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Polyhedra in History: Part-1 (Göbekli Tepe - Etruscans)

Updated: Aug 22, 2021


Harold Scott Macdonald Coxeter, one of the greatest geometers of the 20th century, in his book, Regular Polytopes (ref 1), wrote:


"The early history of these polyhedra is lost in the shadows of antiquity. To ask who first constructed them is almost as futile to ask who first used fire"

Coxeter was right, it is probably futile to invest time in exactly pinning down the person who first constructed polyhedra. But, having said that, here is something that is not futile (and actually quite worthwhile, as we will see): In tracing the history of polyhedra, instead of trying to pin down the exact person who constructed the first polyhedra, we will go back in time to as far as the archaeological evidence takes us and move forward from that point. This is the best we can do in our effort of tracing the history of polyhedra. This will take us back in time 11,500 years (9500 BCE) and from there we will move forward to the present times.


The knowledge of conceptualizing and constructing polyhedra is either sustained or (in other cases) continuously rediscovered through out history across various cultures and geographies. In some other places, we see that Polyhedra appear sporadically, an example is the Medieval Islamic world (ref 2). We will (chronologically) explore all of this and much more in this four part series of blog posts called: Polyhedra in History.


In these posts, I attempt to trace the origins of various polyhedra in an exhaustive manner. I will look into all available Archaeological (and other) evidence available on polyhedra. In doing so, we will explore polyhedral motivations in: Architecture, Jewelry, Mesopotamian accounting tokens, an Iranian grave of a burnt and abandoned city among other sources. If there is a possible polyhedra origin story in history, I will hunt that down.


For each of the polyhedra we find in various points in history, we will learn about its geometry and how it relates to other Polyhedra, which family does it belong to and much more. Along the way, I will shed light on the major figures/cultures in history who devoted extensive time to studying, making and providing Mathematical descriptions of these polyhedra.


In Part-1 we will start exploring the earliest polyhedra motivations (in the form of cube and cuboidal structures) in the earliest human made structures like the Göbekli Tepe, c. 9500 BCE (11,500 years ago), a Neolithic archaeological site near the city of Şanlıurfa in Southeastern Anatolia, Turkey. From there we move on to exploring the polyhedra of the Egyptian civilization in ~5000 BCE (~7000 years ago) and move forward in time ending at the Etruscan civilization (900 BCE - 500 BCE).


In Part-2 will start exploring the Greek civilization (from 460 BCE - 320 CE). Greeks contributed extensively to the study of polyhedrons. They were the first to provide a Mathematical description of these objects and contributed extensively towards the field, continuously for a millennia. In this part we also look at a few isolated examples of Polyhedra that appear sporadically elsewhere (Ancient Egypt, Elephantine Egypt, Greco-Roman Egypt, Roman civilization, Ancient Chinese, etc.) during the Ancient Greece era (12th century BCE - 6th century CE). This will take us to 7th century CE.


Part-3, starts in the post Ancient Greece period exploring the Polyhedra of the Medieval Islamic world (starting from 800 CE) and moving on to exploring the plethora of polyhedra artists of the Renaissance (15th and 16th century CE).


Finally Part-4 (last part of the series) starts from Post-renaissance (beginning of 17th century CE) and move forward to the present times.


Please Note: While tracing polyhedra in history, I will introduce and explain new concepts. Many topics that will come up, ideally deserve a separate blog post of their own to be properly discussed. For such topics, I will write separate blog posts in the future. For the purposes of this series (in which our focus is on tracing the history of Polyhedra) I will briefly touch on and explain various concepts, which will get their separate individual attention in posts that will follow the polyhedra in history series of blog posts. I encourage the readers to share their suggestions (about specific topics in these posts that you would like to be more comprehensively explored) in the comments section.



 


The earliest known written record of polyhedra comes from Classical Greek authors who also gave a Mathematical description of them. But, archaeological findings shows us that enquiry into polyhedra for meeting artistic and architectural ends, started much earlier and dates back to as early as 9500 BCE (11,500 years ago). On the other hand, the earliest evidence of Mathematical enquiry into polyhedra can be traced back to the Egyptian civilization (1850 BCE, ~3850 years ago).


Making of a polyhedra, with building materials (or in art) requires substantial knowledge of its geometry. Various civilizations before Greeks were intimately familiar with the geometry of various Polyhedra, but it were the Greeks who were the first to view polyhedra as an entire field (or a subfield of Geometry) of Mathematics, to be studied in and of itself. Before the Greeks, enquiry into polyhedra was more artistic than Mathematical and limited to specific contexts (examples of contexts include: Architecture, Gaming, jewellery, Art etc.) and was not properly studied as a Mathematical Discipline in and of itself. Greeks were the first to view polyhedra as a field of study (a discipline), one that they rigorously investigated.


One thing that remains true for everyone (irrespective of the fact of whether the enquiry was Mathematical or purely artistic), including Greeks, their predecessors, and their successors is aptly summarized in the following quote by Coxeter:


"The chief reason for studying regular polyhedra is still the same as in the time of the Pythagoreans, namely, that their symmetrical shapes appeal to one's artistic sense"

With the above quote in mind, we trace the history of polyhedra by going back in time to 9500 BCE (11,500 years ago) to one of the first manifestations of human made monumental architecture, the Göbekli Tepe. Göbekli Tepe is a Neolithic archaeological site near the city of Şanlıurfa in Southeastern Anatolia, Turkey.





Dated to the Pre-Pottery era, between c. 9500 and 8000 BCE, the site comprises a number of large circular structures supported by massive stone pillars – the world's oldest known megaliths.





A megalith is a large prehistoric stone that has been used to construct a structure or monument, either alone or together with other stones. The image shown below is one of the megalith pillars (pillar 18, Building D) from Göbekli Tepe, on which some stabilization work is being performed.





The slabs that went into making of these limestone pillars were transported from bedrock pits located approximately 100 metres (330 ft) from the hilltop, with workers using flint points to cut through the limestone bedrock.


In these structures (pillar number 18 above), we see the most early form of polyhedral motivations in the form of shapes (of the various megalithic pillars) that are roughly hexahedrons. In most cases they are roughly cuboidal (a cuboid is a polyhedron made up of 6 quadrilateral faces) structures with 6 faces.


A Hexahedron is any polyhedron with 6 flat faces.





On the other hand, a regular hexahedron (more commonly known as a Cube) is the one in which each (of the total 6) faces is a square, and where three square faces meet at each corner (vertex).





These Hexahedrons crop up in some of the oldest surviving architecture, elsewhere in the world at different points in time. A few examples are as follows:

  • Tarxien temples of Malta.

  • Dolmens: a type of single-chamber megalithic tomb, usually consisting of two or more vertical megaliths supporting a large flat horizontal capstone or "table".

  • Dolmen of Menga, a megalithic burial mound called a tumulus, a long barrow form of dolmen, dating from the 3750-3650 BCE approx. It is near Antequera, Málaga, Spain.

  • Dolmens of North Caucasus in Russia.

  • Ġgantija (3700 BCE), a megalithic temple complex from the Neolithic times on the Mediterranean island of Gozo, in Malta.





The architects from these early human settlements would probably not have realized that in making such roughly flat faced pillars out of stone, they were setting the foundations of artistic enquiry into the world of polyhedra.


Moving forward in time to the Pre-Dynastic period (5000 BCE - 3100 BCE, approximately 7000 years ago) of Egypt, we find tombs, called Mastaba (plural Mastabas, precursor to the Pyramids), which take the form of a flat-roofed rectangular structure with inward sloping sides constructed out of mudbricks. These edifices, marked the burial of many eminent Egyptians during that time.





Mastabas are similar to a family of Polyhedra called Frustums (singular: frustum). A polyhedron belonging to the frustum family contains two similar (but not congruent) polygons (called 'bases') that face each other, with their corresponding vertices (corners) connected to each other by straight lines, which makes the rest of the faces (other than the two bases) trapezoids. Given an infinite number of possible base polygons, an infinite number of frustums are possible, each distinguished (and named) given its base polygon.





In the above example, we saw a Frustum with a Trapezoidal Base. Another example of a polyhedra from the frustum family is a frustum with a Square base. This looks like an Egyptian Pyramid (see below) that is sliced from the top. In general, all polyhedra that belong to the frustum family, are cut out portions of some other Polyhedra. Below is an example of a frustum with a square base.


Frustum (Square base)


In general, the frustum family of polyhedra is a subset of a family of polyhedra called Prismatoids. A polyhedron belonging to the prismatoid family is recognized as follows: all of its vertices (corners) lie in 2 parallel planes. Other than the frustum family of polyhedra, the prismatoid family includes these other families of polyhedra: Pyramids, Wedges, Parallelepipeds, Prisms, Antiprisms, Cupolae. All of these aforementioned families are part of the prismatoid family of polyhedra.





Ahead in this blog post, we will see Polyhedra in history from three of these above Prismatoid families: Pyramids, Parallelepipeds and Prisms. The rest of the families of polyhedra will be discussed in greater detail in a separate blog post. Below mentioned are examples of polyhedra (one example each for each family) from all of the above families.





Egyptian Mastabas were pre-cursor to the Egyptian Pyramids (a polyhedron that we will explore in greater detail ahead in this post). The first Mastaba appeared at least 2300 years before the first Pyramid (Pyramid of Djoser in 2670 BCE).


Before we get to the Pyramids, we travel North-West from Egypt to Mesopotamia, specifically: Tepe Gawra (in Iraq). Here, accounting tokens ( from around 4000 BCE) have been excavated. In ancient Mesopotamia, almost 10,000 years ago, scribes started using counters, to represent certain quantities, units or goods (ref 3). Thousands of these tokens have been found in archaeological sites across the Middle East. The ones excavated at Tepe Gawra come in variety of shapes, of which one is roughly a Tetrahedral token (second from the left in the below image).





A Tetrahedron is a polyhedron composed of 4 Triangular faces, 6 straight edges and 4 vertex corners). A regular tetrahedron is a tetrahedron in which all four faces are equilateral triangles.





Later on (around 3200 BCE) these counters, were pressed into clay tablets (while the clay was still soft) to create a lasting record. Two such tablets, from Susa in Iran (created around 3200 BCE) are shown below.





Triangular and Circular impressions can be clearly seen on the tablets. These simple markings, laid the foundation for Cuneiform, one of the first writing systems in History.


One important thing to note is the following: A Tetrahedron has a triangle at its base. The Pyramids at Egypt are not Tetrahedrons (more detail later in the post), because they have a square base. In other words, a Tetrahedron is a pyramid whose base, and also all other faces are triangles.


Next stop in our hunt for polyhedra takes us to Kincardineshire, a historic county on the coast of Northeast Scotland. The solids found here were carved sandstone balls, roughly 3 inches in diameter, dating back to (c. 4000 BCE - c. 1400 cal. BCE) and are currently at display in the Ashmolean Museum at the University of Oxford (ref 4).







The knobs (the rounded bulges) in the above balls are roughly comparable to the vertices (corners) of a polyhedron. Furthermore, if we connect the midpoints of all the knobs (connecting each midpoint to the its adjacent midpoint in all directions by straight lines, that cut through these balls), the resulting skeleton is a polyhedron.


Next, we look at the Step Pyramids that have shown up in various places in History. These structures were similar to the Trapezoidal Prisms that we saw earlier in this post. They look like various Trapezoidal base frustums stacked one over the other, with the highest one having the smallest base area. Egyptian Step pyramids were the pre-cursor to the "True Pyramids" (with smooth sides and a pointed apex).


In the First Dynasty at Saqqara (beginning around 3100 BCE), a large step pyramid like structure was found within the interior of Mastaba 3808 dating to the reign of the pharaoh Anedjib. This is supposedly the first step pyramid of Egypt, and is considered by some Egyptologists as the forerunner of the Step Pyramid of Djoser (earliest colossal stone building in Egypt).





Step Pyramids are not confined to Egypt. Mesopotamian Ziggurats were huge religious monuments, built in ancient Mesopotamian valley and western Iranian Plateau, having the form of terraced step pyramid of successively receding stories or levels. The oldest Ziggurat is the Sialk Ziggurat in Kashan, Iran, which dates back to early third-millennium BCE. There are 32 Ziggurat known at and near Mesopotamia. Some notable ones include the Great Ziggurat of Ur, near Nasiriyah, Iraq, the Ziggurat of Aqur Quf near Baghdad, Iraq.





The step pyramid of Great Ziggurat of Ur is a frustum with a Trapezoidal base (the same shape that we saw above in the Egyptian Mastabas). Furthermore, if we look closely, we notice another polyhedra at its entrance. The staircase of the Ziggurat, is shaped like a right triangular prism.





Earlier in the blog post, we learned that, prisms is a family of polyhedra that itself belongs to the prismatoid family of polyhedra. A prism is a polyhedron comprising of a base that is a polygon, another base that is a translated copy of the first base, and the remaining faces are all parallelograms that join corresponding sides of the two bases without rotation. In the above example of the right triangular prism, the 2 bases are the two right angled triangles (one is an translated copy of the other) and the rest of the faces are rectangles. The right triangular prism is one of the infinitely many possible prisms (corresponding to the infinitely many polygon bases) in the prism family.


It is important that we note the distinction between frustums and prisms. Both prisms and frustums have 2 bases. But, in a frustum the 2 bases are not translated copies of one another. In a frustum, the 2 bases are similar polygons ( their sides are proportional in length) but not congruent. On the other hand, the 2 bases of prisms are congruent and translated copies of one another. To further clarify the distinction, in the image below, a pentagonal base prism is shown alongside a pentagonal base frustum.





Another step pyramid, dating back to 4000 BCE is present at the archaeological site of Monte d'Accoddi, in Sardinia, Italy. It is a trapezoidal platform on an artificial mound reached by a sloped causeway and is relatively much smaller in size than its counterparts in Egypt and Mesopotamia.





As we will see later in this series of posts, various civilizations and cultures at various points in time (after the Egyptians and Mesopotamians), built step pyramids. Some of the civilizations and places where these structures were built are: Mayans, Nsude Pyramids of Africa, South American Step Pyramids (Moche and Chavin culture), Step pyramids of North America, Cambodia and Indonesia (ref 5).


From Step Pyramids, we move to the "True Pyramids" of Egypt. In the Fourth Dynasty (starting around 2613 BCE, around 4600 years ago), the Egyptians began to build these pyramids with smooth sides and a pointed Apex. The earliest of these Pyramids, located at Meidum was first constructed as a finished "tower-shaped" step pyramid like structure and later converted to a true pyramid. Sneferu (the founding Pharaoh of the fourth dynasty of Egypt during the old kingdom) was responsible for converting the earlier step pyramid construction into the True Pyramid with smooth sides.


We have already seen a pyramid with a triangular base (Tetrahedron). The Egyptian Pyramids (like the ones built by Sneferu and his successors) are roughly Square base pyramids. In general, a Pyramid is a polyhedron, which contains a polygon for its base and the rest of the faces (total number of which depends on the base Polygon) are triangles, all meeting at a common point (the apex). Given the infinite number of possible polygon bases, infinite number of pyramids are possible.



Egyptian Pyramid (Square base)


Sneferu is also credited with building two other pyramids, the Bent Pyramid and the Red Pyramid at Dahshur, which were the first true pyramids to be built as such from the beginning. It was Sneferu's son and successor Khufu, the second pharoah of the fourth dynasty who is generally accepted as the pharoah who commissioned the construction of the Great Pyramid of Giza (ref 6).





The Bent Pyramid (top left in the above images) is not a typical Egyptian pyramid (see above), given the bend right around the middle of its height. It is more accurately defined as: a square base frustum (lower section) with a square base pyramid (upper section) sitting over it. The Bent Pyramid rises from the desert at a 54-degree inclination, but the top section (square base pyramid) is built at the shallower angle of 43 degrees, lending the pyramid a visibly 'bent' appearance. As a result, the Bent Pyramid is a Polyhedron with 4 triangular faces, 4 trapezoidal faces, and a square base. This distinguishes it from the typical Egyptian pyramid which has four triangular faces and a square base.


Our knowledge of Egyptian mathematics comes largely from the mathematical problems contained in the papyri that have been preserved by the dryness of the desert climate. Papyrus is a material similar to thick paper that was used in ancient times as a writing surface. It was made from the pith of the papyrus plant, Cyperus papyrus, a wetland sedge.


Peter R. Cromwell, in his book Polyhedra (ref 7), writes:


"These problems contain various problems of practical nature, ranging from plane geometry to solid geometry (volumes of cylinders, barns, beams, pyramids). It is generally accepted (on indirect evidence) that the Egyptians could find the volume of a pyramid, and that they also had a rudimentary notion of trigonometry which allowed them to express the degree of slope (or gradient) of a pyramid. Since the Egyptians did not possess a symbolic notation in which to express their formulae, they had to give a series of worked examples—special cases with particular numbers rather than the general case. The same example would be used in different contexts. For instance, problems 57 and 58 of the Rhind mathematical papyrus both deal with the same pyramid. In the former problem, the height and base are the data and the slope is calculated; in the latter, the height is the unknown quantity"

The fourteenth problem of the Moscow mathematical papyrus (dating to around 1803 BCE) is concerned with the volume of a truncated Pyramid (also known as the frustum). The papyrus itself is 8cm wide and over 5m long. It contains 25 problems, many of which are unclear because the papyrus is damaged. The text containing problem 14 is shown below. It is written in hieratic script, a simplified form of hieroglyphics.





How did the Egyptians discover their formulae and algorithmic procedures? We cannot properly answer this yet, since the papyri are not concerned with such things. They neither contain any motivation, nor any justification to validate the methods they describe. Their only motivations is to communicate the technique or rules to be followed in particular circumstances. The Egyptians did not possess a notation in which they could express general formulation of their results.


Cromwell further writes,


"This was a civilization who had no plus, minus, multiplication or division signs, no equals or square root signs, no zero, no decimal point, no coinage, no indices, and no means of writing the common fraction"

This lack of notation was overcome by giving several worked examples of a similar

nature using different numbers to illustrate the method of solving a particular

type of problem. They expected that the general procedure would be abstracted

from sufficiently many specific examples. But, description of procedures, cannot establish the universal validity of a formula (ref 8).


Babylonians (an ancient civilization from Mesopotamia) considered similar problems (around the same time, i.e. 1800 BCE) to Egyptians, namely problems concerning cones, pyramids, frustums etc. Their records were preserved in the form of Clay tablets, which were inscribed by wedge shaped signs by pressing a stylus into the soft clay. The script is now called cuneiform. Even, later ancient Chinese texts (dating to around first century CE) deal with similar problems. The oldest surviving and most influential of the ancient Chinese mathematics texts is the Chiu-Chang Suan-Shu (Nine Chapters on the Mathematical Art), a series of 246 problems together with their solutions.


The Mathematics of the Egyptians, Babylonians and the Chinese all shared similar features, most of their content is a description of a process, often given as a series of worked examples. It were the Greeks (as far as the evidence dictates) who were the first civilization to introduce a notion of proof in Mathematics. It is uncertain whether early civilizations originating in different places on the globe, could even formulate propositions in a general context. There are no traces of deductive arguments being used to justify methods in any pre-hellenic culture. The Greeks not only stated general propositions but furnished them with rational arguments to demonstrate their universal validity.


Greeks also contributed extensively to the study of Polyhedra. We will get back to them in the second part of this series of blog posts. But, still it is interesting to note that how much progress a civilization (like the Egyptians) can make with a lots of worked examples alone.


Moving back to Mesopotamia, a gaming board (Royal Game of Ur), along with its Tetrahedral dice (1.7 centimeters in length and width and 1.3 centimeters high) made of stone was found in the Royal cemetry at Ur and has been dated to 2600 BCE (~4600 years old).


Ur was an important Sumerian city-state in ancient Mesopotamia, located at the site of modern Tell el-Muqayyar in southern Iraq's Dhi Qar Governorate.



Ruins of the town of Ur


The regally adorned board (full of beautiful and intricate geometric patterns) along with its Tetrahedral dice (see below, the dice is placed in front of the board) is currently at display at the British Museum (Museum item # 120840).



Royal Game of Ur (Gaming board and Tetrahedral Dice)


We already saw the Mesopotamian Tetrahedral counting tokens from Tepe Gawra (4000 BCE). Now, we have a second context in which the Tetrahedron was used in Mesopotamia, namely as a gaming dice.


The Ur board is identical in form to the Burnt City board (a wooden backgammon like gaming board). The Burnt city board was found in Shahr-e-Sukhteh (meaning "The Burnt City") , an archaeological site in eastern Iran corresponding to an early Bronze age settlement.