41. Excerpts From Sworn Statements Of Witnesses It has nothing to do with the golden mean geometry which he is claiming. It is impossible for rotation of a golden mean spiral about the zaxis perpendicular to the plane in which http://danwinter.com/depoexcerpts.html

EXCERPTS FROM SWORN STATEMENTS OF EXPERT WITNESSES AND FACT WITNESSES The excerpts below are from sworn statements of witnesses, taken prior to June 1998 when USDJ Charles Siragusa issued his Findings of Fact, Conclusions of Law, and Order Sworn Testimony of Expert and Fact Witnesses Excerpted from Affidavits and Depositions, as noted Compiled by Cynthia Tenen, Meru Foundation, November 1997 Copies of the original documents available on request [The following persons were scheduled to be expert witnesses on our behalf in our lawsuit against Mr. Daniel Winter. We have sworn affidavits and depositions from them; below are some quotations from the body of their testimony, and their conclusions.] Louis H. Kauffman, Ph.D. Mathematics: Full Professor, University of Illinois at Chicago. One of world's foremost experts in topology and knot theory. From the Affidavit of Louis H. Kauffman, Ph.D., 11 November 1994: Quotation: Page 2, line 21 through Page 3, line 9: "I have examined Mr. Winter's book, "Alphabet of the Heart," and his CD-ROM for Macintosh, "The Language of Geometry Animated," as well as Macintosh disks which were the precursors of the CD-ROM and contain much of the same material. In my opinion it is abundantly clear that Mr. Winter was inspired by Mr. Tenen's work to create a three dimensional computer image (using the program Aldus Super 3-D) that copies as best he could manage Mr. Tenen's design for a "flame letter" that projects the different letters of the Hebrew alphabet. I am fully aware that this flame letter is Mr. Tenen's individual design, as I had the honor of watching him evolve this design over the decade of the 1980's."

42. Caitlin Kelley Architect The vibrational quality of the Golden Mean geometry gives it strong communication properties thus facilitating resonance with higher realms in prayer and meditation. http://www.caitlinkelleyarchitect.com/soul_whispering.php

Home Casas Villas Dwellings ... Projects On The Board Soul Whispering All of creation is made up of moving light. Energy patterns known as sacred geometry reveal the precise way that nature organizes itself. Every movement or natural pattern of growth adheres to one or more geometric shapes. The beauty of the patterns in nature is wonderfully revealed through sacred geometry. It shows up in the molecules of our DNA, the cornea of our eye, in the delicacy of a snow flake, a pine cone, flower petals, diamond crystals, the branching of a tree, in the nautilus shell. All life forms come out of timeless geometric codes. These codes connect us, the universe, and the universal mind. Sacred geometry touches our soul. These patterns and codes reflect our inner world, our consciousness and the structure and mystery of our awareness. From the mystery schools of the Greeks 2,500 years ago, the Platonic Solids were taught as the five perfect 3-dimensional forms. The tetrahedron, hexahedron, octahedron, dodecahedron and icosahedron are the foundation of everything in the physical world. The Periodic Table of Elements is further proof that literally everything in the physical world is based on these five forms. Sacred geometry reveals the inherent proportion, balance and harmony that exists in any situation. These geometric shapes actually represent the process of manifestation. Understanding geometry can abolish our false notion of separateness from nature and from each other.

43. Arabic Geometry – Page 2/2 Fivepoint geometry is also relatively easy to construct and, with it, the setting for golden mean geometry, mentioned below. However, if you combine the two geometries, it is http://catnaps.org/islamic/geometry2.html

Home Site map Updates Bullecourt ... Contact a collection of notes on areas of personal interest Islamic design menu for this section of the site Search the Islamic design study pages Introduction Arabic / Islamic design Islamic architecture Islamic gardens ... Links to other sites menu for notes relating to this section on Islamic design Geometric basis of patterns In the context of the notes on this and the preceding page it should be noted that it was with the defeat of the Ummayad empire based in Damascus and the development of the Abbasid dynasty in Baghdad in 762 that a distinct Islamic form of art is said to have developed. Prior to this, Ummayad buildings such as the Grand Mosque in Damascus, shown here, were characterised by the Byzantine style of mosaic work, although geometric designs had been a common feature of ornamental work from the later Roman period. The move from the Byzantine style to one of a more geometric character marked a significant change in applied ornamental designs. The construction of patterns for the purpose of decoration has been the subject of research and investigation for centuries. While I have witnessed the use of straight edges and string in the setting out of Islamic patterns, in watching craftsmen work it is evident that much is carried in their heads as they set out the details governing the patterns they are creating. Just as with this research, albeit over a much shorter time, the knowledge craftsmen have developed has accumulated over centuries. Nowadays we have greater resources available to investigate pattern conceptualisation and construction; we also have the capability to examine non-Euclidian geometry and create designs which would not have been possible six hundred years ago, as has been effected in

44. Golden Mean Find out more about constructions for the golden mean Geometry I am interested in Geometry, including finite geometric structures, like finite projective planes and finite http://www.math.uwaterloo.ca/~ljdickey/geometry_corner/golden_mean/index.html

The Golden Mean The Golden Mean (or golden ratio ) is a famous number that has been know from ancient times. For example some ancient buildings were built with this ratio used for its proportions. Here is an algebraic conditions that defines the golden ratio. It is positive and subtracting 1 from it gives a number that equals its reciprocal. So, it satisfies the equation x-1 = 1/x. It is not hard to show that the golden mean is one half of the sum of one and the square root of 5. Since the square root of 5 is about 2.236, and one plus the square root of 5 is 3.236, the golden ratio is approximately 1.618. The geometric construction that I have seen most often golden ratio uses a square A, B, C, D. Find the midpoint of the side AB and call it M. Make the circle with center M that passes through C (and D). Let E be the point where this circle meets the ray AB. Then the ratio AE / AB equals the ratio AB / BE. It is easy from this that that AE/AB is the golden ratio. Click here to see a picture (coming soon). A recent and surprising construction is one given by George Odom . George constructs an equilateral triangle ABC with vertices on a circle. Let M be the midpoint of AB and N be the midpoint of AC. Then the ray MN meets the circle at a point P. The ratio MP/MN is the golden ratio. Click here to see a picture (coming soon). Another pleasant surprise construction is given by Gabriel Bosia . Gabriel starts with the famous pythagorean triangle with sides 3, 4, and 5. Construct the circle of diameter 5 with center at the midpoint of the hypotenuse.

A  B  C  D  E  F  G  H  I  J  K  L  M  N  O  P  Q  R  S  T  U  V  W  X  Y  Z

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