Fano plane

In finite geometry, the Fano plane (after Gino Fano) is the finite projective plane of order 2, having the smallest possible number of points and lines, 7 each, with 3 points on every line and 3 lines through every point. The standard notation for this plane, as a member of a family of projective spaces, is PG(2,2) where PG stands for "Projective Geometry", the first parameter is the geometric dimension and the second parameter is the order.

Homogeneous coordinates

The Fano plane can be constructed via linear algebra as the projective plane over the finite field with two elements. One can similarly construct projective planes over any other finite field, with the Fano plane being the smallest.

Using the standard construction of projective spaces via homogeneous coordinates, the seven points of the Fano plane may be labeled with the seven non-zero ordered triples of binary digits 001, 010, 011, 100, 101, 110, and 111. This can be done in such a way that for every two points p and q, the third point on line pq has the label formed by adding the labels of p and q modulo 2. In other words, the points of the Fano plane correspond to the non-zero points of the finite vector space of dimension 3 over the finite field of order 2.

Fano

Fano [ˈfaːno] is a town and comune of the province of Pesaro and Urbino in the Marche region of Italy. It is a beach resort 12 kilometres (7 miles) southeast of Pesaro, located where the Via Flaminia reaches the Adriatic Sea. It is the third city in the region by population after Ancona and Pesaro.

History

An ancient town of Marche, it was known as Fanum Fortunae after a temple of Fortuna located there. Its first mention in history only dates from 49 BC, when Julius Caesar held it, along with Pisaurum and Ancona. Caesar Augustus established a colonia, and built a wall, some parts of which remain. In 2 AD Augustus also built an arch (which is still standing) at the entrance to the town.

In January 271, the Roman Army defeated the Alamanni in the Battle of Fano that took place on the banks of the Metauro river just inland of Fano.

Fano was destroyed by Vitiges' Ostrogoths in AD 538. It was rebuilt by the Byzantines, becoming the capital of the maritime Pentapolis ("Five Cities") that included also Rimini, Pesaro, Senigallia and Ancona. In 754 it was donated to the Popes by the Frank kings.

Fano (Gijón)

Fano is a parish of the municipality of Gijón / Xixón, in Asturias, Spain. In 2012, its population was 220. Located in the south-east of the municipality, Fano is a rural area which borders the municipality of Siero in the south, and with the district of Valdornón in the east.

Toponym comes from Latin Fanum, a kind of temples ancient romans built in pre-Roman cults sacred places. A Benedictine monastery existed in Fano from 12th to 17th centuries. Its front romanesque façade is nowadays part of the San Juan Evangelista de Fano church.

Villages and their neighbourhoods

Notes

External links

Coordinates: 43°27′18″N 5°37′33″W / 43.4551°N 5.6259°W / 43.4551; -5.6259

Fano (disambiguation)

Fano may refer to: