The Pythagorean Theorem as a Rooted In-tree Dependency Graph
نویسنده
چکیده
We look back to concept and dependency graphs of Euclid’s Elements, Book 1, that show the deductive relationships among its propositions. We claim that Book 1 does not have one overall coherent structure but rather is organized around two propositions—1.45 and 1.47. In other words, Book 1 has a dual core. For the latter proposition, the Pythagorean Theorem, we constructed a rooted in-tree graph to help visualize its role and proof.
منابع مشابه
Cauchy-binet for Pseudo-determinants
The pseudo-determinant Det(A) of a square matrix A is defined as the product of the nonzero eigenvalues of A. It is a basis-independent number which is up to a sign the first nonzero entry of the characteristic polynomial of A. We prove Det(FG) = ∑ P det(FP)det(GP) for any two n×m matrices F,G. The sum to the right runs over all k × k minors of A, where k is determined by F and G. If F = G is t...
متن کاملNote on the Pythagorean Triple System
We investigate some combinatorial aspects of the “Pythagorean triple system”. Our motivation is the following question: Is it possible to color the naturals with finitely many colors so that no Pythagorean triple is monochromatic? This question is open even for two colors. A natural approach is to search for a nonbipartite triple system that can be realized as a family of Pythagorean triples. S...
متن کاملCounting rooted forests in a network
If F,G are two n×m matrices, then det(1+xFG) = ∑ P x |P det(FP )det(GP ) where the sum is over all minors [19]. An application is a new proof of the Chebotarev-Shamis forest theorem telling that det(1 + L) is the number of rooted spanning forests in a finite simple graph G with Laplacian L. We can generalize this and show that det(1 + kL) is the number of rooted edge-k-colored spanning forests....
متن کاملSe p 20 08 Pythagorean Partition - Regularity and Ordered Triple Systems with the Sum Property Joshua Cooper
Is it possible to color the naturals with finitely many colors so that no Pythagorean triple is monochromatic? This question is even open for two colors. A natural strategy is to show that some small nonbipartite triple systems cannot be realized as a family of Pythagorean triples. It suffices to consider partial triple systems (PTS’s), and it is therefore natural to consider the Fano plane, th...
متن کاملColoring so that no Pythagorean Triple is Monochromatic
We address the question of the “partition regularity” of the Pythagorean equation a+b = c; in particular, can the natural numbers be assigned a 2-coloring, so that no Pythagorean triple (i.e., a solution to the equation) is monochromatic? We prove that the hypergraph of Pythagorean triples can contain no Steiner triple systems, a natural obstruction to 2-colorability. Then, after transforming t...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2016