron, and the icosahedron all have faces that are equilateral triangles. Determining the area of a single triangular face will allow the student to calculate rather easily the surface areas of these three regular solids. One of the Montessori Geometry Insets for use by elementary students demonstrates how an equilateral triangle with an edge (base) of 10 centimeters can be converted geometrically into a rectangle. Calculating the area of the rectangle produced by this transformation is, again, rather easy. The student should be shown how the width of this rectangle contains a fraction of a centimeter. Rather than teaching students to round off the width number to the nearest centimeter, I advise having them measure the fraction of the centimeter in millimeters. This will produce a number with a decimal point such as 4.3 cm.
The only regular solid that presents a real difficulty with regard to calculating its surface area is the dodecahedron. Each face is a regular pentagon. Once again, the Montessori Geometry Insets present the solution of this problem. The Regular Pentagon Insets demonstrate how a regular pentagon can be transformed into a rectangle if the apothem is determined first. The apothem is the line from the exact center of the regular pentagon to the midpoint of any one side. Once the apothem is determined as the width of the rectangle, the length of the rectangle will be seen to be 2 ½ times the side lengths of the pentagon. Another one of the Montessori Pentagon Insets shows an elongated rectangle, which has a width of exactly half of the apothem and a length of the perimeter (all five sides or edges) of the pentagon. Both rectangles have the same area. These mathematical calculations add a considerable depth to the activity of constructing models of the five regular solids, a depth that Claude Claremont and Maria Montessori would concur as “expanding the mathematical mind” of the student so involved.
I hope my description of the five regular solids and their construction has convinced you to add these models to your classroom mathematical materials. However, I would also like to propose adding models of the 13 semi-regular solids to your geometry lessons, especially lessons for upper elementary students.
Models of the semi-regular solids do not appear in any old or new catalog of Montessori materials, so you may not find any justification for creating them with your students. Maria Montessori did not describe them in her Psico Geometria, and Dr. Claremont never mentioned them in my presence during the 3 years I studied with him. In fact, I did not know they existed until I began to research Archimedes for the purpose of impersonating this ancient Greek scientist and mathematician. The 13 semi-regular solids were first described in the writings of Archimedes. This is why they are often called the Archimedean polyhedra.
The regular solids have regular polygonal faces of the same type for each solid. Also, the internal dihedral angles are all the same within each solid. No one has ever been able to create a regular solid with regular polygons of more than six sides. “Three regular polygons of more than six sides cannot meet at a point without overlapping. Evidently the tetrahedron, cube, octahedron, dodecahedron, and icosahedron are the only regular solids” (Beard, 1973). Some may believe that Buckminster Fuller discovered or created new regular solids when he created geodesic domes, but Fuller’s geodesic structures were not created with regular polygons. Some of his “domes” were two-frequency structures, meaning that they had isosceles triangular faces. But, most were three-frequency structures that had scalene triangular faces. At first glance, the triangles of these geodesic structures appear to be equilateral, but they are not.
Archimedes had one of the most inventive minds in the ancient world. He came close to devising a mathematical calculus almost 2,000 years before Newton and Leibniz successfully did so. Archimedes was born in Sicily about 287 BCE and died in the sack of Syracuse by Roman soldiers in 212 BCE. He lived at a time when Rome and Carthage were contending for control of the Mediterranean Sea and the lands bordering this sea. King Hiero of Syracuse, a kinsman of Archimedes, skillfully kept Sicily safe from invasion for almost 50 years by playing one power off against the other. Archimedes spent part of his life studying with Eratosthenes and other thoughtful teachers at the Library of Alexandria, in Egypt. While in Alexandria, Archimedes undoubtedly read Plato’s Timaeus and pondered the mathematical order of the universe. Archimedes might have made models of the regular solids. Most ancient Greek philosophers or thinkers rejected the idea of materializing abstractions, but there is some evidence that Archimedes made models to help himself conceptualize. Whether or not he made models of the regular solids, he imagined truncating the apices or points of these solids in such a fashion that the faces remained regular polygons.
Thus, he was able to model or visualize a truncated tetrahedron, a truncated hexahedron, a truncated octahedron, a truncated dodecahedron, and a truncated icosahedron. Following his own logical thinking process, Archimedes truncated the apices of these five “semi-regular” solids and discovered eight more semi-regular solids. Try as he might, he was unable to discover any more solids with combinations of regular polygons as faces. In fact, there are only 13 semi-regular solids.
Modern mathematicians have renamed some of the last eight semi-regular solids. It seems to me that the grouping of four into the cubocta group and four into the icosidodeca group makes the most sense. The cubocta group is comprised of the cuboctahedron, the rhombicuboctahedron (small rhombicuboctahedron), the rhombitruncated cuboctahedron (great rhombicuboctahedron), and the snub cuboctahedron (snub cube). The icosidodeca group is composed of the icosidodecahedron, the rhombicosidodecahedron (small rhombicosidodecahedron), the rhombitruncated icosidodecahedron (great rhombicosidododecahedron), and the snub icosidodecahedron (snub dodecahedron).
If you think that elementary students are intimidated by these names, think again! Elementary students are enchanted by superlatives. Long words fascinate children. Broken down into their syllables, these names are not hard to pronounce.
The networks or patterns for these 13 semi-regular solids are published in many books devoted to solid geometry (Beard, 1973; Critchlow, 1970). I indicated above that I fabricated a series of regular polygons in Plexiglas many years ago. All of these templates have 10-centimeter sides or edges. To be able to construct all 13 semi-regular solids out of poster board you must have an equilateral triangle, a square, a pentagon, a hexagon, an octagon, and a decagon. During the 2001-02 and 2002-03 school years, one boy, Alex Schopp, in my elementary class at the Claremont Montessori School, constructed models of all 13 semi-regular solids. He received help from time to time, but the driving force to create all 13 models was his alone. Needless to say, he was quite proud of his accomplishment.
It would be pointless to try to describe the procedure for fabricating each and every semi-regular solid in this article. Once the network is traced on a piece of poster board, the edges must be carefully scored and folded. Any careless tracing or scoring will result in a model that will not be close properly. Very few of the semi-regular solids can be networked on a single piece of poster board. The solids with octagonal or decagonal faces may require a dozen pieces or more, if you adopt my templates with 10centimeter edges.
Over the years, I have had students calculate the surface areas of the semi-regular solids they created, but I must say the primary feat was simply the construction of the model. If any student seemed reluctant to calculate a surface area, I did not press the issue.
One other arithmetical activity seemed to engender enthusiasm whenever I introduced it: Euler’s Theorem. Leonhard Euler discovered a simple mathematical relationship among the apices, edges, and faces of all polyhedra. Apices plus faces equal edges plus two: A + F = E + 2. While it is not difficult to prove Euler’s Theorem with the regular solids, the semi-regular solids present a student with the challenge of counting the apices, faces, and edges accurately. These parts may have to be labeled with pieces of numbered masking tape.
In conclusion, I need to state clearly that we must not abandon the many pedagogical materials that Maria Montessori created with the assistance of her trusted colleagues such as Claude Claremont. The Stamp Game and the Bead Frames continue to fascinate children today as they diddecades ago. However, we must also not lose sight of the fact that we are continuing a tradition of scientific pedagogy begun by Maria Montessori at the end of the 19th century. A scientific pedagogy must grow and evolve as any scientific discipline does. If you are timid about introducing the five regular solids into your prepared environments, know that they were approved by Maria Montessori for use in English elementary programs in the 1920s and 1930s. Surely they can be used to the children’s advantage in the 21st century.
Once you have made it possible for the students to create their own models of the five regular solids, the logical extension is to the 13 semi-regular solids. The advantages for the development of manual dexterity in boys who accept the discipline of constructing models of the solids will be immediately apparent. It is no accident that Alex Schopp is not only the most adept fabricator of geometric solids, but he is also the most adept lefthanded calligrapher of the Italic Chancery Hand whom I have ever directed.
Beard, R. S. (1973). Patterns in space. Creative Publications: Palo Alto, California.
Critchlow, K. (1970). Order in space. Viking Press: New York.
Fresco, H. (1993). Montessori material: Contained in some of the catalogues published in New York, London, Bucharest, Berlin, Gonzaga, from 1910 to the 1930s. Edizioni II Quaderno Montessori: Varese, Italia.
Lawlor, R. (1982). Sacred geometry: Philosophy and practice. Crossroad Publishing: New York.
Montessori, M. (1934). Psico geometerica. Casa Editorial Araluce: Barcelona.
Pearce, P. & Pearce, S.P. (1978). Polyhedra primer. Van Nostrand Reinhold: New York.
Wade, C. (ed.). (1991). The streets of belize. Camden History Society: London.
HARVEY HALLENBERG is head of Claremont Montessori School in Boca Raton, FL.
Copyright American Montessori Society Spring 2004
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