By Marshall Clagett
This quantity maintains Marshall Clagett's experiences of a few of the points of the technological know-how of historical Egypt. the amount offers a discourse at the nature and accomplishments of Egyptian arithmetic and in addition informs the reader as to how our wisdom of Egyptian arithmetic has grown because the book of the Rhind Mathematical Papyrus towards the top of the nineteenth century. the writer fees and discusses interpretations of such authors as Eisenlohr, Griffith, Hultsch, Peet, Struce, Neugebauer, Chace, Glanville, van der Waerden, Bruins, Gillings, and others. He additionally additionally considers experiences of more moderen authors resembling Couchoud, Caveing, and Guillemot.
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Extra resources for Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics
Inverse Element The inverse of a given braid is its mirror image. In Fig. 7c the inverse of the braid in Fig. 4 is represented. It is clear that combining the two we obtain the trivial braid, see Fig. 8. The fact that Bn is a group is a deep result, in particular because it implies that each braid is an element of this group. We can present Bn with a set of generators and relations. Then, it will be possible to decompose any n-braid as a composition of the generators and their inverses. We can identify n − 1 generating braids: r1 ; .
3. 4. Closure 8 a; b 2 G; a Á b 2 G: Associativity 8 a; b; c 2 G; ða Á bÞ Á c ¼ a Á ðb Á cÞ: Identity Element 9 id 2 G such that 8 a 2 G; a Á id ¼ id Á a ¼ a: Inverse Element 8 a 2 G; 9 aÀ1 2 G such that a Á aÀ1 ¼ aÀ1 Á a ¼ id: So, a crucial aspect of braids, as mentioned, is that they form an algebraic structure: Theorem 1 The n-braids form a group, for all n. This is Bn, the braid group of order n. Intuitively, it is very easy to see that all the braids with a ﬁxed number of strings form a group.
In Fig. 9 are represented r1 and r2 as generators of B3. It is straightforward that these braids actually generate all the braid group. In fact, all braids can be decomposed into single twists. Therefore, by composing the ri s and their inverse we can create any braid. The generators and their inverses are the atomic building blocks with which we can build any braid. Figure 10 represents the braid in Fig. 4 as the composition of the generator r1 and the inverse of the generator r2 . So, Bn, the braid group in n strands, is generated by n-1 simple braids.