shape with finite volume and infinite surface area
Gabriel's horn
title: Octacube (sculpture) text: the Platonic solids are the regular polytopes. These shapes were first described in the mid-19th century by a Swiss mathematician, Ludwig Schläfli. In four dimensions, there are six of them: the pentachoron (5-cell), tesseract (8-cell), hexadecachoron (16-cell), octacube (24-cell), hecatonicosachoron (120-cell), and the hexacosichoron (600-cell). The 24-cell consists of 24 octahedrons, joined in 4-dimensional space. The 24-cell's vertex figure (the 3-D shape formed when a 4-D corner is cut off) is a cube. Despite its suggestive name, the octacube is not the 4-D analog of either the octahedron or the cube. In fact, it is the only one of the title: Infinity text: Common examples are the infinite-dimensional complex projective space K(Z,2) and the infinite-dimensional real projective space K(Z/2Z,1). The structure of a fractal object is reiterated in its magnifications. Fractals can be magnified indefinitely without losing their structure and becoming "smooth"; they have infinite perimeters—some with infinite, and others with finite surface areas. One such fractal curve with an infinite perimeter and finite surface area is the Koch snowflake. Leopold Kronecker was skeptical of the notion of infinity and how his fellow mathematicians were using it in the 1870s and 1880s. This skepticism was developed in the philosophy of mathematics called finitism, title: Double bubble conjecture text: Double bubble conjecture In the mathematical theory of minimal surfaces, the double bubble conjecture states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a "standard double bubble" — three spherical surfaces meeting at angles of 2/3 on a common circle. It is now a theorem, as a proof of it was published in 2002. According to Plateau's laws, the minimum area shape that encloses any volume or set of volumes must take a form commonly seen in soap bubbles in which surfaces of constant mean curvature meet in threes, forming title: Equable shape text: its surface area is numerically equal to its volume. As with equable shapes in two dimensions, you may find an equable solid, in which the volume is numerically equal to the surface area, by scaling any solid by an appropriate factor. For example a cube with side length six. Equable shape A two-dimensional equable shape (or perfect shape) is one whose area is numerically equal to its perimeter. For example, a right angled triangle with sides 5, 12 and 13 has area and perimeter both have a unitless numerical value of 30. An area cannot be equal to a length title: Double bubble conjecture text: of surface area per cell. Double bubble conjecture In the mathematical theory of minimal surfaces, the double bubble conjecture states that the shape that encloses and separates two given volumes and has the minimum possible surface area is a "standard double bubble" — three spherical surfaces meeting at angles of 2/3 on a common circle. It is now a theorem, as a proof of it was published in 2002. According to Plateau's laws, the minimum area shape that encloses any volume or set of volumes must take a form commonly seen in soap bubbles in which surfaces of constant mean