Tic

A tic is a sudden, repetitive, nonrhythmic motor movement or vocalization involving discrete muscle groups. Tics can be invisible to the observer, such as abdominal tensing or toe crunching. Common motor and phonic tics are, respectively, eye blinking and throat clearing.

Tics must be distinguished from movements of other movement disorders such as chorea, dystonia, myoclonus; movements exhibited in stereotypic movement disorder or some autistic people, and the compulsions of obsessive-compulsive disorder (OCD) and seizure activity.

Classification

Tics are classified as either motor or phonic, and simple or complex.

Motor or phonic

Motor tics are movement-based tics affecting discrete muscle groups.

Phonic tics are involuntary sounds produced by moving air through the nose, mouth, or throat. They may be alternately referred to as verbal tics or vocal tics, but most diagnosticians prefer the term phonic tics to reflect the notion that the vocal cords are not involved in all tics that produce sound.

Tongue-in-cheek

The tongue-in-cheek figure of speech is used to imply that a statement or other production is humorously or otherwise not seriously intended, and it should not be taken at face value. The facial expression typically indicates that one is joking or making a mental effort. In the past, it may also have indicated contempt, but that is no longer common.

By 1842, the phrase had acquired its contemporary meaning, indicating that a statement was not meant to be taken seriously. Early users of the phrase include Sir Walter Scott in his 1828 The Fair Maid of Perth.

History

Putting one's tongue into a cheek was formerly used to signify contempt. For example, in Tobias George Smollett's The Adventures of Roderick Random, which was published in 1748, the eponymous hero is taking a coach to Bath and apprehends a highwayman. This provokes an altercation with a less brave passenger:

A similar usage appears in 1828 in The Fair Maid of Perth by Sir Walter Scott, "The fellow who gave this all-hail thrust his tongue in his cheek to some scapegraces like himself."

Truncated cube

In geometry, the truncated cube, or truncated hexahedron, is an Archimedean solid. It has 14 regular faces (6 octagonal and 8 triangular), 36 edges, and 24 vertices.

If the truncated cube has unit edge length, its dual triakis octahedron has edges of lengths 2 and .

Area and volume

The area A and the volume V of a truncated cube of edge length a are:

Orthogonal projections

The truncated cube has five special orthogonal projections, centered, on a vertex, on two types of edges, and two types of faces: triangles, and octagons. The last two correspond to the B₂ and A₂Coxeter planes.

Spherical tiling

The truncated cube can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Cartesian coordinates

The following Cartesian coordinates define the vertices of a truncated hexahedron centered at the origin with edge length 2ξ: