Craig Lang and Patrik Levlin

Math 371: Modern Geometry

March 27, 2008

Outline

General Facts Numerology and Sacred Geometry Constructions Various Types of Crop Circles

DefinitionA crop circle is an area in a field of crops where the plants have been mysteriously flattened into the shape of a circle or a more complex pattern.

Required Tools

The following tools are commonly used to aid in the construction of crop circles:

A plank of wood with knotted holes through each end used to flatten the crops.

Measuring tapes > 100’ in length. Marking poles Pre-measured ropes

Crop Circles Most crop circles were based on sacred

geometry until the year 2000. Recent formations appear to be based on

natural science and mathematical designs, such as fractals.

Crop circles range from simple to complex patterns of which most are believed to be man-made.

A number of patterns have been created that are unexplained due to the complexity of the design and the projected difficulty of completing it over the span of the night.

Crop Circle Facts The earliest recorded image of a crop circle came from a

1678 news pamphlet.

Someone who studies crop circles is called a cereologist.

David Chorley and Doug Bower earned an IG Nobel Prize in 1992 for crop circle hoaxing since 1976.

A farmer set up an admission booth to see a crop circle created near Stonehenge in 1996 and earned £30,000 in four weeks. The damaged crop held a value of £150.

Over 12,000 crop circles have been documented to date. Most were found in England.

Documented LocationsCrop circles are primarily found in England; however, they have “cropped up” all over the world.

Numerology Mystical relationships between numbers

and physical objects/living things Plato and Pythagoras, Chinese Examples

666 (34 Natural numbers and 7 first primes² - Hexakosioihexekontahexaphobia

13 - TriskaidekaphobiaDigit summing

Patrik = 16 + 1 + 20 + 18 + 9 + 11 = 75 = 7 + 5 = 12 = 1 + 2 = 3Craig = 3 + 18 + 1 + 9 + 7 = 38 = 3 + 8 = 11 = 1 + 1 = 2

Numbers

Monad – God Dyad – Dividing unity Triad – Restores balance Tetrad – Matter Pentad – Life Hexad – Economy and efficiency Heptad – Irrational, feminine Octad – End and beginning

Numbers cont’d

Ennead – Cardinal power Decad – Earth and Heaven Hendecad – Exists only in an ideal world Dodecad – Solar number Triskaidecad - Transformation

Sacred Geometry Geometry, mathematical ratios and

proportions exist in reality Music, light, cosmology, art Sacred structures such as temples, mosques,

monuments, churches, altars etc. Geometric forms: Sine wave, sphere, vesica

piscis, 5 platonic solids, torus, golden spiral and tesseract

Platonic Solids: Tetrahedron, Cube, Octahedron, Dodecahedron and Icosahedron

Two (= Dyad) Two-ness Dividing unity => Variety (Light – Dark etc) Can be seen in nature; human egg, atoms, big bang... Vesica Piscis – God’s offspring, Ichthys,

http://www.halexandria.org/dward097.htm

How many Triads do you see?

Three (= Triad)

Tetrad - Squaring the Circle

How many pentagrams do you see?

Five (= Pentad)

Six (= Hexad)

Eight (= Octad)

Can you find the error?

Ten (= Decad)

Twelve (= Dodecad)

Dodecagon and dodecagram

Thirteen (= Triskaidecad)

Triskaidecagon and triskaidecagram Triskaidekaphobia

AirplanesFloorsFormula OneMicrosoft® Office®

Transformation

Crop Circle Constructions

Designs range from days to months to complete

The most complex involve intricate patterns, mathematical logic, scientific meaning, and often reference to the unknown.

The most genuine crop circles leave no trace of their construction

Often creators will leave artificial trails that point to the extraterrestrial.

Reconstruction of the Chilcomb Down Formation of 2007

http://www.zefdamen.nl/CropCircles/Reconstructions/2007/ChilcombDown07/chilcombdown2007en.htm 1

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Julia Set Crop circle found near

Stonehenge Had over 15,000

visitors – most ever to a crop circle

The complex formation is a spiraling Julia Set

Dimensions of the design are 915’ x 508’.

151 Circles

Wiltshire – July 7, 1996

The Triple Julia

194 Circles “Triple” Julia Set ~ 700’

Windmill Hill, Wiltshire – July 30, 1996

Julia Set

Milk Hill, Wiltshire – August 13, 2001

Binary Fission

Represents the geometric expression:

200 = 1 211 = 2 222 = 4 233 = 8 244 = 16 255 = 32

~ 200’

Wiltshire – July 28, 2001

Sound Wave Interference

Two sources are separated by 1.5 times the wavelength of the sound to form this interference pattern.

Typical of a tuning fork

3 interference lines on each side

Constructed with 20 concentric circles

250’Shalbourne, Wiltshire – August 7, 2005

The Golden Tree The tree has four

steps of branching and a total of 16 fruits representing 24 = 16

Branched at 120° Branch lengths make

decreasing iterations of the golden ratio

~ 120’

Hampshire – August 15, 2002

Sine and Cosine This design

resembles the Human DNA helix

Also representative of an overlapped sine and cosine function.

89 Circles ~ 650’

Alton Barnes, Wiltshire – June 17, 1996

Various other crop circles…

Fractals

Labyrinths

Spirals

Yin Yang

Keltic Knot

Scarab

Dolphins

Dolphins

Scarab

Butterfly

Designed for a Nike Commercial

Advertising for the Olympics

Mushroom

Yggdrasil

3-D Skyscrapers

3-D Corridor

Possible Exam Question

Name 3 of the various types of crop circles.

Answer: Numerological, fractal, labyrinths, spirals, living things, knots etc.

Kiitoksia!!

Resources Crop Circles: Signs, Wonders, and

Mysteries - Steve and Karen Alexander www.temporarytemples.co.uk www.alien-ufo-pictures.com http://en.wikipedia.org/wiki/Crop_circle www.zefdamen.nl/CropCircles/en/

Crop_circles_en.htm www.hypermaths.org/cropcircles/

frontpage/index.html www.lucypringle.co.uk/

These are old slides with more information in case you are very interested in learning more…

One (= Monad) Unity and Eternity No beginning, no end Eternal, wholeness & perfection Divine Perimeter = 2 π – Irrational => Divine

Three (= Triad)

Trinity Isosceles triangle – born from Vesica The only number to equal the sum of all

the terms below it, and the only number whose sum with those below equals the product of them and itself

Beginning, middle and end (humans)

Four (= Tetrad)

Stability, equality and strength Matter, substance and mass Four elements Squaring the circle

Five (= Pentad) Pentagon and Pentagram Life, power and invulnerability Quintessence – Fifth being Human Pentagram

WitchcraftFractalsDivine Proportion & Fibonacci Sequence

Six (= Hexad)

Hexagon and hexagram Economy and efficiency

Storage

StrengthIce molecules

24 + 12 = 36

Seven (= Heptad)

Heptagon and hexagram Rainbow, musical notes and sins Does not divide into 360 => cannot be

born from the vesica Irrational, feminine

Eight (= Octad)

Octagon and octagram Seven notes => Octave, resonance End and beginning Infinity Can you find the

error?

Nine (= Ennead)

Nonagon and enneagram Three equilateral triangles Nine months of pregnancy Nine muses in the ancient world

Ten (= Decad)

Decagon and decagram Also means all numbers 1-10 Decimals, fingers, lobsters