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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
2
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9
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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