Fourth Year Students


In this post, I’m going to try my best to explain the first, second, and third dimensions, and why fractals don’t belong to any of them, as well as what happens when you get into higher dimensions. But before getting into the nitty-gritty of the subject, I think it’s worth prefacing this post with a short note on the nature of mathematics itself:

Alain Badiou said that mathematics is a rigorous aesthetic; it tells us nothing of real being, but forges a fiction of intelligible consistency. That being said, I think it’s interesting to think about whether or not mathematics were invented or discovered – whether or not numbers exist outside of the human mind.

While I don’t have an answer to this question (and there are at least three different schools of thought on the subject), I do think it’s important to keep in mind that we only use math as a tool to measure and represent ‘real world’ things. In other words, our knowledge of mathematics has its limitations as far as understanding the space-time continuum goes.


1 Traditional Dimensions

In physics and mathematics, dimensions are used to define the Cartesian plains. The measure of a mathematical space is based on the number of variables require to define it. The dimension of an object is defined by how many coordinates are required to specify a point on it.

It’s important to note that there is no ‘first’ or ‘second’ dimension. It’s a bit like pouring three cups of water into a vase and asking someone which cup is the first one. The question doesn’t even make sense.


Except from ‘minutephysics’

We usually arbitrarily pick a dimension and calling it the ‘first’ one.


1.1 – Zero Dimensions


Something of zero dimensions give us a point. While a point can inhabit (and be defined in) higher dimensions, the point itself has a dimension of zero; you cannot move anywhere on a point.

1.2 – One Dimension


A line or a curve gives us a one-dimensional object, and is typically bound by two zero-dimensional things.
Only one coordinate is required to define a point on the curve.
Similarly to the point, a curve can inhabit higher dimension (i.e. you can plot a curve in three dimensions), but as an object, it only possesses one dimension.
Another way to think about it is: if you were to walk along this curve, you could only go forwards or backward – you’d only have access to one dimension, even though you’d be technically moving through three dimensions.


1.3 – Two Dimensions


Surfaces or plains gives us two-dimensional shapes, and are typically bound by one-dimensional shapes (lines/curves).

A plain can be defined by x&yy&z or x&z; more complex surfaces are commonly defined by u&v values. These variable are arbitrary, what is important is that there are two of them.

A surface can live in three+ dimensions, but still only possesses two dimension. Two coordinate are required to define a point on a surface. For example a sphere is a three-dimensional object, but the surface of a sphere is two-dimensional – a point can be define on the surface of a sphere with latitude and longitude.


1.4 – Three Dimensions


A volume gives us a three-dimensional shape, and can be bound by two-dimensional shapes (surfaces).

Shapes in three dimensions are most commonly represented in relation to an x, y and z axis. If a person were to swim in a body of water, their position could be defined by no less than three coordinates – their latitude, longitude and depth. Traveling through this body of water grants access to three dimensions.


2 Fractal Dimensions

Fractals can be generally classified as shapes with a non-integer dimension (a dimension that is not a whole number). They may or may not be self-similar, but are typically measured by their properties at different scales.

Felix Hausdorff and Abram Besicovitch demonstrated that, though a line has a dimension of one and a square a dimension of two, many curves fit in-between dimensions due to the varying amounts of information they contain. These dimensions between whole numbers are known as Hausdorff-Besicovitch dimensions.


2.1 – Between the First & Second Dimensions

A line or a curve gives us a one-dimensional object that allows us to move forwards and backwards, where only one coordinate is required to define a point on them.

Surfaces give us two-dimensional shapes, where two coordinate are required to define a point on them.

Here is a shape that cannot be classified as a one-dimensional shape, or a two-dimensional shape. It can be plotted in two dimensions, or even three dimensions, but the object itself does not have access the two whole dimensions.

If you were to walk along the shape starting from the base, you could go forwards and backwards, but suddenly you have an option that’s more than forwards and backwards, but less than left and right.

You cannot define a point on this shape with a single coordinate, and a two coordinate system would define a point off of the shape more often than not.

Each fractal has a unique dimensional measure based on how much space they fill.


2.2 – Between the Second & Third Dimensions


Developing Koch Snowflake

The same logic applies when exploring fractals plotted in three dimensions:

Surfaces give us two-dimensional shapes, where two coordinate are required to define a point on them.

A volume gives us a three-dimensional shape where a point could be defined by no less than three coordinates.

While these models live in three dimensions, they do not quite have access to all of them. You cannot define a point on them with two coordinates: they are more than a surface and less than a volume.


Fourth Iteration Menger Sponge

The Menger Sponge for example has (mathematically) a volume of zero, but an infinite surface area.


2.3 – Calculating Fractal Dimensions

The following are three methods of calculating Hausdorff-Besicovitch dimension:
• The exactly self-similar method for calculating dimensions of mathematically generated repeating patterns.
• The Richardson method for calculating a dimensional slope.
• The box-counting method for determining the ratios of a fractal’s area or volume.


On another note:

In theory, higher (non-integer) dimensional fractals are possible.
As far as I’m concerned however, they’re not particularly good for anything in a three-dimensional world. You are more than welcome to prove me wrong though.


3 Higher Dimensions

Sadly, living in a three-dimensional world makes it especially difficult to think about, and nearly impossible to visualise, higher dimensions. This is in the same way that a two-dimensional being would find it impossibly hard to think about our three-dimensional world, a subject explored in the novel ‘Flatland’ by Edwin A. Abbott.

That being said, it’s plausible that we experience much higher dimensions that are just too hard to perceive. For example, an ant walking along the surface of a sphere will only ever perceive two dimensions, but is moving through three dimensions, and is subject to the fourth (temporal) dimension.


3.1 – The Fourth Dimension (Temporal)

If we consider time an additional variable, then despite the fact that we live in a three dimensional world, we are always subject to (even if we cannot visualize) a fourth dimension.

Neil deGrasse Tyson puts it quite plainly by saying:
“[…] you have never met someone at a place, unless it was at a time; you have never met someone at a time, unless it was at a place […]”

Suppose we call our first three dimensions x, y & z, and our fourth t: latitude, longitude, altitude and time, respectively. In this instance, time is linear, and time & space are one. As if the universe is a kind of film, where going forwards and backwards in time will always yield the exact same outcome; no matter how many times you return to a point in point time, you will always find yourself (and everything else) in the exact same place.
However time is only linear for us as three-dimensional beings. For a four-dimensional being, time is something that can be moved through as freely as swimming or walking.

Except from ‘Seeker’


3.2 – The Fourth Dimension (Spacial)

If we explore spacial dimensions, a four-dimensional object may be achieved by ‘folding’ three-dimensional objects together. They cannot exist in our three-dimensional world, but there are tricks to visualise them.


We know that we can construct a cube by folding a series of two-dimensional surfaces together, but this is only possible with the third dimension, which we have access to.



If we visualise, in two dimensions, a cube rotating (as seen above), it looks like each surface is distorting, growing and shrinking, and is passing through the other. However we are familiar enough with the cube as a shape to know that this is simply a trick of perspective – that objects only look smaller when they are farther away.

In the same way that a cube is made of six squares, a four-dimensional cube (hypercube or tesseract), is made of eight cubes.

  • A line is bound by two zero-dimensional things
  • A square is bound by four one-dimensional shapes
  • A cube is bound by six two-dimensional surfaces
  • A hypercube, bound by eight three-dimensional volumes

It looks like each cube is distorting, growing and shrinking, and passing through the other. This is because we can only represent eight cubes folding together in the fourth dimension with three-dimensional perspective animation.


3D representation of eight cubes folding in 4D space to form a Hypercube

Perspective makes it look like the cubes are growing and shrinking, when they are simply getting closer and further in four-dimensional space. If somehow we could access this higher dimension, we would see these cubes fold together unharmed the same way forming a cube leaves each square unharmed.

Below is a three-dimensional perspective view of hypercube rotating in four dimensions, where (in four-dimensional space) all eight cubes are always the same, but are being subjected to perspective.




3.3 – The Fifth and Sixth Dimensions

On the temporal side of things, adding the ability to move ‘left & right’ and ‘up & down’ in time gives us the fifth and sixth dimensions.

(For example: x, y, z, t1, t2, t3)

This is a space where one can move through time based on probability and permutations of what could have been, is, was, or will be on alternate timelines. For any one point in this space, there are six coordinates that describe its position.

In spacial dimensions, it is theoretically possible to fold four-dimensional objects with a fifth dimension. However, it becomes increasingly difficult for us to visualise what is happening to the shapes that we’re folding.
In theory, objects can keep being folded together into higher and higher spacial dimensions indefinitely. (R1, R2, R3,R4,R5, R6, Retc.)

There’s a terrific explanation of what happens to platonic solids and regular polytopes in higher dimensions on Numberphile:


3.4 – Even Higher Dimensions

If we can take a point and move it through space and time, including all the futures and pasts possible, for that point, we can then move along a number line where the laws of gravity are different, the speed of light has changed.

Dimensions seven though ten are different universes with different possibilities, and impossibilities, and even different laws of physics. These grasp all the possibilities and permutations of how each universe operates, and the whole of reality with all the permutations they’re in, throughout all of time and space. The highest dimension is the encompassment of all of those universes, possibilities, choices, times, places all into a single ‘thing.’

These ten time-space dimensions belong to something called Super-string Theory, which is what physicist are using to help us understand the universe work.


Cross section of the quintic Calabi–Yau manifold

There may very well be a link between temporal dimensions and spacial dimensions. For all I know, they are actually the same thing, but thinking about it for too long makes my head hurt. If the topic interests you, there is a philosophical approach to the nature of time called eternalism, where one may find answers to these questions. Other dimensional models include M-Theory, which suggests there are eleven dimensions.

While we don’t have experimental or observational evidence to confirm whether or not any of these additional dimensions really exist, theoretical physicists continue to use these studies to help us learn more about how the universe works. Like how gravity affects time, or the higher dimensions affect quantum theory.



Dream Catchers by Nick Huard

Legend of the Dream Catcher

‘The legend of a dream catcher began long time ago, when the child of a Woodland chief fell ill. Unsettled by fever, the child was plagued with bad dreams and unable to sleep. In an attempt to heal him, the tribe’s Medicine Woman created a device that would ‘catch’ these bad dreams. Forming a circle with a slender willow branch, she filled the centre with sinew, using a pattern borrowed from our brother the Spider, who weaves a web. This dream catcher was then hung over the bed of the child. Soon the fever broke, and the child slept peacefully.

It is said that at night, when dreams visit, they are caught in the dream catcher’s web, and only the good dreams are able to find their way to the dreamer, filtering down through the feather. When the warmth of the morning sun arrives, it burns away the bad dreams that have been caught. The good dreams, now knowing the path,visit again on other nights.’ (Oberholtzer, 2012, p9).



Dreamcatchers originated with the Ojibwe, a tribe of Native Americans scattered throughout the areas of the lake country in northern Michigan, Wisconsin, and Minnesota, and along the southern border of Canada, along the shores of Lakes Huron, Superior and Michigan, whose survival relied on fishing, hunting and trapping.  

Traditionally, the dream catchers were made by tying sinew strands onto a few inches in diameter round or tear-shaped frames of willow and were often wrapped in leather.

The spiritual life of the Ojibwa centred around the Midewiwin, the Grand Medicine Society and focused on the individual spiritual growth, gaining the insight through their dreams or visions.



Grey Owl repairing an Ojibwa-style shoe

Mystical Experience

My project is a re-interpretation of the beliefs that dreams have magical qualities with the ability to change or direct one’s path in life. The bawaajige nagwaagan intends to create a mystical experience, where people are caught inside, similar to the way that bad dreams are caught in the dreamcatcher’s web, and good dreams escape through the centre. The participants are encouraged to climb through the centre and escape their bad dreams and feelings, releasing their spirit through the enclosure. Now they can sleep in the peaceful environment, stimulated by the fantasy of glowing feathers and luminescent rope structures. The pavilion aims for people to sleep, relax and free themselves from stress while being protected by the magical webs of the dream catcher.



The Bawaajige Nagwaagan at night


Close-up render at night

Romantic essence of the Native American Culture

The proposal is a celebration of the romantic essence of the Native American Culture. The large scale, three dimensional net is inspired by the native methods and techniques of making dream catchers. It is a manifestation of the traditions and significance of the Native Americans, paying respect and pledging support to the indigenous people of America.

The structure situated in the Burning Man festival commemorates the ceremonies of Native Americans, dedicated to acquiring an insight through dreams and visions. Fasting, or giving up of certain necessities for a certain length of time was a common practice used to enhance one’s ability to access different dreams or visions. Another method was to pour water over hot rocks to produce steam, which enhanced the occurrence of dreams, used as source of introspection. These rituals relate to the festival’s assertion of disconnecting from the necessities of our contemporary world, supplemented by the extreme weather conditions, which are hoped to encourage reflection.

The pavilion responds and works together with the Black Rock desert’s environment, and adds to the wider cultural context of leaving behind the essentials and expectations of the contemporary world while creating a moment for contemplation and tranquility in the magical weaves of the dream catcher.



The Bawaajige Nagwaagan during the day

Proposal Development_System





1.Aleksandra Wojciak_90gsm_A1_Merylbone_west

Form Experimentation_Platonic Forms



Development Model



Diagrams explaining model assembly




1:10 Model



Diagrams explaining model assembly


Physical Description

The structure will be composed out of three, seven meters in diameter, dream catchers, tilted to form a tetrahedron. Each dream catcher’s net will be made out of 275 meters, 18mm, synthetic hemp rope which will be entwined in 1320 meters of 3mm fluorescent cord. Attached to the frame uv lights will make the fluorescent rope glow at night. Three rings hold the net structure together, with the bottom ring anchored to the ground, made out of T-shape plywood frames. The web of the frame will be 4 layers of 15mm ‘banana’ shaped pieces which will create a circle, together with 4 layers of 230mm x 2400mm x 9mm flange pieces bent in shape of the banana edge. Smaller rings, supporting the centre of the dreamcatcher net will be of similar structure, with 2 layers of banana pieces and 2 layers of 150mm x 2400m x 9mm flange pieces, bent in shape. The frame will be wrapped in 13500 meters of 8mm synthetic hemp with attached fluorescent fabric feathers.



Axonometric View-Construction Development


assembly 1

Frame’s web assembly

assmbly 2

Frame’s flange assembly


assembly 3

Initial assembly diagrams


UV lighting-Construction Development

Testing Ideas in 1:1 Scale



1:1 Scale Test Model exploring the possibilities of glowing net structure and its connections.

Assembly of the net is inspired by a macrame knotting technique rather than weaving which means that the net could be made out of smaller 15 meters long pieces, rather than one 275 m coil of rope, making it easier to assemble and repair. Rope is anchored to the frame with thimbles and shackles, attached to the bolted staple on the plate. The rope is connected with simple S-shape stainless steel hooks. After testing the net I found that although these are easy to assemble, they can create some movement in a connection, therefore I am planning on exploring the idea of ferrules, which could be crimped in place.



Photographs of 1:1 Scale Model




This project is a physical exploration of anamorphosis in three dimensions centred around the theme of duality. It aims to combine two widely recognisable figures into a pavilion that will attract burners, provoke debate, and catalyse interaction.

Duality DayThe theme of this project arose from the realisation that even the most widely recognisable symbols contain multiple layers of meaning and mystery.  Social, historical and sometimes even spiritual contexts give a symbol its perceived meaning. For example, while the Christian cross is a symbol of hope it is literally a scaled representation of an ancient torture device – an icon synonymous with good carries with it a darker elucidation. This interpretation led to the emergence of duality as a topic and a title. 
There are many symbols which have multiple meanings and nuances to those who interpret them.

pages-for-blog-re-systemI began by looking at the Ankh, the Egyptian symbol for life/fertility. The Loop of the Ankh represents the feminine discipline or the womb, while the elongated section represent the masculine discipline or the penis. These two sacred units then come together and form life. This is a perfect representation of man and woman in perfect union. I then was led to study the symbol for mercury, which is used in botany to indicate a flower with both male and female reproductive organs.

This duality of meaning in symbols led me to the desire to study how I could physically combine other symbols and forms to create one form. Anamorphosis, from the Greek anamorphōsis meaning ‘transformation,’ from ana- ‘back, again’ + morphosis ‘a shaping’, became an interesting opportunity to do just this.




I want to explore this theme using the iconic faces of Donald Trump and Kim Kardashian as instigators. From a random vantage point or even from up close, the subject matter of the piece is evidently unclear, the image changes until the viewer arrives at a specific pre-set location, only then does the likeness reveal itself. This echoes our warped perception of figures in limelight; anything the media choose to present to the world is an engineered production and if taken out of its context it becomes incomprehensible. My aim is to stir ambivalence among the burners, for them to engage in discussion with one another about these two incredibly famous personalities and what they seemingly represent.

As a physical entity, the sculpture is purposefully made durable enough to be able to endure the brunt of any elicited reactions. Its exposed surfaces are smooth, an open invitation to graffiti, carve or deface in any manner possible. It is large enough to climb and to gather within as a group – it only takes a spontaneous suggestion from a creative festival goer to give the sculpture another unforeseen use.

pages-for-blog-re-system-new pages-for-blog-re-system-new2

The aim of my proposed sculpture is to provoke an exchange of opinions and interactions between burners. It depicts two iconic and highly controversial public figures who personify two tremendously important issues that we as a society face today; political and social change.

As festival goers approach the installation, and the two widely recognisable faces reveal themselves, comments about the likenesses will spiral inevitably highlighting or at least touching upon the shift that these two personalities represent.

The sculpture’s physical form comprises of several spatial elements that lend themselves to fostering the kind of debates that I wished to promote. The hollow centre creates an enclosure, to enable hosting or housing for a meeting, it gives its participants a sense of protection; this is an open forum, please take part. The raised base on the peripheries can act as stages or podia. The expansive smooth external surfaces can act as billboards or banners, the skin of the sculpture will bear the physical outcome of the issues discussed here.

Whether people get photographed with it, or whether they deface, damage or even burn it to the ground, I will have succeeded if among any of the interactions the agenda was heard and a heartfelt reaction was made.


The sculpture will be made of 8mm CNC routed plywood sheets fixed to a heavy plywood formwork. Standing at 6m tall, one side will represent a 25:1 scale stencilled portrait of president-elect Donald Trump, the other side; the likeness of reality television personality and socialite Kim Kardashian. Much like the oblique anamorphosis incorporated in Holbien’s The Ambassadors, the sculpture’s subject matters will reveal themselves only from some 60m away, but from close up, the installation will seem like a mass of abstract wooden extrusions, something suggestive of an adult-sized climbing frame. Fluorescent LEDs recessed into junctions of the outer plywood skin layer will illuminate the piece at night.


The pavilion achieves the incredible feat of allowing the viewer to have a personal and intimate connection with it whilst also allowing for reflection. The two images are intended to bring moments of delight to viewers to allow for interaction even from a distance.

Combined with its symbolic and evocative power, it should indeed conjure a deeper sense of place and self, and bring a subtlety and complexity to what might have been just another pavilion.


heartThe Heart is an internationally known Symbol for Love.

The Love Nest is a Pavilion designed for Burning Man as a destination to express your Love to another. Wedding ceremonies will take place within the Heart structure, this is the biggest gesture of Love, joining together as one, declaring your Love for everyone to see

The wooden hearts floating toward the sky create a Tower of Love



Inside the Love Nest


1:20 Model and Heart Bending


Love Nest – Night Time


Developing the Love Nest

The origin of the Heart Symbol


Symbols as a System


At Burning Man within the Temple (also known as the Temple of Love) people leave messages to remember there loved ones that they have lost, the temple is then burnt so the messages can get to those who are being remembered. The Idea of the below design was inspired by a book, being a place of words, people add to the pages of the book of love.7

Interlocking Hearts – Heart Tower

910The Tower

Every time someone is married within the Tower a coloured heart is added to the Structure. 11

Progression of the Form

Developing the design to be more fluid and natural, as love isn’t hard and spiky.



Bending Hearts

Looking at different ways of bending wooden Hearts to be able to work with the newly designed form and being able to attach to the ‘Ribs’ of the design. I looked at two ways of achieving this:

Laser Cutting

By laser cutting a line pattern into the heart I achieved a material that bends easily

Cutting and folding

Cutting a slit from the bottom of the heart to the centre and taking the two half and bending one on top of the other

Results – The Laser Heart bent easily but did not stay in place and became more fragile whereas the folded heart keeps its shape and is a more solid form15


The Love Nest – Initial Renders


The Love Nest – Initial Model



Folding Hearts – Further Research

Looking at what effects the curvature of the heart.

Variable – Length of cut for bending

Results – The Longer the cut the smoother the curve



Final Design – Love Nest

The Final Design looks at creating a form by connecting the folding hearts, removing the structural ‘Ribs’ from the previous design and creating a system to achieve a Form


Love Nest – Final Model


Love Nest – Internal Views


Love Nest – Lighting


Love Nest – Night Time


Narrative | ‘Hayam’: a filigree temple of light and shelter, a spiritual retreat resting lightly on the Playa, a tiny tessellated palace named for love and open to the sky, a miniature caravansary to welcome the weary traveller.

The Hayam embodies the spirit of Islamic geometry: intricately interwoven patterns and repeating themes that speak of infinity. Geometry is the language of the universe; in the very small the infinite can be found.

Physical Description | Erupting flowers of perforated plywood seamlessly joined together to form a beautiful curvilinear structure. Reminiscent of muqarnas and moucharaby but stripped back to the pure essential fretwork and form, leaving behind only what is necessary. Enamels, glazes and precious metals are replaced by the gold of scattered light filtering through the delicate tracery of the screen, elevating the spirit. The treasures are not material things; they are spiritual. A place of illumination, intended for contemplation.

Emerging from a study into the geometry of Islamic art the pavilion references motifs and arabesques traditionally found in mosques and other sacred places though in itself the Hayam has ties to no religion; it transcends time and space, language and culture.

Interactivity | The structure provides a refuge from the heat of the sun and an intimate spiritual place for people to gather and rest. During the night the four pillars illuminate like a giant lantern with gas fires and the flames can be seen dancing behind the filigree patterns. The gas fires heat the area during the cold night so the space continues to function as a comfortable retreat.

More Info:

JH_Hayam Sun Temple_SECTIONS.jpg

Construction sequence and prefabrication:12 122


Small scale test model:a

Large 1:1 Scale Test Model:b

So I have been trying out a variety of software’s via my smartphone to enable the projection of architecture related 3d models onto surfaces which the user can orient and move around. The three strongest were;

AndAR   Augment   Aurasma

AndAr had the most consistant viewport, but could view only very low poly models

Augment could view more complex models, but was prone to crashing and cut parts of the model out

Aurasma I found to be the most successful. I joined as a developer, and after working through a lot of new software’s was able to create material maps, lighting and orientation (in Maya) to a level of control that the other apps do not have.

So if you want a go, download Aurasma on your smart device, search for University of Westminster’s channel, point the viewfinder at the playing card picture in the gallery and you can have a look at my model yourself!

I own the University of Westminster’s developer account it seems, so if anyone is interested in having a go then feel free to ask.