The complex beauty of fractal geometry

Dr Matteo Sommacal

Department of Mathematics, Physics and Electrical Engineering at Northumbria University


Dr Sommacal was born in Roma, Italy, on November 18, 1977. Currently, he is Senior Lecturer in Applied Mathematics in the Department of Mathematics, Physics and Electrical Engineering at Northumbria University, Newcastle upon Tyne, UK, where he arrived in January 2012. He earned a M.Sc. in Physics at the Università degli studi di Roma “La Sapienza”, Italy (2002). Subsequently, he received a Ph.D. in Mathematical Physics at the International School for Advanced Studies in Trieste, Italy, with Professor Francesco Calogero as advisor (2005), discussing the thesis “The Transition from Regular to Irregular Motions, Explained as Travel on Riemann Surfaces”.

After the Ph.D. studies, he got several academic positions as post-doctoral fellow and research scholar. He was “Chercheur post-doctorant” at the Laboratoire J.-L. Lions, Université Pierre et Marie Curie-Paris VI, France (November 2005 – October 2006). He then received a three-year post-doctoral fellowship (“assegno di ricerca”) from the Department of Mathematics and Computer Science, Università degli Studi di Perugia, Italy (November 2006 – October 2009). After one year as post-doctoral fellow at the Department of Physics, Università degli Studi di Roma “La Sapienza”, Italy (November 2009 – October 2010), he got an invitation as “Visiteur de longue durée” at the Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette, France (November 2010 – April 2011). He then was appointed a post-doctoral research scholarship at the Department of Mathematics, North Carolina State University, Raleigh, NC, USA (May 2011 – December 2011).


In this lecture, Dr Sommacal explained, with the aid of some computer graphics, how fractals can be defined as geometrical objects characterised by two properties: self-similarity, and non-integer dimension. Differently from the ‘smooth’ figures of classical Geometry, such as circles or triangles, fractals turn out to be ‘rough’ and infinitely complex.

The following are notes from the on-line lecture. Even though I could stop the video and go back over things there are likely to be mistakes because I haven’t heard things correctly or not understood them. I hope Dr Sommacal, and my readers will forgive any mistakes and let me know what I got wrong.

Fractal geometry is a very different type of geometry that most school children (or physics teachers. i.e. me) experience.

Sierpiński triangle

Dr Sommacal started his talk by playing a sort of “on-line” computer game.

He put the following image on the screen.


He randomly chose three points by clicking on the plane and these became the vertices of a triangle.


Next, he selected a random point on the plane inside the triangle and he labelled it 0 (zero)


The game didn’t involve changing the vertices of the triangle, but working with points inside the triangle and he wanted to generate other points. He first set down a little rule and that was that any of points A, B or C had to be chosen randomly by the computer program.

The computer selected vertex B


He drew a line between 0 and B and half way along this line he added a new point and labelled it 1.


The computer then selected the next vertex randomly (it could have picked A, C or B again)

The computer selected C as the next vertex


A line was then drawn from 1 to C and a point put midway on this line was labelled 2.

This process was combining a random rule with a deterministic rule. So, part of the rule is out of human control as the vertices are chosen randomly (A, B or C). But once the point was picked the second part of the process was down to human intervention.

So, the midline point was joined to the randomly chosen vertex. Halfway along this line a new point was created and a line was joined between it and another randomly joined vertex. The process was continued.

A was the next randomly chosen vertex and a line was drawn between A and 2. The midpoint was labelled 3.


The process was repeated over and over again.


C was selected again. A line was drawn between 3 and C and the midpoint was labelled 4


C was selected again. A line drawn from 4 to C with the midpoint labelled 5. Dr Sommacal thought this was rather boring because not much was happening (but then he is a mathematician).

So, what is the point (no pun intended)?

Well to understand what is happening the game needs to be played for longer. So, more points were needed. The process was repeated ten times.

Ten additional points added and the numerical labels were removed to make it easier to follow the pattern


Just a triangle with spots. Still no pattern

100 additional points added


Still no pattern

Is anything going to emerge from this process? Initially it simply looks like the points were roaming around.

The computer could have selected the vertices more than once.

There was no restriction in the selection of the vertices, the random part of the process.

Why should a pattern emerge from this process? Because the choice of the vertices is so random there is no expectation that a specific structure will emerge.


Random spots inside the triangle although there didn’t appear to be spots in the centre of the triangle. This was very strange. There also seemed to be other regions that didn’t seem to have points.

The game was played a little more with a further 100 points added


Maybe there is going to be regions that won’t be visited by spots at all, but there was still no clear structure coming out from the exercise.

So maybe more points needed to be added.

1000 more points added.

A structure is emerging


10000 additional points added


100000 points added


A very precise structure started to occur with a process that started with three completely random points. The rules of the game always produces figures like this ne.

There is central blank region that divided the triangle into three sections and each of these triangular sections had a central blank section that divided the triangle into three sections, and so on.

If the structure could be magnified over and over again the sequence of triangles would be seen over and over again.

This was surprising because the game was carried out using a combinatopn of randomicity and determinism.

The triangular figure had a lot of interesting properties that pops up in many branches of mathematics.

The above figure is called a Sierpinski triangle. Named after the first person to “discover it”.


Wacław Franciszek Sierpiński (14 March 1882 – 21 October 1969) was a Polish mathematician. He was known for contributions to set theory (research on the axiom of choice and the continuum hypothesis), number theory, theory of functions and topology. He published over 700 papers and 50 books.

Three well-known fractals are named after him (the Sierpiński triangle, the Sierpiński carpet and the Sierpiński curve), as are Sierpiński numbers and the associated Sierpiński problem.

The Sierpiński triangle (sometimes spelled Sierpinski), also called the Sierpiński gasket or Sierpiński sieve, is a fractal attractive fixed set with the overall shape of an equilateral triangle, subdivided recursively into smaller equilateral triangles. Originally constructed as a curve, this is one of the basic examples of self-similar sets—that is, it is a mathematically generated pattern that is reproducible at any magnification or reduction. It is named after the Polish mathematician Wacław Sierpiński, but appeared as a decorative pattern many centuries before the work of Sierpiński.

Questions and answers 1

1) How many times do we need to play in the game before we see the pattern emerging? Is there a mathematical correlation?

If you play the game 100 thousand times you will have a dred thousand points and you will get the structure, but in order to see the structure even if you keep magnifying parts of the structure you would need to play the game an infinite number of times to see the pattern infinitely.

What mathematicians do by abstraction is that once they have an idea of what the shape is, they extend this idea to all the possible scales.

2) Does this “game” work with other shapes or is it only with triangles?

This question will be answered in the talk.

3) Does the program use true random code or computer random generator

It uses a computer random generator. It is an imitator of a random process. In this experiment it doesn’t change much.

You can play the game in a determinist form without randomly choosing the 3 points, for instance choosing in a rota i.e. A then B the C and back to A etc. and you still get the same picture/structure.

Back to talk

Sierpinski triangle


The game started with an equilateral triangle with sides of length 1


Then it made three shrunk copies of it, each one with sides of length of 1/2



The program positioned the three shrunk triangles so that each triangle touched the other two triangles at a corner, leaving a central triangular hole.


The procedure was repeated ad infinitum and at each stage each triangle was substituted with three smaller triangles, each one touching the other two at a corner, and leaving a triangular hole in the middle.








The process:

Take a copy of the last stage

Takes 3 copies and shrink them to so that each side is half the original size

Reconstruct the triangle using the copies

Continue the construction forever and ever to reach stage infinity

The Sierpinski triangle is what you get at stage infinity

The same shape at all scales.

The Sierpinski triangle is a classical example of a self-similar object, i.e., a mathematical object that, at all scales of magnification or reduction, is exactly (or approximately) similar to a part of itself (in other words, the whole has the same shape as one or more of the parts).

Sometimes you get an exact self-similarity but sometimes you get an approximate self-similarity, this is observed in nature. In nature self-similarity can only occur up to the atomic scale. Below that we enter the world of quantum mechanics.

Quantum mechanics is the science of very small things. It explains the behaviour of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behaviour of astronomical bodies such as the Moon. Classical physics is still used in much of modern science and technology.


The Sierpinski triangle appeared as a decorative pattern many centuries before the work of Sierpinski

Sierpinski triangles in two Rome churches

image (above left)

The Basilica of Saint Clement (Italian: Basilica di San Clemente al Laterano) is a Latin Catholic minor basilica dedicated to Pope Clement I located in Rome, Italy. Archaeologically speaking, the structure is a three-tiered complex of buildings: the present basilica built just before the year 1100 during the height of the Middle Ages; beneath the present basilica is a 4th-century basilica that had been converted out of the home of a Roman nobleman, part of which had in the 1st century briefly served as an early church, and the basement of which had in the 2nd century briefly served as a mithraeum; the home of the Roman nobleman had been built on the foundations of republican era villa and warehouse that had been destroyed in the Great Fire of 64 AD. (Above right)

The Basilica of Saints John and Paul on the Caelian Hill (Italian: Basilica dei Santi Giovanni e Paolo al Celio) is an ancient basilica church in Rome, located on the Caelian Hill. It was originally built in 398.

It is home to the Passionists and is the burial place of St. Paul of the Cross. Additionally, it is the station church of the first Friday in Lent.

Similar patterns appear in the 13th-century Cosmati mosaics in the cathedral of Anagni, Italy, and other places of central Italy, for carpets in many places such as the nave of the Roman Basilica of Santa Maria in Cosmedin, and for isolated triangles positioned in rotae in several churches and basilicas. In the case of the isolated triangle, the iteration is at least of three levels.

A medieval triangle, with historically certain dating has been studied recently. It is in porphiry and golden leaf, isolated, level 4 iteration.

Generated using a random algorithm

Animated creation of a Sierpinski triangle using the chaos game

Animated construction of a Sierpinski triangle

The Sierpinski triangle appears in many different areas of mathematics, in particular the theory of dynamical systems, and finds applications in physics, biology and engineering.

Expanding a binomial power

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial (x + y)n into a sum involving terms of the form axbyc, where the exponents b and c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b.

Coefficients appearing in the polynomial expansion of the binomial power (a + b)n


In view of what was going to happen the coefficient 1 was specified even though there was no need to do this.


Now everything was cancelled down except the coefficients


Each entry of each subsequent row was constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0.

Take any two neighbouring numbers on one row. Sum them and you get the number immediately below them. In fact, by using this property and starting with the first three rows you can construct the whole triangle.


This is a very famous triangle, known as Pascal’s triangle in the UK but it has been discovered many times in the history of mathematics, by many different people.



Halayudha was a 10th-century Indian mathematician who wrote the Mṛtasañjīvanī, a commentary on Pingala’s Chandaḥśāstra. The latter contains a clear description of Pascal’s triangle (called meru-prastaara).

Omar Khayyam (18 May 1048 – 4 December 1131) was a Persian mathematician, astronomer, philosopher, and poet. He was born in Nishabur, in northeastern Iran, and spent most of his life near the court of the Karakhanid and Seljuq rulers in the period which witnessed the First Crusade.

Yang Hui (ca. 1238–1298), courtesy name Qianguang, was a Chinese mathematician and writer during the Song dynasty. Originally, from Qiantang (modern Hangzhou, Zhejiang), Yang worked on magic squares, magic circles and the binomial theorem, and is best known for his contribution of presenting Yang Hui’s Triangle. This triangle was the same as Pascal’s Triangle, discovered by Yang’s predecessor Jia Xian.

Jia Xian (ca. 1010–1070) was a Chinese mathematician from Kaifeng of the Song dynasty.

Petrus Apianus (April 16, 1495 – April 21, 1552), also known as Peter Apian, Peter Bennewitz, and Peter Bienewitz, was a German humanist, known for his works in mathematics, astronomy and cartography. His work on “cosmography”, the field that dealt with the earth and its position in the universe, was presented in his most famous publications, Astronomicum Caesareum (1540) and Cosmographicus liber (1524. His books were extremely influential in his time, with the numerous editions in multiple languages being published until 1609. The lunar crater Apianus and asteroid 19139 Apian are named in his honour.

Michael Stifel or Styfel (1487 – April 19, 1567) was a German monk, Protestant reformer and mathematician. He was an Augustinian who became an early supporter of Martin Luther. He was later appointed professor of mathematics at Jena University.

Niccolò Fontana Tartaglia (1499/1500 – 13 December 1557) was an Italian mathematician, engineer (designing fortifications), a surveyor (of topography, seeking the best means of defence or offence) and a bookkeeper from the then-Republic of Venice (now part of Italy). He published many books, including the first Italian translations of Archimedes and Euclid, and an acclaimed compilation of mathematics. Tartaglia was the first to apply mathematics to the investigation of the paths of cannonballs, known as ballistics, in his Nova Scientia (A New Science, 1537); his work was later partially validated and partially superseded by Galileo’s studies on falling bodies. He also published a treatise on retrieving sunken ships.


Blaise Pascal (19 June 1623 – 19 August 1662) was a French mathematician, physicist, inventor, philosopher, writer and Catholic theologian.


More rows were added to the triangle of numbers


And yet more rows were added

Dr Sommacal focused on the first 32 rows


The numbers had to be tiny to fit on the slide. But if the slide could be magnified


Numbers in a Pascal triangle can get very big very quickly.

Dr Sommacal then carried out an experiment as the actual numbers in the triangle were not important. He played with the parity mathematics, identifying the odd and even numbers in the triangle and replaced every even number with a 0, and every odd number with a 1


At the above resolution it is very difficult to tell a 0 from a 1 so no structure is emerging. The 1s were then made darker and the 0’s lighter


The above image looks oddly familiar! The next stage involved replacing every 1 with a dot and every 0 with a blank.



64 rows instead of 32


256 rows

The more rows that are added, the more Pascal’s triangle resembles Sierpinski’s triangle! Apparently, this is an unexpected result coming from the expansion of a binomial.

You take many, many binomials and you “magically” get Sierpinski’s triangle.

So there must be a self-similar structure in the kind of algebra behind the binomials – and this is indeed the case.

Questions and answers 2

1) In the Sierpinski triangle activity you took the middle point pf the line. In the Pascal triangle you used the binomial expansion. Is there a correlation between the two?

No there is no correlation in the background. It is one of those interesting facts in mathematics that you see certain structures appearing ubiquitously in circumstances where there are no evident corelations and no corelations at all, as in this case.

Dr Sommacal selected the two examples exactly because there is no corelation. The two are completely different processes. The Sierpinski triangle emerges from a prime number structure as well.

So, by studying the distribution of certain exponents in the natural numbers the Sierpinski triangle appears in a lot of other situations. It appears in physics, i.e. the study of the crystals in snowflakes,

2) There are lots of patterns in Pascal’s triangle. What property of it produces these patterns?

Pascal’s triangle has a lot of properties. Dr Sommacal picked this for his talk as it is less well known. Other structures would need talks of their own.

Back to the talk

Self-similarity and iterated functions systems

The Sierpinski triangle is an example of a self-similar object by means of an iterative process (iterated function system).


The first 5 stages of the construction of the Sierpinski triangle

The self-similarity games with the abstract construction can be played with other shapes. When the game is played in mathematics it is called an iterated function system.


The first 5 stages of the construction of the Sierpinski carpet

The Sierpiński carpet is a plane fractal first described by Wacław Sierpiński in 1916. The carpet is one generalization of the Cantor set to two dimensions; another is the Cantor dust.


The first 5 stages of the construction of the Viscek snowflake

In mathematics the Vicsek fractal, also known as Vicsek snowflake or box fractal, is a fractal arising from a construction similar to that of the Sierpinski carpet, proposed by Tamás Vicsek. It has applications including as compact antennas, particularly in cellular phones.


Tamás Vicsek (born 10 May 1948, Budapest) is a Hungarian scientist with research interests in numerical studies of dense liquids, percolation theory, Monte Carlo simulation of cluster models, aggregation phenomena, fractal growth, pattern formation (computer and laboratory experiments), collective phenomena in biological systems (flocking, oscillations, crowds), molecular motors, cell locomotion in vitro. He held the position of professor of physics at the Eötvös Loránd University, Budapest, Hungary, and was visiting scientists in various academia.

Vicsek snowflake uses less copies and is organised differently.

At this point in the talk Dr Sommacal hadn’t explained why this object is called a fractal.

There is potentiality for an infinite number of structures.


The first 4 stages of the construction of the Von Koch snowflake

The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a fractal curve and one of the earliest fractals to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled “On a Continuous Curve Without Tangents, Constructible from Elementary Geometry” by the Swedish mathematician Helge von Koch.


Niels Fabian Helge von Koch (25 January 1870 – 11 March 1924) was a Swedish mathematician who gave his name to the famous fractal known as the Koch snowflake, one of the earliest fractal curves to be described.


The first 4 stages of the construction of the Cesaro snowflake


The first 4 stages of the construction of the Pentaflake snowflake


The first 5 stages of the construction of the Gosper Island


Ralph William Gosper Jr. (born April 26, 1943), known as Bill Gosper, is an American mathematician and programmer.


The first 4 stages of the construction of the exterior snowflake

The iterative process can also be done in 3 and 4 dimensions (perhaps even 4 or 5?)

Self- similar structure in 3-dimensions.


Menger sponge

In mathematics, the Menger sponge (also known as the Menger cube, Menger universal curve, Sierpinski cube, or Sierpinski sponge) is a fractal curve. It is a three-dimensional generalisation of the one-dimensional Cantor set and two-dimensional Sierpinski carpet. It was first described by Karl Menger in 1926, in his studies of the concept of topological dimension.


Karl Menger (January 13, 1902 – October 5, 1985) was an Austrian-American mathematician. He was the son of the economist Carl Menger. He is credited with Menger’s theorem. He worked on mathematics of algebras, algebra of geometries, curve and dimension theory, etc. Moreover, he contributed to game theory and social sciences.


Sierpinski pyramid

The Sierpinski tetrahedron or tetrix is the three-dimensional analogue of the Sierpinski triangle, formed by repeatedly shrinking a regular tetrahedron to one half its original height, putting together four copies of this tetrahedron with corners touching, and then repeating the process.


Von Koch pyramid

A three-dimensional fractal constructed from Koch curves. The shape can be considered a three-dimensional extension of the curve in the same sense that the Sierpiński pyramid and Menger sponge can be considered extensions of the Sierpinski triangle and Sierpinski carpet.

Nature also likes to play with shapes


Above left is a computer graphic rendition of Romanesco Broccoli. A self-similar structure. Above right is a real Romanesco Broccoli


Above left is a computer graphic rendition of Brassica oleraccea. A self-similar structure. Above right is a real Brassica oleraccea.



The structure is made from a big spiral (arrow on the above right image is tracing the spiral)

The big spiral is made up of tiny spirals and each one of these tiny spirals are made up of even tinier spirals etc. (magnification would be needed to see them). Theoretically the structure would go on forever (infinity) but in nature a point is reached at the atomic/molecular threshold. You can’t get any smaller due to the limits caused by quantum mechanics.

There is something in the object’s DNA that controls the 3-D structure that makes it behave with an iterative behaviour like the ones met earlier.

What happens if the rules of the game that generated the Sierpinski triangle are changed slightly?

Barnsley fern

A slightly different form of the chaos game played at the start of the talk. It was invented by a British mathematician Michael Barnsley.


Michael Fielding Barnsley, born in 1946, is a British mathematician, researcher and an entrepreneur who has worked on fractal compression; he holds several patents on the technology. He received a BA in Mathematics from Oxford in 1968 and his Ph.D. in theoretical chemistry from University of Wisconsin–Madison in 1972. In 1987 he founded Iterated Systems Incorporated, and in 1988 he published a book entitled Fractals Everywhere and in 2006 SuperFractals.

Instead of adding three vertices of a triangle you have to extract a number randomly between 1 and 100 and then make a decision on the next point based on the number you have extracted.

If the first chosen number is 1 you have a certain rule. If the next chosen number is between 2 and 86 you have a different rule. If the next chosen number is between 87 and 93 you have a different rule and if the final chosen number is between 94 and 100 you have yet another different rule.

So you have four different rules depending on which number you have extracted and then there is an identity procedure. So, once you have generated a known number e.g. a point in the plane then the new point in the plane is determined using the rule you decide on the number you have extracted. Play the game for the first 50 iterations starting from a random point in the plane.

The process:

Given a point in the plane, of coordinates (x,y), create a new point of coordinates (xold,yold) -→ (xnew,ynew) i.e. (x,y) = (xold,yold) -→ (xnew,ynew)

In this game, extract a random number n between 1 and 100 and then act as follows





Above shows the first 50 iteration


Above shows the first 100 iterations


Above shows the first 1000 iterations


Above shows the first 10000 iterations


Above shows the first 100000 iterations


Above shows the first 1 million iterations

Choosing different numbers in the four rules you can get different fern shapes and amazingly they all resemble existing species of fern.


The top layer of the above image is the mathematical version of the ferns and the bottom layer show ferns that resemble them.

There must be something in the DNA of the organisms for iterative procedures.

Mathematicians just found it when studying fractal geometry.

Self-similarity in nature is ubiquitous


Above left is an Osgood curve and above right is a natural version, a satellite image of the Alps

In mathematics, an Osgood curve is a non-self-intersecting curve (either a Jordan curve or a Jordan arc) of positive area. More formally, these are curves in the Euclidean plane with positive two-dimensional Lebesgue measure.

The first examples of Osgood curves were found by William Fogg Osgood (1903) and Henri Lebesgue (1903). Both examples have positive area in parts of the curve, but zero area in other parts; this flaw was corrected by Knopp (1917), who found a curve that has positive area in every neighborhood of each of its points, based on an earlier construction of Wacław Sierpiński. Knopp’s example has the additional advantage that its area can be controlled to be any desired fraction of the area of its convex hull. (below left)


William Fogg Osgood (March 10, 1864, Boston – July 22, 1943, Belmont, Massachusetts) was an American mathematician, born in Boston. (above centre)

Henri Léon Lebesgue ForMemRS (June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of a function defined for that axis. His theory was published originally in his dissertation Intégrale, longueur, aire (“Integral, length, area”) at the University of Nancy during 1902.

Konrad Knopp – Wikipedia

Konrad Hermann Theodor Knopp (22 July 1882 – 20 April 1957) was a German mathematician who worked on generalized limits and complex functions.


Above left is an L-system trajectory and above right is a natural version, lightning on the sea

An L-system or Lindenmayer system is a parallel rewriting system and a type of formal grammar. An L-system consists of an alphabet of symbols that can be used to make strings, a collection of production rules that expand each symbol into some larger string of symbols, an initial “axiom” string from which to begin construction, and a mechanism for translating the generated strings into geometric structures. L-systems were introduced and developed in 1968 by Aristid Lindenmayer, a Hungarian theoretical biologist and botanist at the University of Utrecht. Lindenmayer used L-systems to describe the behaviour of plant cells and to model the growth processes of plant development. L-systems have also been used to model the morphology of a variety of organisms and can be used to generate self-similar fractals.


Aristid Lindenmayer (17 November 1925 – 30 October 1989) was a Hungarian biologist. In 1968 he developed a type of formal languages that is today called L-systems or Lindenmayer Systems. Using those systems Lindenmayer modelled the behaviour of cells of plants. L-systems nowadays are also used to model whole plants.

Ball Lightning as a SelfOrganized Complexity

Fractal dimension and fractals

A plane is a two-dimensional object. Any point on a line can be given a distance from a reference point. A distance is enough to locate all the points on the line and because of this the line is a one-dimensional object.

What does dimension mean?

A point has 0 dimensions: roughly speaking, it has no length, no width and no height.


A line has one dimension: If a point is fixed on a line it serves as the origin. Then any other point on the line can be found by giving its distance from the origin (with a sign indicating if it falls before or after the origin)


A plane has two dimensions: If a point is fixed on a plane it serves as the origin for two lines intersecting. This means any other point on the plane can be found by giving its coordinates with respect to the lines.


So the new game is very consistent because we now take a plane and we trace two lines on that plane with a point of reference which is given as the intersection of the two lines. Then if you take any other point on the plane and you measure the distance from this point and one of the lines, and this point and the other line you have two pieces of information and that is the two distances between this point and the two lines. By using these two distance you can locate any point on the plane,

So two distances gives you two pieces of information . The object has two dimensions because you have two distances.

For three dimensions you get something similar with three lines instead of two.

A space has 3 dimensions: A point in space serves as the origin, and three lines intersect at the origin, then any other point in this space can be found by giving its coordinates with respect to the lines.


So you need three pieces of information to locate anything in three-dimensional space.


Moving about the surface of the Earth is a good example of the use of two dimensions. The surface has two dimensions so it is a two-dimensional object and the two pieces of information are latitude and longitude. You can find any object on the surface of the Earth if you know the latitude and longitude.

This idea of taking dimensions and measuring distances is embedded in the human mind.

The definition of dimension involves the idea of measuring distances.

However, for the self-similar objects that have been built so far, there is a problem with this definition.

For example. The von Koch curve.


We need to measure its length.

The segment length is 1


To begin to form the shape the central part needs to be removed and be replaced by two sides of a triangle

Each little part of the arrangement below is 1/3 of the original length


So, four pieces each of length 1/3


Then produce an iteration. For each one of the segments remove the central part and replace with two sides of a triangle.


Each segment is 1/9 of the original length and there are sixteen of them


Repeat over and over again



The length of the piecewise curve is becoming larger and larger, at each step of the construction of the von Koch curve. The total length of the self-similar structure is increasing.

After an infinite number of steps, the length has become infinite. Consequently, the length has become infinite, that is, the curve obtained after an infinity of steps, will be infinite.


In fact, it is not just that the total length of the von Koch curve is infinite. The von Koch curve is self-similar in that any part of the curve is similar to the whole curve. As the length of the whole curve is infinite, then the length of any part of the curve must be infinite too. This entails that, on the von Koch curve, the distance between any two points, taken along the curve, is infinite.

So, you can hold something finite in your hand that has an infinite length.

All of this sounds counter-intuitive. The idea of distance along the von Koch curve doesn’t make any sense. This means any definition of dimension based on the idea of distance doesn’t make any sense.

In other words, there is a big problem with dimensionality. To measure a length of a line you measure the distance between the point of reference, at the beginning of the line, and the point at the end of the line.


However, with a point of reference on a von Koch curve, every other point on it will be at an infinite distance from it. The reason for this is because the curve is self-similar. If you magnify any portion of the curve it will have the same properties as the whole. So every single portion, no matter how small, will have an infinite length at the infinite scale of the partition.

Then, what is the dimension of the von Koch curve? We can’t use the old definition of dimension. We need a new definition of dimension.

The person who solved this problem was Felix Hausdorff.


Felix Hausdorff (November 8, 1868 – January 26, 1942) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, and functional analysis.

He came up with a definition of dimension that didn’t rely on distance.

The following outlines his method:


Take a segment and split it up into two halves. This gives two little copies of the original segment.

Rescaling is halving the original length and having two copies.


The segment could be split up into 3 or 4 instead of 2. In this case there is a direct proportionality as the number of copies and the scale factor are the same. n = r

A similar process takes place with a square.


If you repeat the game with a square you need 4 copies when the scale factor is 2. In this case n = r2.


If you repeat the game with a cube you need 8 copies when the scale factor is 2. In this case n = r3

There is a very strange coincidence here because the segment is a one-dimensional object and the scale factor is raised to the power of 1.


However, Felix Hausdorff didn’t regard it as a coincidence at all. He defined a dimension to be equal to the exponent.



So, the exponent is the power you need to raise the scale factor to in order to give the number of copies needed to cover the original object.

A little use of logs

The Hausdorff dimension of a geometrical object is





It’s easy to see that regular shapes such as line (one dimension), square (two dimensions) and cube (three dimensions) are in fact 1, 2 and 3 dimensions but in the case of self-similar structures the oddity is that the dimensions are not integers.

So, what actually happens if we use Felix Hausdorff’s solution to the von Koch curve?



In general, at the stage k of the construction, we get:

Scale factor (r) = 3k

Number of copies (n) = 4k

Therefore, the dimension of the von Koch curve is


1.2619 is surprising as dimensions are expected to be an integer.

The dimension of the von Koch curve is not an integer number!. It lives between the line (dimension 1) and the plane (dimension 2). It is midway between them.

So the von Koch curve is not a one dimensional object (it is not an integer number). It is something between a line (dimension 1) and a plane (dimension 2) and this is why objects like these are called fractals. The dimension is fractional.

Fractal geometry is the branch of mathematics that studies fractals and their properties.

The Hausdoff dimension of a geometrical object is


What is the dimension of the Sierpinski triangle?


If the scale factor (r) is 2 then the number of copies (n) = 3

In general, at the stage k of the construction of the triangle, we have:

Scale factor (r) = 2k and the number of copies (n) = 3k

Therefore, the dimension of the Sierpinski triangle is


The above result shows the Sierpinski triangle is more plane like than the von Koch curve because its dimension is closer to 2 than 1.

This “game” can be extended to natural objects.

How long is the coast of Britain?

You want to measure the coastline of Britain and you have a very large measuring stick.

If the stick is 200km long the coastline is found to be about 2400km

If a shorter stick of length 100km is used the coastline is found to be about 2800km

If an even shorter stick of length 50km is used the coastline is found to be about 3450km


So, what would the length of the coastline of Britain be, if you could use an infinitely small measuring stick?

What is the dimension of the coastline of Britain?

Using the maths outlined earlier the dimension of Britain’s coastline is not an integer.

A mathematician, Mandelbrot, asked the question about the dimension of Britain’s coastline in 1967. He found is to be about 1.25.

Mandelbrot introduced the term “fractal” in 1975.


Benoit Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labelled as “the art of roughness” of physical phenomena and “the uncontrolled element in life”. He referred to himself as a “fractalist” and is recognized for his contribution to the field of fractal geometry, which included coining the word “fractal”, as well as developing a theory of “roughness and self-similarity” in nature.

Fractal landscapes


The above are computer generated fractal landscapes. These sort of things are found in most if not all computer graphics, from computer games to movies.

The level of the images are a very clsoe reproduction of what our eyes perceive as reality. So all the images on the above slide have been generated with some kind of iterative process.

Even putting the memory of the images into a computer is, in fact, an iterative process.

You can play games with fractals which are very sophisticated and if you play a certain iterative game using complex numbers you would get complex fractals like a Mandelbrot set and by changing the game slightly you would get a Julia set. (below left) (below right)



Above is another Julia set

Questions and answers 3

1) Looking back at the binomial activity would highlighting prime numbers instead affect the pattern differently?

It is possible to have a Sirpinski triangle appear even when using prime numbers and this was discovered only a few years ago, and is still under investigation.

It is a very complicated game. It is not played just by hiding or highlighting certain prime numbers according to a certain rule.

You have to go through a much more complicated and involved abstract kind of game playing more with prime representation of the integers.

Surprisingly, you can get a Sirpinski triangle even in the context of the study of prime numbers.

2) Are there any links between the Sirpinski triangle and the Mandelbrot set?

Dr Sommacal didn’t really know the answer. If you browse the web there are people claiming that by magnifying the Mandelbrot set an enormous number of times in certain regions, they actually managed to have the Sirpinski triangle appearing in a corner.

Dr Sommacal has never checked if the claims are correct.

Both the Sirpinski triangle and the Mandelbrot set can be generated in terms of some kind of iteration process although it is a discrete process involving discrete stages in the case of Sirpinski triangle and in the case of the Mandelbrot set you have a normal polynomial in the context of complex numbers. In principle they are separate processes.

3) I know these patterns don’t have to be purely geometrical – I’ve heard of something called the “logistic map” which applies to things like population size. Could you comment on these other instances?

The idea of of fractal dimension in the contect of self-similarities is easier to understand or present in simple terms.

Fractal dimensions are not a unique perogative of self similar objects or objects which share some properties of self-similarities.

They are approximately self-similar or they can be generated in the context of studying self-similarities of similar objects, but they are not self-similar per se.

This is an example of what you can get by studying a logistic map which means, in other words, when you have a logistic process which is driven by a certain parameter and you change the values of the parameter then at some point you reach a threshold value for which you observe physically that the behaviour modelled by your logistic model is different.

So you are undergoing a transition. The place where transitions occur is called the parameter space.

If you have obtained the transitions then at some point you start recovering fractal objects. So, you need fractal geometry to study these kinds of things and this is very important in terms of applications. This makes up a very important chapter of a branch of mathematics called dynamical systems. This is one of the things Dr Sommacal studies as a researcher, as well as chaos theory.

Fractal geometry is one of the tools that is needed to investigate chaos theory and dynamical systems as very often you get fractal objects emerging from the kind of models that are studied in this discipline.

The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May, in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst,_Baron_May_of_Oxford (below left)


Robert McCredie May, Baron May of Oxford, OM, AC, FRS, FAA, FTSE, FRSN, HonFAIB (8 January 1936 – 28 April 2020) was an Australian scientist who was Chief Scientific Adviser to the UK Government, President of the Royal Society, and a professor at the University of Sydney and Princeton University. He held joint professorships at the University of Oxford and Imperial College London. (above right)

Pierre François Verhulst (28 October 1804, Brussels – 15 February 1849, Brussels) was a Belgian mathematician and a doctor in number theory from the University of Ghent in 1825. He is best known for the logistic growth model.

The parameter space is the space of possible parameter values that define a particular mathematical model, often a subset of finite-dimensional Euclidean space. Often the parameters are inputs of a function, in which case the technical term for the parameter space is domain of a function. The ranges of values of the parameters may form the axes of a plot, and particular outcomes of the model may be plotted against these axes to illustrate how different regions of the parameter space produce different types of behaviour in the model.

In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in a geometrical space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each springtime in a lake.

Chaos theory is a branch of mathematics focusing on the study of chaos — dynamical systems whose apparently random states of disorder and irregularities are actually governed by underlying patterns and deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behaviour is that a butterfly flapping its wings in Texas can cause a hurricane in China.

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