## Mathematical Sculpture

There is a relationship between Mathematics and all the Arts. Mathematical Sculpture makes use of concepts relative to many mathematical fields: Geometry, Differential Calculus or Vector Calculus, Algebra, Topology, Logic, etc. sculptures for which the use of Mathematics becomes essential in their conception, design, development or execution will belong to this typology.

This relationship can even be extended to most artistic manifestations, taken in its broader sense. The great advances in Mathematics in the Modern Age and Contemporary Age have made the development of a conceptually mathematical art possible. Sculpture is also related to Mathematics. This relationship becomes more evident in the sculpture developed in the 20th century and at present time.

Some sculptures explicitly show their mathematical nature; a clear example could be a work based on the figure of a polyhedron or other specific geometrical shape, so it will be easier classifying it. However, in other works mathematics is only present in an implicit or hidden way, in which the mathematical conception is implicit in the design.

Geometrical Sculpture. This is the widest main group, as a consequence of the relationship between plastic arts, particularly Sculpture, and Geometry. It is a type of sculpture with a great tradition, especially in the 20th century. By the beginning of this century we find some works in Cubism. Also, some authors belonging to Abstract, Minimal and Conceptual movements used Geometry as well. The following subgroups are included in this group: Polyhedral Sculpture. The Platonic Polyhedrons are one of the solids most widely used by sculptors due to their beauty and simplicity. The truncated polyhedrons and a specific case, the Archimedean or semiregular polyhedrons, are commonly used as well. The transformations on these solids, such as deforming, star-shaping or rounding their sides, or any other that may result in aesthetic effects, are interesting.

Mathematical Curved Surfaces.: Quadrics, Revolution Surfaces, Ruled Surfaces and Other Surfaces. A commonly used surface is the hyperbolic paraboloid, which is a Quadric and a Ruled Surface simultaneously.

Fractal Geometry. Nowadays, the use in Mathematical sculpture of “new geometries” like fractal, different to the classical Euclidean, is not widespread.

Classification of Mathematical Sculpture:
Types of Mathematical Sculpture: Classic and Polyhedral, Geometry, Non-oriented Surfaces, Topological Knots, Quadric and Ruled Surfaces, Modular and Symmetric Structures, Boolean Operations, Minimal Surfaces, Transformations and Others.

Geometrical Sculpture
This is the widest group in the classification, as a consequence of the relationship between plastic arts, specially Sculpture, and Geometry. This kind of classification is so general that it could include most of the mathematical sculpture, from the simplest ones like cubes, spheres, cones, cylinders, prisms, etc., to the most complex solids, like irregular polyhedrons or surfaces defined by highly complex mathematical equations. In addition, in some works the most relevant element is not a particular type of solid or a combination of them, but some property or properties, like a curved surface, etc.

Polyhedral Sculpture. It is the first type included in the group of Geometrical Sculpture. The first polyhedrons analyzed will be Platonic Solids. This kind of solids is one of the geometrical figures more widely used by mathematical sculptors and by many other artists due to their beauty and simplicity.

Although their description is well-known, it is worth mentioning some characteristics of these regular polyhedrons. A convex polyhedron is regular if it is limited by regular polygons of a single type and if the same number of aristae converge at each vertex. There are only five solids of this type, known as Platonic (after the Greek geometrician and philosopher Plato) or cosmic. These five solids are: tetrahedron (4 sides); hexahedron or cube (6 sides); octahedron (8 sides); icosahedron (20 sides) and dodecahedron (12 sides).

In the same way as the Platonic polyhedrons, the truncated polyhedrons have been the source of inspiration of many mathematical sculptures. The possible cases of this type of polyhedrons are infinite. In addition, if the sides converge at every and each of the vertices of a regular polyhedron, they cut each other in such a way that the resulting plane sections are regular and congruent, and the rest of the solid is a new polyhedron known as semiregular or Archimediane. These have also been widely used in sculpture.

Another type of figures commonly used by mathematical sculptors are those resulting from transformations on the polyhedrons, such as deforming, star-shaping or rounding their sides, or any other geometrical transformation that may result in aesthetic effects.

Mathematical Curved Surfaces forms the following type of the classification within the general group of Geometrical Sculptures; this type has been sub-divided into other non-excluding types. For example, a commonly used surface in Art is the hyperbolic paraboloid, also called saddle, which is a quadric and a ruled surface simultaneously.

Quadrics are surfaces defined by a two-degree (at most) algebraic equation, in the three variables. Non-degenerated quadrics are: spheres, cones, cylinders, ellipsoids, hyperboloids (with one or two sheets) and paraboloids (elliptic and hyperbolic).

Non-oriented Surfaces. Unlike the surfaces mentioned above, they are characterized by a concept of vector calculus, that of orienting surfaces. The simplest surface is Moebius strip, one of the first objects of this kind that appeared in sculpture.

Sculpture with Algebraic Concepts:
This second general group of the classification comprises sculptures that make use of some algebraic concept in their design. These works can also adopt some of the geometric figures included in the other types of sculpture, but if the algebraic property is the dominant aspect in the sculpture, then we have classified it within this group.

Sculptures with Symmetries. One of the properties with more applications in Art is symmetry.

Transformations and Modular Sculptures. In other cases, the work will consist of a number of simple mathematical solids, like prisms or simple polyhedrons, to which some kind of algebraic transformation, such as translations, rotations, etc., has been applied.

Modular Sculptures are those sculptures in which a given pattern is repeated; the modules thus formed can present very different figures.

Boolean Sculpture. Other sculptures are created using diverse operations with the shape of one or several solids, based on a specific algebraic structure, for instance, Boolean Algebra in this group.

Topological Sculpture:
Mathematicians have studied “knots” for many centuries. This interesting and fascinating category of topological objects presents a wide range of possibilities to be used in sculpture.

Sculpture with Different Mathematical Concepts:
Novel concepts in mathematical sculpture, such as fractals, chaotic attractors, etc. Mathematical sculpture of non-Euclidean, elliptic and hyperbolic geometries. The works generated using these new concepts.

Sculpture with Concepts of Differential Calculus. It is divided in Other Concepts of Differential Calculus and Minimal Surfaces or Zero-Mean Curvature; that is, local-area minimising surfaces resulting from the adoption of the minimum possible value of area for the given boundary curve.

Sculpture with Algebraic Concepts. Make use of some algebraic concepts, processes and/or methods. Most sculptures can also adopt some geometric figures included in other groups, but if the algebraic property is the dominant aspect, then I will classify them within this group. Divided it into Symmetries, Transformations, Modular Sculptures and Boolean Operations.

Symmetries. One of the algebraic properties with more applications in Art is symmetry. It is especially notorious in Architecture. In mathematical sculpture its utilisation is also rather usual.

Transformations. There are sculptures made with a mathematical solid (or a set of them) in which some algebraic transformation, such as movements, rotations and/or translations, have been applied.

Modular Sculptures, a motive of “mathematic type”, is successively repeated. Boolean Operations; that is, operations that fulfill the properties of Boolean Algebra. The works created using diverse transformations of the shape of one or several solids, based on Boolean Algebra.

Topological Sculpture, base on a specific area of Mathematics: Topology. This subject deals with properties that are not affected by continue deformations, such as “flexion”, “stretching” and “warping”. Most important mathematical sculptors have made works of this type with very different designs. The subgroups included in Topological Sculpture are: Non-Oriented Surfaces. These shapes are characterised by a vector calculus concept.

Knots and Interwoven Figures. Mathematicians have studied “knots” for many centuries. This category of fascinating topological objects presents a wide range of possibilities to be used in Sculpture. Most mathematical sculptors have made use of them.

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