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.