. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. 176 UNICURSAL PROBLEMS [CH. IX The Konigsberg bridges lead to a network with four odd nodes; hence, by Euler's fourth proposition, it cannot be described unicursally in a single journey, though it can be traversed completely in two separate routes. The first and second diagrams figured below contain only even nodes, and therefore each of them can be described uni- cursally. The first of these is a regular re-entrant pentagon; the second is the so-called sign-ma
. Mathematical recreations and essays. Mathematical recreations; Geometry; Bees; Cryptography; Ciphers; String figures; Magic squares. 176 UNICURSAL PROBLEMS [CH. IX The Konigsberg bridges lead to a network with four odd nodes; hence, by Euler's fourth proposition, it cannot be described unicursally in a single journey, though it can be traversed completely in two separate routes. The first and second diagrams figured below contain only even nodes, and therefore each of them can be described uni- cursally. The first of these is a regular re-entrant pentagon; the second is the so-called sign-manual of Mohammed, said to have been originally traced in the sand by the point of his scimetar without taking it off the ground or retracing any part of the figure—which, as it contains only even nodes, is possible. The third diagram is taken from Tait's article: it contains only two odd nodes, and therefore can be described unicursally if we start from one of them, and finish at the The re-entrant pentagon, figured above, has some interest from having been used by the Pythagoreans as a sign—known as the triple triangle or pentagram star—by which they could recognize one another. It was considered symbolical of health, and probably the angles were denoted by the letters of the word vyieia, the diphthong ei being replaced by a 6. Iamblichus, who is our authority for this, tells us that a certain Pythagorean, when travelling, fell ill at a roadside inn where he had put up for the night; he was poor and sick, but the landlord, who was a kind-hearted fellow, nursed him carefully and spared no trouble or expense to relieve his pains. However, in spite of all efforts, the student got worse. Feeling that he was dying and unable to make the landlord any pecuniary recompense, he asked for a board on which he inscribed the pentagram star; this he gave to his host, begging him to hang it up outside so that all passers by might see it, and assuring him that the result. Please
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