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Geometry

The Geometry of Hope

I believe that a future ecology will be based on an understanding of interrelationships and patterns in nature. I have therefore involved myself for many years in interdisciplinary research in geometry, the science of relationships per se. The results of my research are various discoveries, some of which have been published in 'Mathematisch-Physikalische Korrespondenz' (International Journal for Mathematics and Physics) Nr. 197 & Nr. 203. Many of my discoveries reveal self-restorative laws within geometry and patterns of perfection behind irregular figures.

Extended Morley's Theorem

Figure 1, 2 & 3 Figure 4 Figure 5 & 6 Figure 7 & 8 Figure 9 Figure 10, 11, 12 & 13

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I have occuppied myself with Morley's Theorem and researched its application in the realm of synthetic geometry. So extended, Morley's Theorem reveals several properties that might, on account of their geometrical beauty, be of interest to many actively pursuing this science.

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Reoccurring equilateral triangles on the circle at the periphery

Figure 1 Figure 2 Figure 3 - 8 Figure 9 -16 Figure 17 - 24

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In my research into regular patterns I have come across the re-occurrence of the equilateral triangles on the circle at the periphery of a triangle. This phenomena has its counterpart in Equilateral Triangles on the Inscribed Circle (see below).

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Reoccurring equilateral triangles on the inscribed circle

Figure 1 Figure 2 - 7 Figure 8 Figure 9 -16

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This phenomena shows the re-occurrence of the equilateral triangles on the inscribed circle of a triangle. It has its counterpart in Equilateral Triangles on the Circle at the Periphery(see above).

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Immanent Triangles

Figure 1 & 2 Figure 3 Figure 4 Figure 5 Figure 6 & 7 Figure 8

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Immanent Triangles is one of two related phenomena which have hitherto gone unobserved in the literature in geometry (the other is Immanent Squares). This phenomenon allows the construction of equilateral triangles from irregular ones and illustrates the presence of the perfect in the imperfect. Related to Immanent Triangles is a wealth of other phenomena some of which are included.

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Immanent Squares

Figure 1 & 2 Figure 3 & 4 Figure 5 & 6 Figure 7 & 8 Figure 9 Figure 10 Figure 11

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'Immanent Squares' is one of two related phenomena which illustrate the presence of the perfect in the imperfect (the other is Immanent Triangles). This phenomenon allows the construction of squares from quadrilaterals of every shape.

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