DIY Fractals

It’s been a while since I’ve posted anything mathematical.

Fractals are everywhere in nature. In this video I show some examples of fractals you can find in your own garden, how computers generate fractals and finally some fun examples you can construct with nothing more than some paper and colouring pens.

Hyperspace and La Grande Hypercube

Hyperspace.  What does it mean?  Science fiction has largely hijacked the word but its real foundations are embedded in the mathematics of higher or n-dimensional space pioneered by Bernhard Riemann in the late 19th century.  Riemann lay the foundations for extra dimensional spaces and so-called non-euclidean geometries that were self consistent and logical.  His Riemann tensor generalised geometry to curved surfaces, breaking free from over two millennia of Euclid’s dominance over the subject.  In this new geometry of the curved surface, parallel lines can intersect and the angles of a triangle do not necessarily add to 180 degrees.

Bernhard Riemann broke past the established doctrine of 'flat' Euclidean geometry.

Bernhard Riemann broke past the established doctrine of ‘flat’ Euclidean geometry.

‘Therefore, either the reality on which our space is based must form a discrete manifold or else the reason for the metric relationships must be sought for, externally, in the binding forces acting on it’ – Bernhard Riemann

Riemann’s curvature tensor was part of a developmental framework encapsulating the idea that forces could be considered as alterations in the fabric or ‘geometry’ of space rather than an instantaneous phenomena acting at a distance.  The reason a three dimensional creature felt the effects of a force could then be explained by a warpage in the fabric of the four dimensional geometry it moved through.  The analogy of a 2D ant residing on an undulating 3D surface is a good way to understand this thought process.  As the ant moves around it cannot see the curvature of the larger 3D space it inhabits yet it feels the effects as a force as it moves across the world.

Hyperspace in Science Fiction

Han-Solo and Chewbacca are seasoned hyperspace users in the film Star Wars.

Han-Solo and Chewbacca are seasoned hyperspace users in the film Star Wars.

‘Traveling through hyperspace ain’t like dusting crops, boy!’ – Han Solo

Hyperspace has been popularised in science fiction since the late 1920s and early 1930s, coincident with Albert Einstein’s special and general theories of relativity which became widely accepted around this time, and involved the idea of time as an additional dimension.  By directly applying the Riemann tensor to his work on universal gravity Einstein was able to powerfully simplify his approach and construct his general theory of relativity.  It was this application of the multi-dimensional to our understanding of gravity, light and space-time that propelled hyperspace into fiction and art. In a science fiction context the word represents a vague notion of transcending into higher dimensions to facilitate quick interstellar travel (Star Wars), or in the case of Howard Phillips Lovecraft’s ‘Dreams in the Witch House’, a gateway into a shadow realm of cosmic horrors.  It also spawned the notion of the worm hole – a higher dimensional bridge permitting a traveler to move between normally disconnected and independent spaces (or possibly times).

Lovecraft used concepts from hyperspace and non-euclidean geometry in his cosmic horror tale 'Dreams in The Witch House'.

Lovecraft used concepts from hyperspace and non-euclidean geometry in his cosmic horror tale ‘Dreams in The Witch House’.

‘One afternoon there was a discussion of possible freakish curvatures in space, and of theoretical points of approach or even contact between our part of the cosmos and various other regions as distant as the farthest stars or the transgalactic gulfs themselves – or even as fabulously remote as the tentatively conceivable cosmic units beyond the whole Einsteinian space-time continuum.’ – The Dreams in the Witch House by H P Lovecraft.

So how far away is the real mathematics of hyperspace from this popularised notion?  And can mathematics let us glimpse into these mysterious spaces?

Mathematical Hyperspace – a journey into the 4th Dimension

Mathematics can represent higher dimensions with relative ease and in doing so can often reduce formerly complex problems to more elegant and simpler forms. But what can mathematics reveal to us visually or conceptually about higher dimensions?   It’s a vast topic but let’s try exploring one geometrical dimension up – the fourth dimension – by constructing one of the simplest 4D objects to conceptualise, the hypercube.  Follow me on an imaginative journey into the fourth dimension!

Journey into the fourth dimension

  • To kickstart our journey into hyperspace lets begin by imagining a single point in space – a singularity.  A singularity has no dimensions so resides in zero dimensional space – the sort of place the universe may have originated from or a black hole will take you.
  • Now take this point and draw an imaginary straight line from it to a second point.  We’re now in 1D space and have 2 points with a line segment connecting them.
  • Now take each point of our line and drag it up or down to make a square with 4 points.  Here we are in two dimensional space.  How many line segments do we have now?
  • Next  grab  those four points and pull them up or down at a right angle to make a cube.   We’re now in familiar 3-dimensional space with 8 points and 12 connecting line segments.

Now for an imaginative leap into hyperspace.  Let’s pull those 8 points out of our reality and into the fourth dimension!  What do you see?  Or let’s put it another way – what should we expect to see? Although we can’t directly visualise it we can infer some of the properties of this four dimensional cube from mathematical extrapolation.  For starters it should have 16 points. How about edges?  Well each point will connect to 4 edges so that’s 4 x 16.  But each vertex will share all its edges with another vertex so we half this value.  Therefore it must have 4 x 16 /2 = 32 edges.

Dimension Descriptor Points Edges
0 Singularity 1 0
1 Line 2 1
2 Square 4 4
3 Cube 8 12
4 Hypercube 16 32

This 4D analogue of a cube is also known as a Tesseract.  Now that we’ve inferred some of its properties can we say anything about what it might look like?  Well in the same way that a sphere only reveals a shadow of itself on a flat plane, we can only visualise a 4D hypercube from its shadow projected into our 3D world.  As it happens such a shadowy denizen from another dimension actually exists on earth.

La Grande Arche a la Hypercube

Most Parisians walking in La Defense are probably oblivious to the fact that there’s a manifestation of the fourth dimension nearby. La Grande Arche was designed by Danish architect Johann Otto von Spreckelsen.  It’s a mystery whether or not this grand design was ever consciously inspired by mathematics but it’s nonetheless a beautiful and radical piece of architecture.

A real earth based projection of a four dimensional cube - The famous Paris arch

A real earth based projection of a four dimensional cube – The famous Paris arch

Despite being only a shadow of a 4D cube this structure contains most of the defining properties of a 4D cube – crucially it has 16 points connecting 32 edges.  One of the properties it lacks is uniformity of distance along its edges.  In the same way a square in two dimensions has equal sides, a real hypercube in 4D space would have all its connecting edges exactly the same length.  But this projection clearly doesn’t (note the lengths of the inner edges are shorter than the outer edges).  That’s because any projection from a higher to a lower dimension doesn’t necessarily preserve the distance between points, or indeed the angles between edges and faces.  Looking at the shadow of a cube on a piece of paper is a good example of this; no orientation can preserve the three dimensional distances between points.  In other words information is lost whenever we try to represent a higher dimensional object in a lower dimensional setting.

Projection of a hypercube  rotation about a single plane bisecting its middle.

Projection of a hypercube rotating about a single plane bisecting its center.

Mathematics can even rotate the hypercube and we can observe the weird and impossible dance made by its three dimensional shadow (above).  The object is seen to move through itself in a way that defies the rational behavior of solids in our 3D world.  The same effect can be seen by rotating a 3D cube in front of a lamp and watching the behavior of its two dimensional projection.  You would really see these  sort of visual oddities if four dimensional objects or beings entered our 3D world.

n-cubes

Of course mathematically cubes don’t stop in 4 dimensions. You can move into five, six or even more dimensions and build n-cubes in all these worlds. For example, a hypercube in n dimensions will have  2^n vertices. From each of these vertices there will be n edges emerging, each of which are counted twice, so the N-dimensional cube has n.2^{n-1} edges.  I’ll leave the task of imagining such n-cubes to the reader!

Dimension Descriptor Points Edges
n n-cube  2^n  n.2^{n-1}

Prime Numbers and Extraterrestrial Communication

A fascinating but little known application of prime numbers involves the search for, and communication with, extraterrestrials.  In the 1997 film Contact (adapted from Carl Sagan’s novel by the same name), Ellie Arroway and her team of radio telescope ET hunters pick up a repeating signal from the nearby star Vega (around 25 light years away).  Tuning in they quickly realise the message contains the sequence of early prime numbers 2, 3, 5, 7, 11.  They excitedly conclude that they’ve stumbled upon a bona fide alien signal.  Why?

A hopeful Dr Ellie Arroway (Jodi Foster) listens for ET in the film adaption of the novel 'Contact'.  Copyright Warner Bros.  1997

A hopeful Dr Ellie Arroway (Jodi Foster) listens for ET in the film adaptation of the novel ‘Contact’. Copyright Warner Bros. 1997

Foremost, because whilst primes are fundamental to the generation of the natural numbers, they’re also extremely difficult to find and predict.  Over 2300 years ago, in one of the most elegant and famous proofs in number theory, Euclid demonstrated that there are infinitely many primes (see below), yet to this day there’s no known polynomial or simple formula for generating the nth prime.  They appear almost randomly embedded in the space of all natural numbers and this elusiveness means that sequences of primes don’t tend to arise from naturally occurring processes.

A visual representation of all the primes from 1 to 62500. Composite (non primes) are rendered in grey with primes shown as bright white squares.  Courtesy of Mode of Expression

A visual representation of all the primes from 1 to 62500. Composite (non primes) are rendered in grey with primes shown as bright white squares. Courtesy of Mode of Expression

Other sequences of numbers frequently occur in nature.  In fact any non-random sequence of numbers we care to think of is probably rooted in some physical process or intuition about our environment. For example, what if we tried to get ET’s attention by broadcasting the following sequence (terminated for some suitable n):

\{2,4,8,16,32,64,...,2^n\}

We might argue such a sequence (doubling the preceding digit) would be interpreted as a clear and deliberate signal.  What else could it be confused with? Quite a lot as it happens.  For starters nuclear chain reactions follow such a  2^n growth.  Fire a neutron at a Uranium 235 atom and energy plus two neutrons are released.  Those two neutrons in turn may collide with two more Uranium atoms generating 4 neutrons, and so on.  Clearly this kind of uncontrolled fission happens routinely within the evolution of stars so such a sequence of numbers wouldn’t be an optimal extraterrestrial handshake.  How about instead we send a simple repeating sequence of fixed amplitude, a sort of cosmic door knock if you like:

\{4, 4, 4, 4, 4, 4, 4, 4\}

A signal with these characteristics was detected in 1967 by Jocelyn Bell Burnell in Cambridge, England.  The signal was uncovered during a study of Quasars and resembled a series of regular peaks of fixed amplitude 11/3 second apart.  The team was so baffled by it they identified it with the codename LGM (little green men).  Yet the signal turned out to have a completely natural source – it was in fact the regular and rapid rotation of a distant pulsar (the first such detected and studied) issuing forth regular bursts of electromagnetic radiation.

Semiprimes and the Drake Cryptograph

In the 1970s Frank Drake of Cornell University proposed a simple yet elegant method for interstellar communication utilising primes.  It involved sending a pictorial binary message (comprising just 0’s and 1’s) and a number for deciphering the message, known as a semiprime.  A semiprime is a number generated from the product of exactly and only two primes.  For example six, fifteen and twenty five are semiprime because 6 = 2 x 3 = 3 x 2,  15 = 3 x 5 = 5 x 3 and 25 = 5 x 5.  Twelve is not semiprime because 12 = 2 x 2 x 3 = 2 x 3 x 2 = 3 x 2 x 2.

“I know perfectly well that at this moment the whole universe is listening to us,” Jean Giraudoux wrote in The Madwoman of Chaillot, “and that every word we say echoes to the remotest star.”  That poetic paranoia is a perfect description of what the Sun, as a gravitational lens, could do for the Search for Extraterrestrial Intelligence. – Frank Drake

The idea is you build a picture array with 0’s and 1’s such that the length (or cardinality) of the data set is a semiprime, l = p x q.  This semiprime is supposed to  suggest to the receiving intelligence to arrange the data in two possible configurations:  p x q or q x p.  One of these configurations contains the message.

Example:  Let’s take a really simple example to illustrate the method.  We’ll pitch this in reverse and pretend we’ve just received a Drake cryptograph from outer space.  Our receivers pick up the following repeated signal from the double star Gamma Andromedae:

1 1 1 1 0 0 1 1 1 1 0 0 1 0 0

On first appearance the data looks pretty uninteresting but if we count the number of elements in the list (the cardinality) we get 15.  What’s special about 15?  Well it happens to be semiprime:  3 x 5 = 5 x 3.  Great.  That suggests two clear ways or arranging the bits.  Let’s first try breaking it into 5 columns by 3 rows:

5x3

Hmmmm….nothing to see here.  What about 3 columns by 5 rows:

Screen Shot 2015-02-15 at 18.45.26

It’s the letter F!  Amazing….we’ve made first contact with an alien civilisation who might join us for a game of interstellar scrabble!

The Arecibo Message

Perhaps the most famous use of Drake’s cryptograph was the Arecibo message broadcast  in 1974 from the recently refurbished Arecibo radio telescope in Puerto Rico towards the globular cluster M13 (25,000 light years away).  The data steam had a cardinality of 1679 bits, which is a semiprime with the following prime factors 73 x 23 or 23 x 73.  The correct orientation for revealing the message is 23 columns x 73 rows.  Voila!

The Arecibo message correctly oriented with descriptions of what each part is supposed to convey to an alien intelligence.

The Arecibo message correctly oriented with descriptions of what each part is supposed to convey to an alien intelligence.

The message (co-authored by Drake, Carl Sagan and others) is supposed to convey a condensed summary of the human species – its technology, biology and location in space (see here for more details).  Whether or not such a message would be understood by another technical civilisation, if successfully decoded, is open to debate.  Any such civilisation would certainly need to share many common traits with the human species at the very least.  But the main motivation behind the message was simply to demonstrate what was possible rather than seriously communicate with denizens who, if they even existed, would take over 50,000 years to call back.


Explore the Mathematics

These sections will define in more detail some of the mathematics used in Modulo Universe.  Readers should consider skipping as desired.

Definition of Prime number:  A natural number p > 1 is prime if it is only divisible by itself and 1.

Definition of Composite number:  A natural number c is composite if it has at least one more divisor other than itself and 1 (i.e. it isn’t prime).

So why are primes so fundamental in mathematics?  This next theorem explains why:

The Fundamental Theorem of Arithmetic:  Every integer > 1 can be represented uniquely (apart from order) as a product of one or more prime numbers.

In other words the prime numbers represent fundamental and indivisible building blocks for the positive integers.  For example 20 = 2 x 10 = 2 x 2 x 5.  32 = 4 x 8 = 2 x 2 x 2 x 2 x 2.

Put another way:  Every positive integer n > 1 can be represented in exactly one way as a product of prime powers.

This leads me to one of the most beautiful and elegant proofs in mathematics – Euclid’s infinite primes.  One of the great appeals of mathematics over other human creations is its eternity.  By this I mean that once you prove something remarkable in mathematics – as Euclid did over 2000 years ago – it’s there forever.  Whereas theories in physics, chemistry and astronomy undergo a constant revision, mathematics is like an indestructible and ever-growing vault.  The literature from this epoch of history might loose some of its original meaning due to the filter of modern culture, yet the logic of greek mathematics cuts through the intervening millennia with a piercing clarity.

I urge you to follow along with this proof as its relatively easy, even without mathematical training.

Theorem (Euclid 300 B.C):  There are an infinite number of primes.

Proof:  Suppose there’s isn’t an infinite number of primes!  Then there must be a biggest prime number – let’s call that P.  We can then make a list of all the known primes including P:

p1, p2, p3, p4,…., P

So far so good.

Now suppose we multiply our list of known primes together to make a number N:

p1.p2.p3…..P = N

Now this number N can’t be prime because it has all these prime divisors.  But what if we add 1 to N?  Let’s call this new number M.

M = N+1

Now M, like any other natural number, must be either a prime or a composite number.  If it’s the former we’ve found another prime larger than N contradicting our assertion.  If it’s a composite then it can be expressed as a product of prime factors.  However none of the primes listed can divide M so there must exist other primes not in our list.  In both cases we’ve contradicted our initial claim so there must be an infinite number of primes!

Definition of Semiprime:  A natural number is semiprime (or bi-prime) if it is the product of two prime numbers p and q (l = p x q = q x p).

From this definition it’s clear that the square of any prime number must be semiprime and therefore the largest known semiprime is also, conveniently, the square of the largest known prime.  Large semiprimes are popular in encryption systems due to the simplicity of making them (grab two primes and multiply them together) but complexity of factorising them with no prior knowledge.