The ''Octacube'' is a large,

stainless steel Stainless steel, also known as inox, corrosion-resistant steel (CRES), or rustless steel, is an iron-based alloy that contains chromium, making it resistant to rust and corrosion. Stainless steel's resistance to corrosion comes from its chromi ...

sculpture Sculpture is the branch of the visual arts that operates in three dimensions. Sculpture is the three-dimensional art work which is physically presented in the dimensions of height, width and depth. It is one of the plastic arts. Durable sc ...

displayed in the mathematics department of

Pennsylvania State University The Pennsylvania State University (Penn State or PSU) is a Public university, public Commonwealth System of Higher Education, state-related Land-grant university, land-grant research university with campuses and facilities throughout Pennsyl ...

State College, PA State College is a borough and home rule municipality in Centre County, Pennsylvania, United States. It is a college town, home to the University Park campus of The Pennsylvania State University. State College is the largest designated boroug ...

. The sculpture represents a mathematical object called the

24-cell In four-dimensional space, four-dimensional geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called C24, or the icositetrachoron, octaplex (short for "octa ...

or "octacube". Because a real 24-cell is

four-dimensional Four-dimensional space (4D) is the mathematical extension of the concept of three-dimensional space (3D). Three-dimensional space is the simplest possible abstraction of the observation that one needs only three numbers, called ''dimensions'' ...

, the artwork is actually a

projection Projection or projections may refer to: Physics * Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction * The display of images by a projector Optics, graphics, and carto ...

into the three-dimensional world. ''Octacube'' has very high intrinsic

symmetry Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...

, which matches features in chemistry (

molecular symmetry In chemistry, molecular symmetry describes the symmetry present in molecules and the classification of these molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explai ...

) and physics (

quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...

). The sculpture was designed by , a mathematics professor at

. The university's machine shop spent over a year completing the intricate metal-work. ''Octacube'' was funded by an alumna in memory of her husband, Kermit Anderson, who died in the

September 11 attacks The September 11 attacks, also known as 9/11, were four coordinated Islamist terrorist suicide attacks by al-Qaeda against the United States in 2001. Nineteen terrorists hijacked four commercial airliners, crashing the first two into ...

Artwork

The ''Octacube's'' metal skeleton measures about in all three dimensions. It is a complex arrangement of unpainted, tri-cornered flanges. The base is a high granite block, with some engraving. The artwork was designed by Adrian Ocneanu, a Penn State mathematics professor. He supplied the specifications for the sculpture's 96 triangular pieces of stainless steel and for their assembly. Fabrication was done by Penn State's machine shop, led by Jerry Anderson. The work took over a year, involving bending and welding as well as cutting. Discussing the construction, Ocneanu said:

It's very hard to make 12 steel sheets meet perfectly—and conformally—at each of the 23 vertices, with no trace of welding left. The people who built it are really world-class experts and perfectionists—artists in steel.

Because of the reflective metal at different angles, the appearance is pleasantly strange. In some cases, the mirror-like surfaces create an illusion of transparency by showing reflections from unexpected sides of the structure. The sculpture's mathematician creator commented:

When I saw the actual sculpture, I had quite a shock. I never imagined the play of light on the surfaces. There are subtle optical effects that you can feel but can't quite put your finger on.

File:OctacCrop.jpg File:OctacCorner.jpg File:OctacSideFull.jpg

Interpretation

Regular shapes

The

Platonic solids In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edge ...

are three-dimensional shapes with special, high,

. They are the next step up in dimension from the two-dimensional

regular polygons In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

(squares, equilateral triangles, etc.). The five Platonic solids are the

tetrahedron In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular Face (geometry), faces, six straight Edge (geometry), edges, and four vertex (geometry), vertices. The tet ...

(4 faces),

cube A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

(6 faces),

octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...

(8 faces),

dodecahedron In geometry, a dodecahedron (; ) or duodecahedron is any polyhedron with twelve flat faces. The most familiar dodecahedron is the regular dodecahedron with regular pentagons as faces, which is a Platonic solid. There are also three Kepler–Po ...

(12 faces), and

icosahedron In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrical tha ...

(20 faces). They have been known since the time of the Ancient Greeks and valued for their aesthetic appeal and philosophical, even mystical, import. (See also the '' Timaeus'', a dialogue of Plato.) In higher dimensions, the counterparts of the Platonic solids are the

regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...

s. These shapes were first described in the mid-19th century by a Swiss mathematician,

Ludwig Schläfli Ludwig Schläfli (; 15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional spac ...

. In four dimensions, there are six of them: the pentachoron (

5-cell In geometry, the 5-cell is the convex 4-polytope with Schläfli symbol . It is a 5-vertex four-dimensional space, four-dimensional object bounded by five tetrahedral cells. It is also known as a C5, hypertetrahedron, pentachoron, pentatope, pe ...

), tesseract (

8-cell In geometry, a tesseract or 4-cube is a four-dimensional space, four-dimensional hypercube, analogous to a two-dimensional square (geometry), square and a three-dimensional cube. Just as the perimeter of the square consists of four edges and ...

), hexadecachoron (

16-cell In geometry, the 16-cell is the regular convex 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the ...

), octacube (

), hecatonicosachoron (

120-cell In geometry, the 120-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also called a C120, dodecaplex (short for "dodecahedral complex"), hyperdodecahedron, polydodecahedron, hec ...

), and the hexacosichoron (

600-cell In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol . It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from ...

). The 24-cell consists of 24

s, joined in 4-dimensional space. The 24-cell's

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a general -polytope is sliced off. Definitions Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...

(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 six 4-D regular polytopes that lacks a corresponding Platonic solid.

Projections

Ocneanu explains the conceptual challenge in working in the fourth dimension: "Although mathematicians can work with a fourth dimension abstractly by adding a fourth coordinate to the three that we use to describe a point in space, a fourth spatial dimension is difficult to visualize." Although it is impossible to see or make 4-dimensional objects, it is possible to map them into lower dimensions to get some impressions of them. An analogy for converting the 4-D 24-cell into its 3-D sculpture is

cartographic projection Cartography (; from , 'papyrus, sheet of paper, map'; and , 'write') is the study and practice of making and using maps. Combining science, aesthetics and technique, cartography builds on the premise that reality (or an imagined reality) can ...

, where the surface of the 3-D Earth (or a globe) is reduced to a flat 2-D plane (a portable map). This is done either with light 'casting a shadow' from the globe onto the map or with some mathematical transformation. Many different types of map projection exist: the familiar rectangular Mercator (used for navigation), the circular gnomonic (first projection invented), and several others. All of them have limitations in that they show some features in a distorted manner—'you can't flatten an orange peel without damaging it'—but they are useful visual aids and convenient references. In the same manner that the exterior of the Earth is a 2-D skin (bent into the third dimension), the exterior of a 4-dimensional shape is a 3-D space (but folded through hyperspace, the fourth dimension). However, just as the surface of Earth's globe cannot be mapped onto a plane without some distortions, neither can the exterior 3-D shape of the 24-cell 4-D hyper-shape. In the image on the right a 24-cell is shown projected into space as a 3-D object (and then the image is a 2-D rendering of it, with perspective to aid the eye). Some of the distortions: *Curving edge lines: these are straight in four dimensions, but the projection into a lower dimension makes them appear to curve (similar effects occur when mapping the Earth). *It is necessary to use semi-transparent faces because of the complexity of the object, so the many "boxes" (octahedral cells) are seen. *Only 23 cells are clearly seen. The 24th cell is the "outside in", the whole exterior space around the object as seen in three dimensions. To map the 24-cell, Ocneanu uses a related projection which he calls ''windowed radial stereographic projection''. As with the stereographic projection, there are curved lines shown in 3-D space. Instead of using semitransparent surfaces, "windows" are cut into the faces of the cells so that interior cells can be seen. Also, only 23 vertices are physically present. The 24th vertex "occurs at infinity" because of the projection; what one sees is the 8 legs and arms of the sculpture diverging outwards from the center of the 3-D sculpture.

Symmetry

The ''Octacube'' sculpture has very high symmetry. The stainless steel structure has the same amount of symmetry as a cube or an octahedron. The artwork can be visualized as related to a cube: the arms and legs of the structure extend to the corners. Imagining an octahedron is more difficult; it involves thinking of the faces of the visualized cube forming the corners of an octahedron. The cube and octahedron have the same amount and type of symmetry:

octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...

, called O_h (order 48) in mathematical notation. Some, but not all, of the symmetry elements are *3 different four-fold rotation axes (one through each pair of opposing faces of the visualized cube): up/down, in/out and left/right as seen in the photograph *4 different three-fold rotation axes (one through each pair of opposing corners of the cube long each of the opposing arm/leg pairs *6 different two-fold rotation axes (one through the midpoint of each opposing edge of the visualized cube) *9 mirror planes that bisect the visualized cube **3 that cut it top/bottom, left/right and front/back. These mirrors represent its reflective dihedral subsymmetry D_2h, order 8 (a subordinate symmetry of any object with octahedral symmetry) **6 that go along the diagonals of opposing faces of the visualized cube (these go along double sets of arm-leg pairs). These mirrors represent its reflective tetrahedral subsymmetry T_d, order 24 (a subordinate symmetry of any object with octahedral symmetry). Using the mid room points, the sculpture represents the root systems of type D4, B4=C4 and F4, that is all 4d ones other than A4. It can visualize the projection of D4 to B3 and D4 to G2.

Science allusions

Many molecules have the same symmetry as the ''Octacube'' sculpture. The organic molecule,

cubane Cubane is a synthetic hydrocarbon compound with the Chemical formula, formula . It consists of eight carbon atoms arranged at the corners of a Cube (geometry), cube, with one hydrogen atom attached to each carbon atom. A solid crystalline substanc ...

(C₈H₈) is one example. The arms and legs of the sculpture are similar to the outward projecting hydrogen atoms.

Sulfur hexafluoride Sulfur hexafluoride or sulphur hexafluoride ( British spelling) is an inorganic compound with the formula SF6. It is a colorless, odorless, non-flammable, and non-toxic gas. has an octahedral geometry, consisting of six fluorine atoms attache ...

(or any molecule with exact

octahedral molecular geometry In chemistry, octahedral molecular geometry, also called square bipyramidal, describes the shape of compounds with six atoms or groups of atoms or ligands symmetrically arranged around a central atom, defining the vertices of an octahedron. The o ...

) also shares the same symmetry although the resemblance is not as similar. The ''Octacube'' also shows parallels to concepts in theoretical physics. Creator Ocneanu researches mathematical aspects of

(QFT). The subject has been described by a

Fields medal The Fields Medal is a prize awarded to two, three, or four mathematicians under 40 years of age at the International Congress of Mathematicians, International Congress of the International Mathematical Union (IMU), a meeting that takes place e ...

winner, Ed Witten, as the most difficult area in physics. Part of Ocneanu's work is to build theoretical, and even physical, models of the symmetry features in QFT. Ocneanu cites the relationship of the inner and outer halves of the structure as analogous to the relationship of spin 1/2 particles (e.g.

electrons The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...

) and

spin 1 particles In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0, 1, 2, ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have half odd-integer ...

(e.g.

photons A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that ...

Memorial

''Octacube'' was commissioned and funded by Jill Anderson, a 1965 PSU math grad, in memory of her husband, Kermit, another 1965 math grad, who was killed in the 9-11 terrorist attacks. Summarizing the memorial, Anderson said:

I hope that the sculpture will encourage students, faculty, administrators, alumnae, and friends to ponder and appreciate the wonderful world of mathematics. I also hope that all who view the sculpture will begin to grasp the sobering fact that everyone is vulnerable to something terrible happening to them and that we all must learn to live one day at a time, making the very best of what has been given to us. It would be great if everyone who views the ''Octacube'' walks away with the feeling that being kind to others is a good way to live.

Anderson also funded a math scholarship in Kermit's name, at the same time the sculpture project went forward.

Reception

A more complete explanation of the sculpture, including how it came to be made, how its construction was funded and its role in

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

and

physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, has been made available by Penn State.News bulletin on the Octacube
Department of Mathematics, Penn State University, 13 October 2005 (accessed 2013-05-06) In addition, Ocneanu has provided his own commentary.The mathematics of the 24-cell
a website maintained by Adrian Ocneanu.

References

Notes Citations

External links

Video from Penn State
about the ''Octacube''
User created video
on imagining a four dimensional object (but a tesseract). Note discussion of projections at ~22 minutes and the discussion of the cells in the model at ~35 minutes. {{coord, 40, 47, 51.5, N, 77, 51, 43.7, W, region:US-PA_type:landmark, display=title Mathematical artworks Quantum field theory 2005 sculptures Mathematics and culture Memorials for the September 11 attacks Pennsylvania State University Steel sculptures in Pennsylvania Group theory Operator algebras