Geometry of the Hebrew Alphabet

Abstract The Geometry of the Hebrew Alphabet.

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The "Hafets Hayim"   is quoted as   saying that even those portims of the Torah constrantly touched upon by everyone - are amenable to study which would induce its buds to swell and its flowers to blossom. This book is a confirmation of the words of the great   sage and   further evidence of the exceptionality of the Hebrew alphabet as an  instrument of the Creator separate from all other alphabets and languages of the world.

1.If the  realizations  of the  letters  (  discussed  in  part  in Rabbi  M. Glazerson's  "Sod Hasimkhah" ) are interpreted in terms of straight lines, then the entire Hebrew alphabet can be encompassed by five such lines. Spiritual affinities of words are paralleled by    projective relationships between planes.

2.   The cross-ratio of a quadruple of collinear points A,B,C,D reads (ABCD) = AC/CB : AD/DB. If it equals -1, the quadruple is called "harmonic". Such quadruples form the basis of the branch of mathematics called Projective Geometry. Computerised analysis brought out 24 harmonic quadruples of 22 letters of the Hebrew alphabet. One such quadruple consists of zayin, gimel, daleth and aleph, whose numerical values produce the cross-ratio ( 1,4,3,7 ) = (1 - 3)/(3 - 4) : (1 - 1)1(1 - 4 ) = -1. Thus, with three  of the letters given as collinear points, the fourth can be constructed with the aid of a  straightedge alone.    The following general theorem is proved:

With three letters of the Hebrew alphabet can be reproduced all the alphabet with the aid of a straightedge alone.

3.   Certain transforms of the Hebrew alphabet ( many of which are illuminated e.g. in Rabbi Glazerson's books ) are piecewise-projective.

By means of these transforms it is possible to bring out spiritual affinities between words and create new interpretations for verses of the Holy Writ. For example, __ : substituting the first and last letters via ____ , we obtain ___ , which can be interpreted as "Sabbath from the Torah" or "Sabbath from the heart" .

4. All transforms, familiar or less familiar, appear in the "Sefer Yetsira" (the book of creating traditionally attributed to the Patriarch Abraham). This circumstance is discussed in the book. Besides the links between the Holy Alphabet and projective geometry, these are references to the Third Temple, to searches for key words and to many other topics.

The book is aimed at an audience acquainted with mathematics at the secondary-school level and interested in the "subsurface" features of the Holy Alphabet. It will prove of use to those interested in the structure of the alphabet, hence in that of our Universe Finally, it familiarizes its readers with the tools of scripture research.