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In graph theory , all graphs with 4 or fewer vertices are planar , however, there is a graph with 5 vertices that is not: K 5 , the complete graph with 5 vertices, where every pair of distinct vertices in a pentagon is joined by unique edges belonging to a pentagram.

By Kuratowski's theorem , a finite graph is planar iff it does not contain a subgraph that is a subdivision of K 5 , or the complete bipartite utility graph K 3,3.

It is most easily described as graph of a pentagram embedded inside a pentagon, with a total of 5 crossings , a girth of 5, and a Thue number of 5. The chromatic number of the plane is at least five, depending on the choice of set-theoretical axioms : the minimum number of colors required to color the plane such that no pair of points at a distance of 1 has the same color.

The plane contains a total of five Bravais lattices , or arrays of points defined by discrete translation operations: hexagonal , oblique , rectangular , centered rectangular , and square lattices. The plane can also be tiled monohedrally with convex pentagons in fifteen different ways, three of which have Laves tilings as special cases.

Five points are needed to determine a conic section , in the same way that two points are needed to determine a line. There are five Platonic solids in three-dimensional space : the tetrahedron, cube, octahedron, dodecahedron, and icosahedron.

There are also five:. Made of five tetrahedra, its Petrie polygon is a regular pentagon and its orthographic projection is equivalent to the complete graph K 5. It is one of six regular 4-polytopes , made of thirty-one elements : five vertices , ten edges , ten faces , five tetrahedral cells and one 4-face. There are also a total of five Coxeter groups that generate non-prismatic Euclidean honeycombs in 4-space, alongside five compact hyperbolic Coxeter groups that generate five regular compact hyperbolic honeycombs with finite facets , as with the order-5 5-cell honeycomb and the order-5 cell honeycomb , both of which have five cells around each face.

Compact hyperbolic honeycombs only exist through the fourth dimension, or rank 5 , with paracompact hyperbolic solutions existing through rank The demipenteract , with one hundred and twenty cells , is the only fifth-dimensional semiregular polytope , and has the rectified 5-cell as its vertex figure, which is one of only three semiregular 4-polytopes alongside the rectified cell and the snub cell.

In the fifth dimension, there are five regular paracompact honeycombs, all with infinite facets and vertex figures. The five Mathieu groups constitute the first generation in the happy family of sporadic groups. There are five non-supersingular primes : 37 , 43 , 53 , 61 , and 67 , all smaller than the largest of fifteen supersingular prime divisors of the friendly giant , All multiples of 5 will end in either 5 or 0 , and vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions because they are prime factors of 10 , the base.

In the powers of 5, every power ends with the number five, and from 5 3 onward, if the exponent is odd , then the hundreds digit is 1 , and if it is even, the hundreds digit is 6. The evolution of the modern Western digit for the numeral 5 cannot be traced back to the Indian system , as for the digits 1 to 4. The Kushana and Gupta empires in what is now India had among themselves several different forms that bear no resemblance to the modern digit. The Ghubar Arabs transformed the digit in several different ways, producing from that were more similar to the digits 4 or 3 than to 5.

While the shape of the character for the digit 5 has an ascender in most modern typefaces , in typefaces with text figures the glyph usually has a descender , as, for example, in. On the seven-segment display of a calculator, it is represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice-versa. From Wikipedia, the free encyclopedia. This article is about the number.

For the year, see AD 5. For other uses, see 5 disambiguation , Number Five disambiguation , and The Five disambiguation. Natural number. List of numbers Integers. Retrieved OEIS Foundation. Also number of regions in 4-space formed by n-1 hyperplanes. Mathematics of Computation. S2CID Guy Unsolved Problems in Number Theory.

ISBN Sometimes called Mersenne numbers, although that name is usually reserved for A Prime Curios! There is probably no other number with this property. Dartmouth College : 1. Sum of two squares. In general, the sum of n consecutive triangular numbers is the nth tetrahedral number. Wichita State University Dept. February Recreational Mathematics Magazine.

The Electronic Journal of Combinatronics. Zbl Nexus Network Journal. The Fifty-Nine Icosahedra 1 ed. New York: Springer-Verlag. OCLC Journal of Combinatorial Theory, Series B. The Petersen Graph.

Cambridge University Press. MR A coloring of the set of edges of a graph G is called non-repetitive if the sequence of colors on any path in G is non-repetitive In Fig. Bibcode : arXivW. Tilings and Patterns. New York: W. Freeman and Company. Section 9. March The Mathematical Gazette.

The Mathematical Association. JSTOR Mathematical Proceedings of the Cambridge Philosophical Society. The Six-Cornered Snowflake. Paul Dry Books. Footnote 18, p. Convex Polyhedra. Archived from the original on Aequationes Mathematicae. Coxeter Regular Polytopes 3 ed. New York: Dover Publications, Inc. The Symmetries of Things. Geometriae Dedicata. M Annals of Discrete Mathematics. North-Holland Mathematics Studies. CiteSeerX Regular Complex Polytopes 2 ed.

London Math. Discrete Comput. Messenger of Mathematics. Griess, Jr. Twelve Sporadic Groups. Springer Monographs in Mathematics. Berlin: Springer-Verlag. Projective and Polar Spaces. Chapter 9 and Chapter 10 are collectively devoted to describing the group structures of this second generation of sporadic groups, and the Leech lattice. Boya Symmetry, Integrability and Geometry: Methods and Applications. David Bellos et al. London: The Harvill Press : , Fig.

Pediatric Hydrocephalus. Macroeconomic Outcomes in Disaster-Prone Countries. International Monetary Fund. Chasing Tornadoes. Twenty-First Century Books. The Planets. The next engine for light-duty trucks from Cummins became more powerful and technologically advanced the ISB 6. The less powerful hp engine was equipped with the 47RE automatic transmission until the model year when 47RE was upgraded to hold more torque capacity and named 48RE.

The all-iron engine provides a long lifespan and opportunity for rebuilding. ISB camshafts are more reliable because an electric fuel pump rotates by electricity, unlike a mechanical fuel pump driven by a camshaft gear on previous B series engines. However, the electronic system left its mark. The electronic VP44 pump had a failure number higher than the older mechanical P injection pump.

The Common Rail injection is very pricey to fix and more sensitive to the quality of fuel. Manual transmission - hp at 2, rpm, lb-ft at 1, rpm. HO version with manual transmission - hp at 2, rpm, lb-ft at 1, rpm.

Cars offered in CA - hp at 2, rpm, lb-ft at 1, rpm. Home Cummins ISB 5. Engine Specs Manufacturer. Production years. Cylinder block material. Cylinder head material. Fuel type. Electronically controlled Bosch VP44 rotary injection pump. Bosch high pre. Number of cylinders. Valves per cylinder.

Valvetrain layout. Bore, mm. Displacement, cc. Type of internal combustion engine.