They are distinct from the Platonic solids, which are composed of only one type of polygon meeting in identical vertices, and from the Johnson solids, whose regular polygonal faces do not meet in identical vertices.

For example, the perimeter of a regular polygon inscribed in a circle approaches the circumference with increasing numbers of sides ( and decrease in the length of one side ).

While at university, Gauss independently rediscovered several important theorems ; his breakthrough occurred in 1796 when he showed that any regular polygon with a number of sides which is a Fermat prime ( and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2 ) can be constructed by compass and straightedge.

Gauss also made important contributions to number theory with his 1801 book Disquisitiones Arithmeticae ( Latin, Arithmetical Investigations ), which, among things, introduced the symbol ≡ for congruence and used it in a clean presentation of modular arithmetic, contained the first two proofs of the law of quadratic reciprocity, developed the theories of binary and ternary quadratic forms, stated the class number problem for them, and showed that a regular heptadecagon ( 17-sided polygon ) can be constructed with straightedge and compass.

The wheels can be any regular polygon except a triangle, but the catenary must have parameters corresponding to the shape and dimensions of the wheels.

For example, every polygon is topologically self-dual ( it has the same number of vertices as edges, and these are switched by duality ), but will not in general be geometrically self-dual ( up to rigid motion, for instance ) – regular polygons are geometrically self-dual ( all angles are congruent, as are all edges, so under duality these congruences swap ), but irregular polygons may not be geometrically self-dual.

In geometry, a Johnson solid is a strictly convex polyhedron, each face of which is a regular polygon, but which is not uniform, i. e., not a Platonic solid, Archimedean solid, prism or antiprism.

Since a regular polygon has angles at least 60 degrees, it follows that at most five faces meet at any vertex.

Although there is no obvious restriction that any given regular polygon cannot be a face of a Johnson solid, it turns out that the faces of Johnson solids always have 3, 4, 5, 6, 8, or 10 sides.

* Star polygon: a polygon which self-intersects in a regular way.

* Regular: A polygon is regular if it is both cyclic and equilateral.

A non-convex regular polygon is called a regular star polygon.

Any polygon, regular or irregular, self-intersecting or simple, has as many corners as it has sides.

The area of a regular polygon is also given in terms of the radius r of its inscribed circle and its perimeter p by

While it is possible to construct analogies to the Penrose triangle with other regular polygons to create a Penrose polygon, the visual effect is not as striking, and as the sides increase, the object seems merely to be warped or twisted.

Carl Friedrich Gauss in 1796 showed that a regular n-sided polygon can be constructed with ruler and compass if the odd prime factors of n are distinct Fermat primes.

Some regular polygons, like the heptagon, become constructible ; and John H. Conway gives constructions for several of them ; but the 11-sided polygon, the hendecagon, is still impossible, and infinitely many others.

In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections.

In this article, D < sub > n </ sub > ( and sometimes Dih < sub > n </ sub >) refers to the symmetries of a regular polygon with n sides.

Dance teachers can respond to President Kennedy's request not only through their regular dance work, but also through the kind of basic gymnastic work that makes for strength and flexibility.

Whereas a high percentage of the regular students can be expected to read other texts which more or less plow the same ground in a little different direction, the married students chose whole books on specific areas and went into much greater detail in their areas of interest.

For example, the inflected forms of a word can be represented, insofar as regular inflection allows, by a stem and a set of endings to be attached.

The fast evaporation of water from regular acrylic paint films can be slowed with the use of acrylic retarders.

This set can be called the regular and semiregular honeycombs.

A team can score a field goal by shooting the ball through the basket during regular play.

These are often open-top buses, but can also be by regular bus or coach.

Each of these methods contributes useful priors for " regular " one-parameter problems, and each prior can handle some challenging statistical models ( with " irregularity " or several parameters ).

The boiling point at regular atmosphere is around − 34 ˚ C, but it can be liquefied at room temperature with pressures above 740 kPa.

In the military sphere, authorities can prosecute both regular crimes and specific acts ( such as mutiny or desertion ) under martial-law codes that either supplant or extend civil codes in times of ( for example ) war.

It demonstrates periods of stability and periods of change and can indicate whether changes follow patterns such as regular cycles.

Additionally, this family of formal languages can be obtained by regular expressions.

The cube can also be called a regular hexahedron and is one of the five Platonic solids.

Given an asymmetric subunit on a triangular face of a regular isosahedron, with three subunits per face 60 such subunits can be placed in an equivalent manner.

Mail can be used under everyday clothes and many reenactors wear a hauberk under their regular clothes to accustom themselves to it.

The language is context-free, however it can be proved that it is not regular.

( In the NFL overtime is decided by a 15-minute sudden-death quarter, and regular season games can still end in a tie if neither team scores.

* Every regular language is context-free because it can be described by a context-free grammar.

Where there is regular feedback, control theory can be used to determine how the system responds to such feedback.

From them, regular sound correspondences between the languages are established, and a sequence of regular sound changes can then be postulated, which allows the proto-language to be reconstructed.

Relation is deemed certain only if at least a partial reconstruction of the common ancestor is feasible, and if regular sound correspondences can be established with chance similarities ruled out.

`` I've served as a counsel for the U.N. for some years, specializing particularly in real estate matters or other problems that the regular U.N. legal staff might not be equipped to handle.

He soon quarreled with all the party leaders in the House, and came to be regarded with detestation by regular Democrats as a professional radical leading a small pack of obedient terriers whose constant snapping was demoralizing to party discipline.

This means an added burden to innumerable postmen, who already are complaining of heavy loads and low pay, and it presumably means an increased postal deficit, but, our correspondent writes, think of the additional junk mail each citizen will now be privileged to receive on a regular basis.

There should be a regular electrical outlet in the shelter as power may continue in many areas.

Blowers should be operated periodically on a regular schedule.

Here may be found regular and impact styrene, cellulose acetate, cellulose butyrate and cellulose propionate, acrylic, vinyl, expandable styrene foam, and polyethylene.

People who have not been able to get relief from regular medical doctors are especially apt to be taken in by quacks ''.

I put in new batteries so as to be certain I'd have plenty of power and on my way out walked over to the regular parking stalls and stood looking at them thoughtfully.

`` You're going to be a regular.

Such vital information, he said, has to be made available to the public frequently and at regular intervals for residents to know.

If Barnett doesn't call a special session in 1961, it will be the first year in the last decade that the Legislature has not met in regular or special session.

Her nose was higher of bridge, her complexion so pale as to be quite susceptible to sunburn, and the fish and vegetable diet of her forebears had given her teeth that were white and regular and strong.

According to Edward Gibbon in The History of the Decline and Fall of the Roman Empire, during the shifting of offices that took place at the beginning of the new reigns, Alaric apparently hoped he would be promoted from a mere commander to the rank of general in one of the regular armies.

In spite of its academic nature, the copious details to be found in the treatise rendered it of the highest value to the army organizers of the 16th century, who were engaged in fashioning a regular military system out of the semi-feudal systems of previous generations.

Let ( m, n ) be a pair of amicable numbers with m < n, and write m = gM and n = gN where g is the greatest common divisor of m and n. If M and N are both coprime to g and square free then the pair ( m, n ) is said to be regular, otherwise it is called irregular or exotic.

If ( m, n ) is regular and M and N have i and j prime factors respectively, then ( m, n ) is said to be of type ( i, j ).

Since 1998, home field advantage has been given to the team that has the better regular season record, unless that team happens to be the Wild Card team.

This attack cannot be used against the regular Blowfish ; it assumes knowledge of the key-dependent S-boxes.