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A sacred geometry of the equilateral triangle

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In this article, we investigate the construction of spirals on an equilateral triangle and prove that these spirals are geometric. In further analysing these spirals we show that both the male (straight line segments) and female (curves) forms of the spiral exhibit exactly the same growth ratios and that these growth ratios are constant independent of the iteration of the spiral. In particular, we show that ratio of any two successive radius vectors from the 'centre' of the spiral as we move inwards towards that 'centre' is always 1/2. This same elegant result is also shown to be true for successive chords. All our results are demonstrated using mostly coordinate and transformational geometry. Finally we look at two methods for constructing these spirals with ruler and compass to maximum accuracy.
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Keywords: coordinate geometry; equilateral triangle; geometric spirals; ruler and compass drawing; sacred geometry

Document Type: Research Article

Affiliations: Trinity College, Dublin, Republic of Ireland

Publication date: 2008-07-01

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