Underweysung der Messung mit dem Zirckel vnd Richtscheyt (Four Books on Measurement). Albrecht Dürer, 1538

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Albrecht Dürer

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Ein schöner kurtzer Extract der Geometriae vnnd Perspectiuae
Paul Pfinzing, 1599

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Anleitung zur Feldmesskunst im Nürnberger Gebiet
(Instructions for surveying, in the Nuremberg area)
Jörg Unger

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Geometria theoretica et practica
Johann Baptist Roppelt, 1772

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— Thank you for sharing, nends.

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$John Arden Hiigli. Chrome 194, 2011 Transparent Oil on Linen Canvas Through a process known as subdivision a tetrahedron can be broken down to infinity. Subdivision produces the cyclical growth of other structures, such as the cube octahedron (CUO). The lowest frequency tetrahedron capable of “growing” a cube octahedron is the four-frequency tetrahedron. The eight-frequency tetrahedron produces a two-frequency cube octahedron as well. The sixteen-frequency tetrahedron produces a nuclear CUO, a nuclear two-frequency CUO and a nuclear four-frequency CUO. Thus we can say that the sub-division of polyhedra manifests a natural space characterized by transformation and change of scale. Scale change involves systems in which the size of individual forms vary incessantly while the proportions and characteristic shapes of these forms remain constant. These forms and their proportions can be grouped in a regular sequence in which the elements of the sequence constitute an infinite series. Such “scale-invariant” sequences occur in nature (the leaf’s nervure, the laceration of the fern, the widening or narrowing line of the snail shell, the shoreline of the continents, etc). This “invariability” is also a characteristic of fractal geometry. In the Isotropic Vector Matrix the potential for scale change is related to infinite transformation. With every doubling (or halving) of edge-length the volume increases (or decreases) by eight (8), equivalent to the octave in music. The artistic strategy of using transparent oil paint makes it possible to explore and communicate this world of higher dimensions. Scale shifting, or scale change, as well as the repetitious logic enabling it, is known in mathematics as “iteration,” hence the term “iterative mathematics/geometry”. It is a particular feature of IVM that is a useful tool in both mathematics and art education. In particular scale change provides a tool of measurement with which to evaluate angle & distance information. via: bridges math art$

John Arden Hiigli. Chrome 194, 2011
Transparent Oil on Linen Canvas

Through a process known as subdivision a tetrahedron can be broken down to infinity. Subdivision produces the cyclical growth of other structures, such as the cube octahedron (CUO). The lowest frequency tetrahedron capable of “growing” a cube octahedron is the four-frequency tetrahedron. The eight-frequency tetrahedron produces a two-frequency cube octahedron as well. The sixteen-frequency tetrahedron produces a nuclear CUO, a nuclear two-frequency CUO and a nuclear four-frequency CUO. Thus we can say that the sub-division of polyhedra manifests a natural space characterized by transformation and change of scale. Scale change involves systems in which the size of individual forms vary incessantly while the proportions and characteristic shapes of these forms remain constant. These forms and their proportions can be grouped in a regular sequence in which the elements of the sequence constitute an infinite series. Such “scale-invariant” sequences occur in nature (the leaf’s nervure, the laceration of the fern, the widening or narrowing line of the snail shell, the shoreline of the continents, etc). This “invariability” is also a characteristic of fractal geometry. In the Isotropic Vector Matrix the potential for scale change is related to infinite transformation. With every doubling (or halving) of edge-length the volume increases (or decreases) by eight (8), equivalent to the octave in music. The artistic strategy of using transparent oil paint makes it possible to explore and communicate this world of higher dimensions. Scale shifting, or scale change, as well as the repetitious logic enabling it, is known in mathematics as “iteration,” hence the term “iterative mathematics/geometry”. It is a particular feature of IVM that is a useful tool in both mathematics and art education. In particular scale change provides a tool of measurement with which to evaluate angle & distance information.

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Closest Packing of Spheres by Buckminster Fuller, 1980

BuckminsterFuller’s geometry shows that any sphere tangentialIy and symmetrically surrounded by spheres of the same radius will always produce an array of twelve balls around one ball. This phenomenon defines what he calls the Vector Equilibrium. The transparent spheres of this sculpture give it an ethereal quality reminiscent of a child’s bubble blowing while lucidly presenting the concept. Faintly visible equators illustrate the tangency of adjacent balls and the red nuclear sphere clarifies the radial symmetry of the structure. Twenty-four rods delineate the edges of the polyhedron uniquely determined by the nuclear packing of spheres. Its shape is unaffected by additional layers of balls. Two layers surround the nucleus which classifies this structure as “two-frequency,” a term that refers to the subdivisions along each edge.

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Ioannis Keppleri Harmonices mvndi libri Kepler, Johannes, 1571-1630.”

In this book Kepler announced the discovery of his third law of planetary motion: the square of the period of time of a planet is proportional to the cube of its mean distance from the sun. Kepler connected planetary motion with musical harmonies, with geometrical figures, with the relations of numbers. Newton’s discoveries were based on Kepler’s three laws.

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