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With a the shorter and b the longer legs of a triangle and c its hypotenuse, the Pythagoras family of triplets is defined by c − b = 1, the Plato family by c − b = 2, and the Fermat family by | a − b | = 1. The Stifel sequence produces all primitive triplets of the Pythagoras family, and the Ozanam sequence produces all primitive triples ...
It can be shown that all Pythagorean triples can be obtained, with appropriate rescaling, from the basic Platonic sequence (a, (a 2 − 1)/2 and (a 2 + 1)/2) by allowing a to take non-integer rational values. If a is replaced with the fraction m/n in the sequence, the result is equal to the 'standard' triple generator (2mn, m 2 − n 2, m 2 + n ...
[6]: p.7 For example, parent (3, 4, 5) has excircle radii equal to 2, 3 and 6. These are precisely the inradii of the three children (5, 12, 13), (15, 8, 17) and (21, 20, 29) respectively. If either of A or C is applied repeatedly from any Pythagorean triple used as an initial condition, then the dynamics of any of a , b , and c can be ...
In molecular biology, a reading frame is a way of dividing the sequence of nucleotides in a nucleic acid (DNA or RNA) molecule into a set of consecutive, non-overlapping triplets. Where these triplets equate to amino acids or stop signals during translation , they are called codons .
The term sister group is used in phylogenetic analysis, however, only groups identified in the analysis are labeled as "sister groups".. An example is birds, whose commonly cited living sister group is the crocodiles, but that is true only when discussing extant organisms; [3] [4] when other, extinct groups are considered, the relationship between birds and crocodiles appears distant.
The spiral is started with an isosceles right triangle, with each leg having unit length.Another right triangle (which is the only automedian right triangle) is formed, with one leg being the hypotenuse of the prior right triangle (with length the square root of 2) and the other leg having length of 1; the length of the hypotenuse of this second right triangle is the square root of 3.
The idea of a tree of life arose from ancient notions of a ladder-like progression from lower into higher forms of life (such as in the Great Chain of Being).Early representations of "branching" phylogenetic trees include a "paleontological chart" showing the geological relationships among plants and animals in the book Elementary Geology, by Edward Hitchcock (first edition: 1840).
For example, in a full tree of life, the entire clade of animals can be collapsed to a single branch of the tree. However, this is merely a limitation of rendering space. In theory, a true and complete tree for all living organisms or for any DNA sequence could be generated. [4] Nevertheless, phylogenies can sometimes appear in a non-treelike form.