Posts tagged ‘Shape’

March 14, 2012

Menger Sponge

In mathematics, the Menger [meng-ersponge is a fractal curve. It is a universal curve, in that it has topological dimension one, and any other curve (more precisely: any compact metric space of topological dimension 1) is homeomorphic to some subset of it.

It is sometimes called the Menger-Sierpinski sponge or the Sierpinski sponge. It is a three-dimensional extension of the Cantor set and Sierpinski carpet. It was first described by Karl Menger (1926) while exploring the concept of topological dimension. The Menger sponge simultaneously exhibits an infinite surface area and encloses zero volume.

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February 9, 2012

Squircle

Squircle

A squircle [skwer-kul] is a mathematical shape with properties between those of a square and those of a circle. It is a special case of superellipse. The word ‘squircle’ is a portmanteau of the words ‘square’ and ‘circle.’ A shape similar to a squircle, called a rounded square, may be generated by arranging four quarters of a circle and connecting their loose ends with straight lines. Although constructing a rounded square may be conceptually and physically simpler, the squircle has the simpler equation and can be generalized much more easily. One consequence of this is that the squircle and other superellipses can be scaled up or down quite easily. This is useful where, for example, one wishes to create nested squircles.

Squircles are useful in optics. If light is passed through a two-dimensional square aperture, the central spot in the diffraction pattern can be closely modeled by a squircle (also called a supercircle). If a rectangular aperture is used, the spot can be approximated by a superellipse. Squircles have also been used to construct dinner plates. A squircular plate has a larger area (and can thus hold more food) than a circular one with the same radius, but still occupies the same amount of space in a rectangular or square cupboard. The same is true of a square plate, but there are various problems (such as wiping up sauce) associated with the corners of square plates.

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February 9, 2012

Superegg

superegg by Piet Hein

In geometry, a superegg is a solid of revolution obtained by rotating an elongated super-ellipse with exponent greater than 2 around its longest axis. It is a special case of super-ellipsoid. Unlike an elongated ellipsoid, an elongated superegg can stand upright on a flat surface, or on top of another superegg.

This is due to its curvature being zero at the tips. The shape was popularized by Danish poet and scientist Piet Hein (1905–1996). Supereggs of various materials were sold as novelties or ‘executive toys’ in the 1960s. A 1-ton superegg made of steel and aluminum was placed outside Kelvin Hall in Glasgow in 1971, on occasion of a lecture by Piet Hein.

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February 21, 2011

Platonic Solid

tetrahedron

cube

A platonic [pluh-ton-iksolid is a three dimensional shape where each face is built from the same type of polygons, and there are the same number of polygons meeting at every corner of the shape.  There are only five Platonic Solids: Tetrahedron, Cube, Hexahedron, Octahedron, Dodecahedron, and Isosahedron. The shapes are often used to make dice, because dice of these shapes can be made fair. 6-sided dice are very common, but the other numbers are commonly used in role-playing games. Such dice are commonly referred to as D followed by the number of faces (d8, d20 etc.).

The tetrahedron (4 sided), cube (6 sided), and octahedron (8 sided), are found naturally in crystal structures. In meteorology and climatology, global numerical models of atmospheric flow are of increasing interest which use grids that are based on an icosahedron (20 sides,refined by triangulation) instead of the more commonly used longitude/latitude grid. This has the advantage of better spatial resolution without singularities (i.e. the poles) at the expense of somewhat greater numerical difficulty.

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