Figurate number
Figured numbers are classes of numbers that relate to geometric figures. If one regular characters from toy bricks and counts the stones, you get figured numbers. Examples of figured numbers are the squares, cubes and pyramid numbers.
All episodes are a series of figurate numbers, because the followers are always sums of numbers in a certain sequence.
The consequences of figurate numbers which are called arithmetic sequences. To determine the explicit formula one examines the differences between adjacent sequence elements, which itself in turn a consequence, the difference sequence form. If there is no other option apparent, this is how the explicit law of each arithmetic sequence with the so-called polynomial algebraically determined.
Even the Greek mathematicians were concerned with figurate numbers.
- 3.1 Pyramidalzahlen or pyramid numbers
- 3.2 buzz centered Polygonalzahlen 3.2.1 Oktaederzahlen
- 3.2.2 cubes
- 3.2.3 -centered cubic numbers
Polygonalzahlen
Depending on the structure, a distinction is decentralized and centered Polygonalzahlen, the former being generally only called Polygonalzahlen. The term Polygonalzahl is also used as an umbrella term for decentralized and centered Polygonalzahlen.
(Distributed ) Polygonalzahlen
→ Main article: Polygonalzahl
A Polygonalzahl is a number for which there is a polygon (polygon ) that can be set with a corresponding number of stones. 16 For example, a Polygonalzahl, since a square can be set from 16 stones.
The 16 is the fourth square number.
22 is the fourth Fünfeckszahl.
The 28 is the fourth Sechseckszahl.
Centered Polygonalzahlen
→ Main article: Centered Polygonalzahl
Another mock-up for regular polygons begins with a stone in the middle. Therearound several polygons are placed, wherein the side lengths increase from the inside outwards in each case by one. The required number of blocks corresponds to a centered Polygonalzahl. The following pictures show some examples:
The 25 is the fourth -centered square number.
The 31 is the fourth -centered Fünfeckszahl.
The 37 is the fourth -centered Sechseckszahl.
Rectangular numbers or numbers pronische
→ Main article: rectangle number
A square number is the product or Proniczahl two consecutive natural numbers. For example, a rectangle number. If one blocks to form a rectangle, one side by 1 is longer than the second, then the number of stones a square number.
Three-dimensional body
The geometric constructions to the Polygonalzahlen can expand from plane figures to three-dimensional body. This creates Pyramidalzahlen and other types of figurate numbers. Since they are polyhedra in the figures, some authors use the term for this Polyederzahl.
Pyramidalzahlen or pyramid numbers
→ Main article: pyramidal number
Adding up the first square numbers one obtains the -th square Pyramidalzahl. Geometric means to stack multiple squares into a pyramid. The following figure shows this for the fourth square Pyramidalzahl.
This design principle can be applied to any of square numbers Polygonalzahlen. This gives rise to the different classes of Pyramidalzahlen.
Buzz centered Polygonalzahlen
Oktaederzahlen
The Oktaederzahlen can be interpreted as the sum of the first -centered square numbers:
The first Oktaederzahlen are
Cubes
The ( decentralized ) cubic numbers are the sum of the first centered Sechseckszahlen. The direct calculation formula is:
Centered cubic numbers
Centered cubic numbers can be defined analogously as
Rhombic Dodekaederzahlen
The rhombic Dodekaederzahlen can be assembled into a rhombic dodecahedron. They have the form
The first numbers of this form are
Regular figured numbers
Figured numbers can be defined for arbitrary dimensions. In general, the te - figured number of order with the binomial coefficients
Identical.
With increasing atomic incurred as from the triangular numbers
The tetrahedral numbers
And Pentatopzahlen
This sequence can be continued recursively in arbitrary dimensions: