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The Pythagoras tree is a plane fractal constructed from squares. Invented by the Dutch mathematics teacher Albert E. Bosman in 1942, [ 1 ] it is named after the ancient Greek mathematician Pythagoras because each triple of touching squares encloses a right triangle , in a configuration traditionally used to depict the Pythagorean theorem .
The version given here is that proven by Nash-Williams; Kruskal's formulation is somewhat stronger. All trees we consider are finite. Given a tree T with a root, and given vertices v, w, call w a successor of v if the unique path from the root to w contains v, and call w an immediate successor of v if additionally the path from v to w contains no other vertex.
(Alternatively, write t = n / m as a fraction in lowest terms and use the formulas from the previous section.) A root node that instead has value t = 1/3 will give the same tree of primitive Pythagorean triples, though with the values of a and b switched.
For instance, the continued fraction representation of 13 / 9 is [1;2,4] and its two children are [1;2,5] = 16 / 11 (the right child) and [1;2,3,2] = 23 / 16 (the left child). It is clear that for each finite continued fraction expression one can repeatedly move to its parent, and reach the root [1;] = 1 / 1 of ...
Can be built with two dragon curves. One of the six 2-rep-tiles in the plane (can be tiled by two copies of itself, of equal size). [13] 1.5850: 3-branches tree: Each branch carries 3 branches (here 90° and 60°). The fractal dimension of the entire tree is the fractal dimension of the terminal branches.
A negative real number −x has no real-valued square roots, but when x is treated as a complex number it has two imaginary square roots, + and , where i is the imaginary unit. In general, any non-zero complex number has n distinct complex-valued n th roots, equally distributed around a complex circle of constant absolute value .
A Bethe lattice with coordination number z = 3. In statistical mechanics and mathematics, the Bethe lattice (also called a regular tree) is an infinite symmetric regular tree where all vertices have the same number of neighbors.
The rule states that if the nonzero terms of a single-variable polynomial with real coefficients are ordered by descending variable exponent, then the number of positive roots of the polynomial is either equal to the number of sign changes between consecutive (nonzero) coefficients, or is less than it by an even number.