Pergamon Printed m Great Britain. All rights reserved 0960.0779/96 615.00 + (I.O(I SO960-0779(96)00022-7 On the Dimension of Micro Space-Time SALEH AL-ATHEL KACST, Riyadh, Saudi Arabia Abstract-A conventional derivation of some basic equations relating to the Cantorian proposal of quantum space-time is given based on the gamma distribution function. Previous work on related subjects is also discussed. Copyright 0 1996 Elsevier Science Ltd 1. INTRODUCTION Questions regarding the dimensionality of space have a very old history reaching back to the times of the great philosophers of antiquity up to the exponents of German idealism, particularly E. Kant. For a very readable review of these and more recent efforts, please refer to the book of Barrow et al. [l]. Attempts to tackle the question of why we are living in three-plus-one-dimensional (three spatial and one temporal) space within the framework of theoretical and mathematical physics are of more recent dates. In this context one may mention important contributions to this subject by Mirman [2], Nielson et al. [3], Svozil 141, El Naschie [5] and Hemion [6]. The present work is concerned mainly with the basic relations developed by El Naschie [5,7] following some ideas advanced initially by Menger [S], Wheeler [9] and Finkelstein [lo]. The main results of El Naschie’s work are three equations with the help of which he was able to fix the dimensionality (n) of a Borel-like random space with an expected value -(n) and found that n is exactly equal to four (n = 4). In addition, a relative average fractal dimension dp’ = (d) was found to be equal to 4 + $3 where 1/@3 = 4 + I$~. In summary, the following equations were derived: &’ = (l/&y (1) (n) = 2/ln (l/d?‘) (2) -(n) = (1 + dLO’)/(l - diO’) (3) (d) = l/1(1 - d:O’)d:O’ (4) pi.“) = -(n) = (d)]l@, qb = @fJ where dr’ is the Hausdorff dimension of n-dimensional Cantorian space and dfp’ is the Hausdorff dimension of the elementary Cantor set living in one-dimensional Eucludian space and possessing a zero Menger-Uhryson topological dimension [5]. Although equations (l)-(5) are, within the assumption, mathematically correct, it is fair to say that the derivations in [5] are rather involved and difficult to follow. On the other hand, it is also quite astonishing that these (superficially) complex relations can be derived easily using standard and rather elementary calculus and probability theorems, as will be 873