By Jay Kappraff
This booklet includes essays that stand all alone yet also are loosely hooked up. half I files how numbers and geometry come up in different cultural contexts and in nature: the traditional musical scale, share in structure, old geometry, megalithic stone circles, the hidden pavements of the Laurentian library, the shapes of the Hebrew letters, and the shapes of organic varieties. the point of interest is on how sure numbers, corresponding to the golden and silver capacity, current themselves inside of those structures. half II exhibits what number of an identical numbers and quantity sequences are concerning the trendy mathematical examine of numbers, dynamical platforms, chaos, and fractals.
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Extra resources for Beyond measure: a guided tour through nature, myth, and number
7 From Beauty of Fractals. H. Richer (1986). -O. 5 Computer generated by R. Langridge. Computer Graphics Laboratory, University of California, San Francisco © Regents University of California. 6 Courtesy of H. Lalvani. 7a Created by Elyse O'Grady from the Mathematics of Design class of Jay Kappraff. 7b Created by Eileen Domonkos from the Mathematics of Design class of Jay Kappraff. 11 Courtesy of Janusz Kapusta. 12 From Chaos, Fractals, and Power Laws by M. Schroeder. H. Freeman. Used by permission.
5 Spiral Forms in Water The book Sensitive Chaos by Theodor Schwenk [Schw] is concerned with the creation of flowing forms in water and air. As Schwenk states: "In the olden days, religious homage was paid to water, for men felt it to be filled with divine beings whom they could only approach with the greatest reverence. Divinities of the 10 Beyond Measure water often appear at the beginning of a mythology (for example, the Australian aborigines). " Water expresses itself in a vocabulary of spiral forms.
In a similar manner, B is transformed to C via point U on m, etc. on line x. Conversely, a co-basal projectivity can be generated from an arbitrary conic (a circle in this figure) m and a base line x intersecting it, by choosing two points O and O' on the conic and relating the intersection points on the base line of pairs from O and O' that meet on the conic. 8 that this co-basal projectivity has two fixed points X and Y. Co-basal projectivities have one fixed point when the base line is tangent to the conic, or no real fixed points when the base line and conic have no real intersection points (we shall see that it then has two imaginary fixed points).