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Section I — Prehistory Section I — Prehistory We look at how and why people did math. We don’t try for a thorough history, just to give a couple ideas. Our source is a nice little book by Miranda Lundy. Why did people do math? Most would say to solve problems … • counting how many sheep one has (number) • Dividing fields (geometry) • Building bridges, temples, etc (number and geometry) There is some truth to this but … • Most mathematics (past and present) has no application at the time it is done. Much mathematics never has an application, or finds unexpected application only long after it is completed. So why did people do mathematics? Likely two reasons closest to the truth are the following • They find it fun and interesting • They are trying to understand their world. The Circle Surely people have known about circles for many thousands of years. Did they know of circles before they could count? When did they understand its many miraculous properties? Two ways of making a circle Tie two sticks together with a rope. Put one stick firmly in the ground (the center), and move the other end around keeping the rope tight. A slightly fancier version of this is the compass. This works because the points on a circle are exactly the points a given distance from the center! A second way of making a circle Place a drop of oil on smooth water! This works because a circle is the 2-dimensional figure having the smallest perimeter for a given surface area! Note! These are two very different ways of thinking about the same thing (a circle). This kind of flexible thinking can be wonderfully useful, and not just in math. Human views of the circle People associate circles with certain themes … • Cycle of life • Perfection • Divinity and gods Can you think of others? Halo Shiva Mayan calendar Chinese Yin-Yang Voodoo circle Gila petroglyph Practical uses of circles The wheel is a big one . Did people know of circles before they invented wheels? Probably they did. Probably a long time before. What else did people do with circles? Two circles Draw two circles so that each passes through the center of the other. The overlapping part is called a vesica pisces, meaning bladder of a fish. Lets do this together on the board … Jesus is commonly drawn inside a vesica. Chalice well gardens (linked to holy grail) The vesica is one of the primary symbols of Christianity. Islamic amulet founding fathers Masonic symbol Other religions and groups also have ties to the vesica. The stories for goddess cults, pagans, etc. are things we are best not to follow here. Uses of the vesica • The vesica is used a great deal in mystic symbolism. • It seems to have few, if any, “practical” uses. • It does have a number of mathematical uses … The vesica and the triangle The vesica easily produces two equilateral triangles. The vesica and the hexagon Make triangles as before, and repeat with another vesica on the other side. Give it a try on the board. Six circles around one An amazing property of circles is that six circles fit perfectly around one. Try this with some pennies! This is closely linked to the hexagon and vesicas we discussed before. Starting with a hexagon … Draw circles at each of the six corners of the hexagon. Each circle should hit the nearest corners. This produces the beautiful figure at right. We build six around one … With the figure we just constructed (dotted lines) Make the centers for the six circles around the one These ideas are common and old Ancient India temple Osiris Egypt Da Vinci Twelve around one Our earlier figure is easily used to create a perfect 12 sided figure called a dodecagon. Twelve around one ̶ another way Just as six circles fit around one, so do twelve spheres fit around one. This is a cuboctohedron. At left is a neolithic (stone age) carving from Scotland dated about 3000 BCE. At right is the rose window from the cathedral at Chartres. Six around one is common in creation stories in many religions (seven days of creation, seven days in a week). Twelve around one is common is many religions as well (twelve disciples) Only a start … We have only discussed a few of the simpler aspects of ancient geometry. There is much, much, more. Try doing a Google search on “sacred geometry” to get an idea of the depth and scope of such things, and how they cut across most major religions and cultural groups. A quick tour around the world Islamic mosque Celtic carving Roman mosaic Taj Mahal Nepal Polynesian tattoo Ukrainian egg Aztec sun Maori carving Common floor plan of prehistoric stone “circles” in Britain. Summary It is not our job to explore “mystic” connections to math. We wanted to show …. • Cultures do not only develop math for practical reasons • Motivation is often from interest, or a sense of beauty • People seek deeper understanding of their world via math This is not a feature of one culture, but a common theme across the world and through many different religions (Christian, Islam, Pagan, Wiccan, etc.). This holds also for our modern scientific culture.