GNOMON FROM PHARAOHS TO FRACTALS PDF

This observation virtually completes the "Pharaohs" end of the story mentioned in the title. Figurate numbers have familiar gnomons. Other figures have analogous gnomons. Continued fractions have obvious gnomonic tendencies, especially those with periodic regular representations. These lead naturally into physics, where ladder networks of resistors or capacitors are gnomonic, and their analysis leads back through continued fractions. Most of us have seen the gnomonic properties relating the golden rectangle, the golden ratio and the Fibonacci numbers.

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This observation virtually completes the "Pharaohs" end of the story mentioned in the title. Figurate numbers have familiar gnomons. Other figures have analogous gnomons. Continued fractions have obvious gnomonic tendencies, especially those with periodic regular representations. These lead naturally into physics, where ladder networks of resistors or capacitors are gnomonic, and their analysis leads back through continued fractions.

Most of us have seen the gnomonic properties relating the golden rectangle, the golden ratio and the Fibonacci numbers. In particular, the golden ratio, called phi in this book, is the limit of the ratios of consecutive Fibonacci numbers. If you construct a rectangle of sides 1 and phi, then the resulting rectangle is gnomonic and its gnomon is a square.

This pentagon is also gnomonic, and its gnomon is an equilateral triangle. Bernoulli loved them, too, but the exposition has more rotation matrices than most readers will enjoy. Finally, we get to the other end of the programme announced in the title, fractals.

He adapts this to form a Kronecker sum of vectors, and then adapts that to give sequences that can be read as instructions on drawing images of fractals.

It is an interesting approach, but again at times he lets the ideas be hidden by the notation. In sum, this reader found the book generally interesting, though at times over-burdened with notation. There is a considerable variety of topics here, all linked by the unifying concept of gnomonicity. Ed Sandifer sandifer wcsu. The table of contents is not available.

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History[ edit ] A gnomon as in Euclid book II A painted stick dating from BC that was excavated at the astronomical site of Taosi is the oldest gnomon known in China. The ancient Chinese used shadow measurements for creating calendars that are mentioned in several ancient texts. According to the collection of Zhou Chinese poetic anthologies Classic of Poetry , one of the distant ancestors of King Wen of the Zhou dynasty used to measure gnomon shadow lengths to determine the orientation around the 14th century BC. This shape may explain its use to describe a shape formed by cutting a smaller square from a larger one. Euclid extended the term to the plane figure formed by removing a similar parallelogram from a corner of a larger parallelogram.

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Gnomon: From Pharaohs to Fractals

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