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the Marinite667
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Figure 1 In my too-numerous-to-count studies of the Golden Section
and its related geometric structures (particularly its square
root) as design tools, I came across an irregular pentagram ( As a way of understanding the tilings, it is first necessary
to understand the Öf rectangle,
as it has a few of its own unique properties. In Figure 2 Point These occult centers have a unique role to play in the geometry
of any rectangle. The four line segments that radiate from a
given occult center will form a geometric progression in the
same proportion as the original rectangle. In the present case,
focusing on occult center Also, Figure 3 The picture is completed in Figure 4 In Figure 5 No other rectangle can claim all of these properties and qualities regarding the golden section, not even the golden section rectangle itself.
It is certainly always possible to find irregular polygons
and stars in virtually any grid, especially if that grid is a
more developed and complex one. Also, pentagons, hexagons and
octagons can all be found mingling together within one perimeter.
I believe that what makes this gridwork so unique and fascinating,
however, is that it uses a variety of numbers and geometric structures
that are all directly related to its original generator, f. In other grids, although we may find
irregular polygons and stars in randomly chosen grids, their
numerical and geometric structures quickly evaporate into a hodgepodge
of numbers not associated with their origins, often not even
remotely. For example, the irregular hexagons that may be found
in a Ö3 rectangle, or the octagons
that result from the Ö2 (not
even to mention any pentagonal systems that can be seen) quickly
move off into unrelated numbers, or numbers that may even be
associated with other, unrelated systems. In the case of R-Tiles,
the f family group of numbers is to
be found. Most notably, we find f, 1/f, f Another aspect of their interest is in their flexibility, whether a periodic pattern or an integrated individuality is sought within the plane. Both work equally well. All that is required is to construct of the Öf rectangle, and then to begin work. The various tiles can easily be found and developed. Figure 1 is the "master tile" for the series, and Figure 6 is a study of R-Tiles within a rectangle, which demonstrates the great variety of tilings to be found within this rectangle. There is infinite variety and beauty in tessellations, and many cultures have explored their possibilities and mathematical relationships. It is my hope that this small group of tiles can join the ranks of those wonders that have been with us for millennia. On the following pages are figures further illustrating the properties discussed above. (In the studies, it may be helpful to look for the Öf rectangles - the "master rectangles" - and their grids for this series - in all their variety.)
return
to text[2]
See [Hambidge 1967] for a clear development of this concept from
both a mathematical and geometric point of view. For the sake
of discussion, I will adopt this term in the description. [3]
In a square the reciprocal to one diagonal is simply the other
diagonal (analogously to the fact that the reciprocal of the
number 1 is the number 1). In other words, the two diagonals
cross at 90° in the center, so the 'four' occult centers
all coexist at the center of the square. [4] One could certainly find an infinite variety of
polygons, but this is true even with a square grid!
Reynolds, Mark. 2000. A
comparative Geometrical Analysis of the Heights and Bases of
the Great Pyramid of Khufu and the Pyramid of the Sun at Teotihuacan.
Copyright ©2005 Kim Williams Books top of
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