Science in Society Archive

Other articles in this series

1) The Story of Phi
2) Watching the Daisies Grow
3) Golden Music of the Brain
4) Golden Cycles & Organic Spacetime
5) Golden Geometry of E-Infinity Fractal Spacetime
6) E-Infinity Spacetime, Quantum Paradoxes and Quantum Gravity
7) Water Structured in the Golden Ratio

Science of the Organism

Golden Geometry of E-Infinity Fractal Spacetime

Story of Phi Part 5

The golden ratio is at the core of our fractal universe of infinite dimensions that looks and feels 4-dimensional. Dr. Mae-Wan Ho

Real processes do not happen at points in a spacetime continuum

Alfred North Whitehead (1861-1947) lived through what must have been a most exciting period in Western science. The very fabric of reality – the flat, smooth, and static absolute spacetime of the Newtonian universe – is being thoroughly ruffled, if not torn asunder, by Albert Einstein’s theories of special and general relativity at large scales and by quantum mechanics at the smallest.  Out goes the proverbial objective scientific observer outside nature, to be supplanted by the knower irreducibly quantum-entangled with the known, possibly with all entities in the entire universe.

These surprising and exhilarating lessons from nature became the basis of Whitehead’s perennial, universal philosophy (a cosmogony) [1], initiating a new age of the organism that has inspired generations of scientists, including me. Most enlightening though least understood was his argument on why the mechanical laws of classical physics and differential calculus fail to describe real processes. Apart from the all-important knowing experiencing organism left out of account, real processes have characteristic times (durations) and spaces (volumes). Nothing, absolutely nothing happens at a point in an instant. In other words, spacetime is not a smooth continuum; it is discrete and discontinuous, as quantum physics already discovered at the smallest scale. Indeed spacetime can more easily be conceived as being created by action, by processes that naturally occur in quantum jumps.

Unfortunately, mathematics - a major tool of thought for western science, especially physical science - had lagged somewhat behind physics. Both relativity theory and quantum theory inherited the predominant mathematics of classical mechanics.

British theoretical physicist Roger Penrose’s monumental tome, The Road to Reality, A Complete Guide to the Laws of The Universe [2] is indeed a “tour de force”, as advertised. It charts the heroic efforts of mathematical physicists to make sense of post Newtonian physics; their ingenuities and successes, as well as failures. More than a century later today, the dream of uniting the two great theories of quantum physics and general relativity has remained stubbornly beyond our grasp. And one main reason is that the two theories are divided by a common ground: a differentiable spacetime manifold.

Differentiable spacetime and differential calculus assume objects have instant point-like ‘simple locations’ in space and time. It is fundamentally at odds with physical reality, and may well be creating more problems than it can solve.  But there is still a tendency to put mathematics (and mathematical physics) on a pedestal as though it has an ‘objective’ independent existence to which physical reality must conform. This neo-Platonist view championed by Penrose [2] is surely a case of misplaced concreteness. It has been more than 80 years since Austrian mathematician and logician Kurt Gödel (1908-1978) proved his incompleteness theorems [3], which essentially state that a set of axioms both complete and consistent is nevertheless unable to guarantee a complete and consistent theory, let alone one that applies to physical reality.

Nevertheless, mathematics is a wonderful, exquisite tool for thought, and I have come to appreciate it more and more in my continuing quest for the meaning of life and the universe (see [4] The Rainbow and the Worm, The Physics of Organisms, I-SIS publication). The mathematics of non-differentiable and discontinuous spaces, which has the potential to describe physical reality more authentically, is among the most significant discoveries/ inventions beginning in the latter half of the 19th century though it did not really blossom until well into the 20th century.

I shall describe an intuitive and ingenious theory of spacetime that makes full use of the newer mathematics. Before proceeding, please read Box 1 for a quick guide to mathematical terms that you will be bumping into a lot in the rest of this article; and return to it as often as is necessary.

Box 1

A quick guide to mathematical terms for non-mathematicians

Set theory
The branch of mathematical logic about collections of mathematical objects. The modern study of set theory was initiated by German mathematicians Georg Cantor (1845-1918) and Richard Dedekind (1831-1916) in the 1870s.

Closed and open sets
A closed set contains its own boundary, its complement is an open set which does not contain its boundary

Borel set
A Borel set is any set in a topological space that can be formed from open sets ( or equivalently from closed sets) through the operations of countable union, countable intersection, and relative complement. A countable set is one with the same number of elements as some subset of the set of natural numbers. The elements of a countable set can be counted one at a time, and although the counting may never finish, every element of the set will eventually be associated with a natural number. Union, denoted by ⋃ of a collection of sets is the set of all distinct elements in the collection. Intersection of sets, denoted by ⋂,  is the set that contains only elements belonging to all the sets. The relative complement of set A in B is the set of elements in B but not in A.

Bijection is a mapping both one-to-one (an injection) and onto (a surjection); it is a function that relates each member of a set S (the domain) to a separate and distinct member of another set T (the range), where each member in T also has a corresponding member in S.

Cantor Set and Random Cantor Set
The classical triadic Cantor set (named after inventor Georg Cantor) is obtained by dividing the unit line into three  equal parts, discarding the middle part except for its end points, and repeating the operation with the two remaining parts ad infinitum. In the random version, it could be any of the three parts that is discarded at random after each division. 

A space is a set with some added structure.

Metric space
A metric space is a set where a notion of distance (a metric) between elements of the set is defined. The metric space that corresponds most closely to our intuitive understanding of space is the 3-dimensional flat Euclidian space.

Riemannian space
A topological space with metric properties that can be defined continuously from point to point (hence also called a Riemannian manifold) including standard non-Euclidean spaces, ie., spaces that are not flat.

Topological space
A topological space is a set of points and a set of neighbourhoods for each point that satisfy a set of axioms relating to points and neighbourhoods. The definition of a topological space relies only on set theory and is the most general notion of a mathematical space.

Topological dimension
Topological dimension is the dimension of a topological space. For example, a point has topological dimension 0 whereas a line has topological dimension of 1, closing up the line into a circle makes no difference; it still has a topological dimension of one. Similarly, a flat sheet has a topological dimension of 2, the same for the surface of a cylinder, a sphere or a doughnut.

Menger-Urysohn dimension
The Menger-Urysohn dimension is a generalized topological dimension of topological spaces, arrived at by mathematical induction. It is based on the observation that, in n-dimensional Euclidean space Rn, (n−1)-dimensional spheres (that is, the boundaries of n-dimensional balls) have dimension n−1. Therefore it should be possible to define the dimension of a space inductively in terms of the dimensions of the boundaries of suitable open sets.

Hausdorff dimension
The Hausdorff dimension generalizes the notion of dimension to irregular sets such as fractals. For example, a Cantor set has a Hausdorff dimension of ln2/ln3, the ratio of the logarithm to the base 2 of the parts remaining to the whole after each iteration.

A fractal is a mathematical set that typically displays self-similar patterns, and has fractional dimensions instead of the usual integer, 1, 2, 3, or 4. Geometric examples are branching trees, blood vessels, frond leaves etc.

Chaos theory
The study of dynamical systems with locally unpredictable behaviour that is highly sensitive to initial conditions, but are nevertheless globally determined, such that the trajectories are confined within a region of phase space called ‘strange attractors’.

A term coined by Cantor; it means beyond finite, but not necessarily the absolute infinite.

E infinity fractal spacetime looks and feels 4-dimensional

The idea that spacetime is fractal originated with Canadian mathematician Garnet Ord [4] who coined the term ‘fractal spacetime’, using a model in which particles are confined to move on fractal trajectories.  Independently, French astrophysicist Laurent Nottale proposed a scale-relativity theory of fractal spacetime [5]. As Ord reminds us [4], American quantum physicist Richard Feynman (1918-1988) had already pointed out [6] that the paths of quantum mechanical particles look more like non-differentiable curves than straight lines when examined on a fine scale.

It is Egyptian-born Mohamed El Naschie, however, who has taken us furthest towards a coherent theory of physical reality that is also closest to our intuitive notion of organic spacetime. For precisely the same reasons, perhaps, El Naschie has attracted admiration and antagonism in equal measure.

As an aside, the journal Nature initiated a smear campaign against El Naschie in 2008 with an article written by its German correspondent [7] that was filled with insinuations and innuendoes, if not outright lies.  It accuses El Naschie of “self-publishing” papers of “poor quality” without proper peer review in a theoretical physics journal of which he has been editor-in-chief. In fact, El Naschie has had hundreds of papers published in other journals; and his prolific output continued throughout the subsequent four-year period during which he brought a libel case single-handedly against Nature. El Naschie turned down all attempts by Nature to settle out of court, until the court ruled against him in 2012. By this time, Nature had spent £5 million in legal fees to defend itself. Given Nature’s shameful record in libeling myself and other scientists over the hazards of genetic modification, I became all the more determined to find out about El Naschie’s work; and have been suitably rewarded as a result.

El Naschie trained and practiced as an engineer while indulging in his hobby of cosmology, and produced a startling new theory of spacetime, which soon took over his life entirely. E-infinity, as he calls it, is a fractal spacetime with infinite dimensions. Yet its Hausdorff (fractal dimension) is 4.236067977... In other words, at ordinary resolution, it looks and feels 4 dimensional (three of space and one of time), with the rest of the dimensions ‘compacted’ in the remaining 0.236067977… “fuzzy tail”[8].

Think of a four dimensional hypercube with further four dimensional hypercubes nested inside like Russian dolls [9]. In fact, the exact Hausdorff dimension is 4 + f3, where f = (√5-1)/2, the golden ratio.  Particularly suggestive is that the dimension 4 + f3 show the following self-similar continued fraction (see Figure 1) which sums to precisely 4 + f3 at infinity:

The 4-dimensional hypercube is the Euclidean representation of the E-infinity universe. It is a challenge to represent E-infinity it in its proper non-Euclidean form.

Figure 1   E infinity fractal spacetime represented as nested four dimensional hypercubes

The E-infinity universe is mathematically a random Cantor set (the simplest fractal) extended to infinite dimensions, and the remarkable result is that the limit of this infinite extension is no larger than 4 + f3.

Figure 2 depicts some of the steps involved in deriving E infinity universe. We start from the 1-dimensional Cantor set, the unit line, which is divided into three equal parts; take out the middle, and carry on the same operation on the two parts remaining, up to an infinite number of steps. At the end, there should be nothing left but isolated points (Cantor dust). Its Menger-Urysohn dimension is 0, but amazingly, its Hausdorff (fractal) dimension is ln2/ln3, for the usual triadic Cantor set; but is f = (√5-1)/2 = 0. 61803398.. for the random Cantor set, in which the section removed is not necessarily the middle one, but c0uld be any of the three parts at random. This important result, proven by American mathematicians Daniel Mauldin and S C Williams in 1986 [10], is what makes it possible to derive the E-infinity universe with all its remarkable properties, as we shall see. Another mathematical property of the Cantor set is that its cardinality (number of points or elements) is exactly the same as the original continuous line. Thus, the Cantor set is a perfect compromise between the discrete and the continuum; it is a discrete structure that has the same number of elements as the continuum.

From the 1-dimensional Cantor set, we can construct the higher dimensional spaces (see Fig.2). The 2-dimensional version is the Sierpinski gasket, with random Hausdorff dimension of 1/f = 1.61803398..; the 3-dimensional version is the Menger sponge, with random Hausdorff dimension 2 + f = 2. 61803398.. The Sierpinski gasket and Menger sponge are both well-known geometric shapes. Not so the 4-dimensional version, with random Hausdorff dimension 4 + f 3 = 4.23606797…, where only an artist’s representation is given. The 4-dimensional version is the same as the E infinity universe constructed from an infinite number of random Cantor sets (as will be made clear later). Note that the diagram is space-filling: it consists of spheres of different sizes representing space-times at different scales so that the entire volume is completely packed. This space-filling is analogous to, if not the same property as quasi-periodic Penrose tiling in 2 dimensions of Euclidean (flat) space where the golden ratio is key (see [11] The Story of Phi Part 1, SiS 62), as it is in E infinity universe. And incidentally, branching processes based on the golden ratio are also space-filling [12], as are spiral leave arrangement patterns with the golden angle between successive leaf primordial (see [13] Watching the Daisies Grow, SiS 62).

Figure 2   Fractals and dimensions in the derivation of E infinity spacetime

El Naschie has presented multiple formal derivations of E infinity spacetime, I shall give the one relying on the mathematical properties of Borel sets to which Cantor sets belong. It is so simple that even a non-mathematician can understand it.

The expectation value of the Hausdorff dimension of the Cantor set extended to infinity is simply a sum over n , for n = 0 to n = ∞, of n multiplied by the Hausdorff dimension of the random Cantor set raised to the power n. This is more concisely encapsulated in the following equation:


where the left hand side is the expectation value of the Hausdorff dimension of the Cantor set extended to infinity; the superscript in dc(0) refers to the Menger-Urysohn dimension of the random Cantor set, which is 0, while the corresponding Hausdorff dimension dc(0) is f. The summing up of the infinite number of terms gives the answer 4 + f3 exactly.


Now, the intersection rule of sets, also known as the bijection formula, which relates the Menger-Urysohn dimension to the Hausdorff dimension, shows that we can lift dc(0)  to any Menger-Urysohn dimension n to arrive at the correct Hausdorff dimension dc(n) as follows:

dc(n) = (1/ dc(0)) n-1                                                                                                      (3)

Taking dc(0)f, and lifting to n = 4 dimensions gives

dc(4) = (1/ dc(0)) 4-1 = 4 + f3 = 1/f3 = 4.236067977…

                                                  = ⟨Dim E ˗ ∞⟩H                                                                        (4)

In other words, the expectation value of the Hausdorff dimension of E infinity universe is the same as that of a universe with a Menger-Urysohn dimension of 4. That is why E infinity is a hierarchical universe that looks and feels 4 dimensional (it is strictly embedded in 5 dimensions, see Fig. 2).

E-infinity, Penrose tiling, and Fibonacci sequence

The E-infinity universe is intimately connected with Penrose tiling and the Fibonacci sequence (see [11]) through E-infinity algebra [14, 15]. In his important book Noncommutative Geometry [16], French mathematician Alain Connes identified Penrose’s fractal tiling as a mathematical quotient space (a space of points ‘glued together’ by an equivalence relationship), with the following dimensional function:

D(a, b) = a + bf                                                                                               (5)

Where a, b are integers (whole numbers) and f = (√5-1)/2. Writing Dn (an, bn) with the Fibonacci sequence, it is easy to see that starting with D0 = D (0, 1) and D1 = D (1, 0), the following dimensional hierarchy is obtained:

D0 = D (0, 1) = 0 + f = f

D1 = D (1, 0) = 1 + (0)f = 1

D2 = D (0+1, 1+0) = 1 + f = 1/f

D3 = D (1+1, o+1) = 2 + f = (1/f)2

D4 = D (1+2, 1+1) = 3 + 2f = (1/f)3

D5 = D (2+3, 1+2) = 5 + 3f = (1/f)4

Dn(an, bn) = D {(an-1, an-2) + (bn-1 + bn-2)}f = (1/f)n-1                                        (6)

Not that for D4 (dimension 4), the Fibonacci number is (1/f)3 = 4 + f3, exactly the Hausdorff dimension of a Menger-Urysohn 4-dimensional space.

By induction,

Dn = (1/f)n-1                                                                                                                 (7)

which is identical to the bijection formula from E-infinity algebra (see Eq (3) above):

dc(n) = (1/f) n-1                                                                                                             (8)

The summing of random Cantor sets to infinity is very suggestive of spacetime being created or constructed by actions over all scales, from submicroscopic to macroscopic and beyond, which is close to what I have envisaged [4], following Whitehead [2] and German theoretical physicist Wolfram Schommers [17].

E infinity spacetime can also resolve major paradoxes within quantum theory and produce new results as we shall see in the next article [18] (E Infinity Spacetime, Quantum Paradoxes and Quantum Gravity, SiS 62).

Article first published 31/03/14


  1. Whitehead AN. Science and the Modern World, Lowell Lectures 1925, Collins Fontana Books, Glasgow, 1975.
  2. Penrose R. The Road to Reality, A Complete Guide to the Laws of the Universe, Vintage Books, London, 2005. 1 099 pp.
  3. Gödel’s incompleteness theorems. Wikipedia, 26 February 2014,
  4. Ho MW. The Rainbow and the Worm, the Physics of Organisms, World Scientific, 1993, 2nd edition, 1996, 3rd enlarged edition, 2008, Singapore and London.
  5. Ord GN. Fractal space-time: a geometric analogue of relativistic quantum mechanics. J Phy A: Math Gen 1983, 16, 1869-84.
  6. Nottale L. Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity, World Scientific, 1993.
  7. Feyman RP and Hibbs AR. Quantum Mechanics and Path Integrals, McGraw-Hill, New York, 1965.
  8. “Self-publishing editor set to retire”, Quirin Schiermeier, Nature 2008 456, 432.
  9. El Naschie MS. A review of E infinity theory and the mass spectrum of high energy particle physics. Chaos Solitons & Fractals 2004, 19, 209-36.
  10. Mauldin RD and Williams SC. Random recursive constructions: asymptotic geometric and topological properties. Trans AM Math Soc 1986, 295, 325-46.
  11. Ho MW. The story of phi part 1. Science in Society 62 2014.
  12. “There is something about phi, Chapter 9 - Fractals and the golden ratio” Javier Romanach, YouTube, accessed 1 March 2014,
  13. Ho MW. Watching the daisies grow. Science in Society 62 2014.
  14. El Naschie MS. The theory of Cantorian spacetime and high energy particle physics (an informal review). Chaos, solitons and Fractals 2009, 41, 2635-46.
  15. Marek-Crnjac L. The Hausdorff Dimension of the Penrose universe. Phys Res Interna 2011, 874302, 4 pages
  16. Connes A. Noncommutative Geometry, Paris, 1994,
  17. Schommers W. Space-time and quantum phenomena. In Quantum Theory and Pictures of Reality (W. Schommers ed.), pp. 217-77, Springer-Verlag, Berlin, 1989.
  18. Ho MW. E infinity spacetime, quantum paradoxes and quantum gravity. Science in Society 62 2014.

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elbl dietmar Comment left 12th May 2014 17:05:00
hi, following the interesting text I finally got into troubles with the values for phi mentioned in the text: ..."with random Hausdorff dimension of 1/phi = 1.61803398..;" but: 1/phi = 0.6180339; later I find: "...with random Hausdorff dimension 2 + phi = 2. 61803398... but 2 + phi = 3.6180339 ... there are some more examples of this kind so please give me a hint how this should be understood. thank you

mae-wan ho Comment left 12th May 2014 17:05:22
Hi elbl, please note that in the notation used in this and other articles in the series, phi (small letter) is 0.6i8..., while phi (capital letter is the reciprocal equal to 1.618.... maewan

"el naschie, mohamed saladin" 7102016068 Comment left 10th December 2014 02:02:04
About 2,900,000 results (0.28 seconds) Search Results [PDF]A short history of fractal-Cantorian space-time - El Naschie by L Marek-Crnjac - ‎2009 - ‎Cited by 31 - ‎Related articles number of scientists worked on fractal space-time notably Garnet Ord in Canada .... of fractal space-time tiling the plane using Klein's-modular-curve with 336 + 3 ... [PDF]Application of chaos and fractals in fundamental physics ... by MS El Naschie - ‎2004 - ‎Cited by 5 - ‎Related articles noncommutative geometry akin to that of Penrose fractal tiling and the related compactified Klein modular curve. This curve with 336 + 3 = 339 hierarchal ... Modular group - Wikipedia, the free encyclopedia For a group whose lattice of subgroups is modular, see Iwasawa group. ..... This tessellation first appears in print in (Klein 1878/79a), where it is credited to ... of this tiling (yielding a tiling on the modular curve), as depicted in the video at right. .... This monoid occurs naturally in the study of fractal curves, and describes the ... E-Infinity - High Energy Communications: The small world of ... 24 Mar 2011 - The diffeomorphic kinship between the Penrose fractal tiling universe and the compactified Klein modular curve is well known from El Naschie's ... Symplectic tiling, hypercolour and hyperflavor E12 ... ABSTRACT This paper will apply symplectic tiling to A. Garrett Lisi's G2 of Color, and to hyperflavor E12. ... Second we need to find the Euler characteristic of our fractal spacetime manifold. .... Comparison of Klein's v(7) modular curve (Fig. HTML - Scientific Research Publishing by MS El Naschie - ‎2013 - ‎Cited by 12 - ‎Related articles We introduce an ultra high energy combined KAM-Rindler fractal spacetime ... noncommutative fractal tiling which constitutes a compactified Klein modular ..... a ramified i.e. compactified Klein modular curve or a fractal Penrose universe [65], ... Full-Text HTML - Scientific Research Publishing by MS El Naschie - ‎2014 - ‎Cited by 1 This is of course a consequence of the fractal nature of Nottale's theory and in ... automorphism of the classical 336 degrees of freedom Klein modular curve, .... (2013) Chaotic Fractal Tiling for the Missing Dark Energy and Veneziano Model. [PDF]Experimentally based theoretical arguments that Unruh's ... by MS El Naschie - ‎2013 - ‎Cited by 10 - ‎Related articles 10 Dec 2013 - Klein modular curve and leading to a Regge-like fractal triangulation ... like in the Penrose fractal tiling universe projection [32]. It is actually a ... More E-infinity communications for your amusement - El ... 24 Mar 2011 - R. Murdzek, Daniela Magop and Roxana Stana: A fractal universe in .... fractal tiling universe and the compactified Klein modular curve is well ... [PDF]Fractal Spacetime and Noncommutative Geometry in ... theories, E-infinity theory, fractal tiling, quantum gravity via fractal Regge calculus, knot theory ...... original Klein modular curve of SL(2,7) simple linear Lie group.

oldbrew Comment left 3rd February 2015 23:11:28
Your number 4.236067977... is in fact Phi³, rather than 4 + 0.236...

Grayham FOrscutt Comment left 2nd June 2015 14:02:01
Great summary of Fractal Fields Mae. Would appreciate feedback if you have the time. I have modelled a Unified Fractal Field, combining Penrose '3D' phi rhomboids with minimum volume phi tetra's that face bond to make the Rhomboids, 5fold geometries and a 'volumetric' golden ratio spiral. Regards.