Re(Calibrator) Phase2 part3

There are two things I am doing simultaneously :
1. Research into multi-dimesional space theory and potential applications
2. Experimentations on different ways of represneting perspective drawings
The idea of multi-dimensional space, and four-dimensional in particular have been haunting mathematicians since 1910s first edition of Theodor Kaluza's paper on "hyperspace" (page 6, "The Great Beyond" By Paul Halpern). Though Einstein defined time to be the forth dimensions, these mathematicians and physicist mention 4th Spatial dimension, that is however hidden from our 3 dimensional view. It is possible that we are right in the middle of the multidimensional space, but can not perceive it, due to our sensory organ limitations. This controversial view is a constant inspiration for me, a soul food, while I am doing my research.
In his "The ontology of physical objects: four-dimensional hunks of matter" , professor of philosophy Mark Heller says "I propose that a physical object is not an enduring spatial hunk of matter, but is, rather, a spatiotemporal hunk of matter. Instead of thinking of matter as filling up regions of space, we should think of matter as filling up regions of spacetime. A physical object is the material content of a region of spacetime." (Chapter 3 "four-dimensional objects", pages 4-5). Heller suggests "If there are any non-conventional objects, then they must not be vague objects. They must have precise boundaries along all dimensions, including the temporal dimension. Moreover, these boundaries must not be a function of our special interests or our arbitrary choice. I propose that the objects that best meet these conditions are four-dimensional hunks of matter" (chapter 10 "nonconventional objects", page 51). More information on Professor Heller can be found here http://thecollege.syr.edu/profiles/pages/heller-mark.html
In short, what Professor Heller is saying is that:

is the four dimensional matter.

The classical spacetime interpretation stands as "space being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions." http://www.ws5.com/spacetime/

It is therefore, safe to say, that most of us already think of a time as the fourth dimension, although Heller seems to differentiate 3D and 4D objects.

Moving forward, this need not to be confused with the idea of the existence of extra dimensions, which string theory is suggesting.

To make contact between string theory and the four-dimensional world of everyday experience, the most straightforward possibility is that six or seven of the dimensions are compactified on an internal manifold, whos size is sufficiently small to have escaped detection. The idea of an extra compact dimension was first discussed by Kaluza and Klein in the 1920. Their goal was to construct a unified description of electromagnetism and gravity in four dimensions by compactifying five dimensional general relativity on a circle. This idea nowadays refers to as "compactification", can be illustrated in terms of the two cylinders below.

A cylinder appearing 2D from distance

The surface of the first cylinder is two dimensional. However, if the radius of the circle becomes extremely small, or equivalently if the cylinder is viewed from a large distance, the cylinder looks effectively one-dimensional. One now imagines that the long dimension of the cylinder is replaced by our four-dimensional space-time and the short dimension by an appropriate six or seven-dimensional compact manifold. At large distances or low energies the compact internal space cannot be seen and the world looks effectively four-dimensional. (page5, chapter 1.2, "String theory and M-theory. A modern introduction." by Katrin Becker, Melanie Becker and John H. Schwarz).
Needless to say that, taking into account our current physiological limitations, most of us finds it difficult to imagine a 4th spatial dimension, let alone the 5th, 6th and more.

Having said that, there are some architects who draw massive inspiration of the idea of hypercube and dare to  portray a geometry resembling to that of a 4 dimensional cube. A Portuguese Architects OODA has come up with a massive "Twisted Hypercubes" concept for the New Taipei Art Museum. (http://www.ooda.eu/)

Twister Hypercubes by OODA

These are two hypercubes,one inside the other. The external cube, with its impressive volumetric shape, holds within another cube at right angle and the reason for this is that the internal slanted steel girders are running from one side of a space to the other and the massive skin is wrapped around them. On the outside these create facilities for water collection, installing solar panels and operable windows for ventilation. So OODA clearly had a great concern for the environment in this design.

The interior of the Twisted Hypercubes

This I find to be such motivating example to pursue the untested ideas and walk through the unknown land in search of new forms. Even though the OODA did not design a 4 dimensional geometry, but they skilfully harnessed the complex geometrical elements of the hypercube in this proposal, incorporating the little known geometry with the existing ideas of sustainability.
Now I will be going a step back to look into curious geometries of slightly different calibre. This geometries I call the "in-between" dimensions, like 2 and a half and 3 and a half dimensional objects.

Mathematical illustration of a Mobius  strip

The Möbius strip, also called the twisted cylinder (page 110, Henle 1994), is a one-sided nonorientable surface obtained by cutting a closed band into a single strip, giving one of the two ends thus produced a half twist, and then reattaching the two ends (pages 322-323 right figure; Gray 1997). Despite the fact that the general opinion is that Möbius invented this in 1858, other sources claim that really Johann Listing was the first to come up with the idea (page 23, The Great Beyond. Paul Halpern). Like the cylinder, it is not a true surface, but rather a surface with boundary. http://mathworld.wolfram.com/MoebiusStrip.html

According to Madachy, the B. F. Goodrich Company patented a conveyor belt in the form of a Möbius strip which lasts twice as long as conventional belts. Because it is one sided, a conveyor belt which is given half a twist, will wear evenly on both sides. (page 152, "The Penguin dictionary of curious and interesting geometry" David Wells)

M. C. Escher Möbius Strip II (Red Ants)

Now, the interesting thing about the Möbius Strip is the paradox of it being a two-dimensional object, while existing in a three-dimensional space only. It is said that "If an ant were to crawl along the length of this strip, it would return to its starting point having traversed every part of the strip (on both sides of the original paper) without ever crossing an edge." http://en.wikipedia.org/wiki/M%C3%B6bius_strip

In this sense, the Möbius Strip is an object that is almost in between dimensions 2 and 3.

Many architects found inspiration in the Möbius Strip's curves and twists.

In 1993, Dutch architect Ben van Berkel, started working on his "Mobius House", inspired by the modern German mathematics. 

Mobius strip study by ven Berkel

It took him 6 years to complete the house.

Mobius house plan by van Berkel

The perception of movement is reinforced by the changing positions of the two main materials used for the house, glass and concrete, which overlap each other and switch places. As the loop turns inside out, the exterior concrete shell becomes interior furniture - such as tables and stairs - and the glass facades turn into inside partition walls.(http://storiesofhouses.blogspot.com/2006/09/mbius-house-in-amsterdam-by-ben-van.html)

van Berkel's Mobius House in Amsterdam

There are other examples, like the National Library in Kazakhstan, designed by BIG Architects. http://www.archdaily.com/33238/national-library-in-astana-kazakhstan-big/

Astana National Library by BIG

The building comprises of four different geometries- the circle, the rotunda, the arch and the yurt, forming a spiral circle around the vertical core. As the museum curves form a möbius strip, the interior becomes the exterior and back again; likewise the walls become the roof and the roof transforms back into the walls. http://inhabitat.com/big-unveil-massive-mobius-strip-library-for-kazakhstan/

Exterior and Interior views of the Library

Through geometric openings in the exterior shell, the natural daylight accesses freely, creating beautifully lit spaces for reading. 

Below is another example of the smart use of mathematical concepts.

Particle accelerator in Sweden designed by Snohetta

This is the work of the Norwegian architectural firm, Snohetta, for a new cyclic particle accelerator, called the “Max IV”, at the Max-Lab to be built in Lund, Sweden. It is a cylindrical, Möbius inspired national laboratory. (http://inhabitat.com/sn%C3%B8hetta-unveils-mobius-strip-inspired-particle-accelerator/)

The  Mobius strip is used for flipping a 2 dimensional objects in 3 dimensional space. Now there is analogue of this, called the Klein Bottle, which is a 3 dimensional object that has been flipped in 4 dimensional space.

Klein Bottle wireframe

A Klein bottle can be produced by gluing two Möbius strips together along their edges; this cannot be done in ordinary three-dimensional Euclidean space without creating self-intersections. The result is not a true picture of the Klein bottle, since it depicts a self-intersection which isn't really there. The Klein bottle can be realized in 4-dimensional space: one lifts up the narrow part of the tube in the direction of the 4-th coordinate axis just as it is about to pass through the thick part of the tube, then drops it back down into 3-dimensional space inside the thick part of the tube. (http://www.math.osu.edu/~fiedorow/math655/Klein2.html)

Such topologies hold intriguing possibilities for architects in search for new spatial configurations.
The Klein Bottle House designed by McBride Charles Ryan, is the "origami" version of the Klein Bottle.

Klein Bottle House in Australia by MCR

The house revolves around a central courtyard, a grand regal stair connecting all the levels. There is a sense of both being near and far to all occupants. http://www.archdaily.com/7952/klein-bottle-house-mcbride-charles-ryan/

Re(Calibrator) Phase2 part2

The paper sketch model was taken to a 3D modelling software called 3Ds Max, for computation and further development of this structure. Here I have built the geometry with slight alteration of the Z axis, which resulted in a slightly different construction.

Constructing the paper model into 3Ds Max

When completed, the structure looks very angular, with harsh corners and a certain rigidity.

A view from the left 

A view from the front

The left and front views reveal a "basement" compartments, which have not been indicated on the paper model. This "ground floor" sprung out during the paper to 3D Max transformation, as a geometric side effect.                                                                                                                                                  

Perspective view with a translucent glass material 

At this point I started comparing the analogue model with this digital one and while the paper model is very much physically present and tactile, yet it is very set in what it is and less allowing for further alterations, whereas the digital equivalent gives an enormous freedom of endless modifications of any of its components, but most importantly, it enables me to have an instant rough representation of different models, like metals, wood, glass etc.

Re(Calibrator) Phase2

The intuitive sketches I have created at this stage, will be helping me to develop a framework for my research and to test some of the techniques. Below are 3D paper models, which I have called variations on 1 point and 2 point perspectives (the division is symbolic at this point). A sketch was drawn on a paper, with no particular structure such as a building, bridge etc in mind , and later I have selected parts of the sketch and assigned them Z value or depth.

"Paper, nail and string" exploratory sketch

I was curious to generate something with relatively complex geometry and to work with intersecting surfaces. Some of the lines have been left 2D on the sketch, while others have been given different heights, thus creating a mixture of "line on the surface" and "line in the space" scenarios.

"Paper and nail" sketch showing the 2D lines

Diagonal view of the paper model    

The "upright stairs" on the abstract geometry

After creating these models I questioned again what is a 2D space? Can we really understand it? Can we describe a space that has no depth/ height? What strikes me, is the fact that anything, which is physically existent in our three-dimensional world, has three dimensional qualities and properties, even an endlessly thin paper has some kind of thickness.

Similarly, anything that is generated with a help of a computer software and appears in front of us on a computer screen, is ultimately two dimensional. (for the exception of the 3D images/videos viewable with 3d goggles i guess).

After studying these abstract geometries, I decided to take the second with more complex structure into a stage further, to develop and manipulate it with various analogue and digital tools and techniques. 

Re(Calibrator) Phase1 part2

In 1783, Kant wrote: "That everywhere space (which is not itself the boundary of another space) has three dimensions and that space in general cannot have more dimensions is based on the proposition that not more than three lines can intersect at right angles in one point. This proposition cannot at all be shown from concepts, but rests immediately on intuition and indeed on pure intuition a priori because it is apodictically (demonstrably) certain." (page 38. Prolegomena to any future metaphysics).
For centuries, many have been contemplating about the other spatial dimension and in recent decades the contemporary science and physicists are developing theories that will explain all the existing forces and the mystery of matter.
In forth, extra spatial dimension, the simplest geometric object is assumed to be the tesseract. In geometry, the tesseract, also called an 8-cell or regular octachoron or cubic prism, is the four-dimensional analogue of the cube.

A square, a cube, a hypercube

However, it is not as easy to visualise a tesseract, or even more complicated multi-dimensional objects. In the past, many scientists tried to picture this with a pen and paper, but now with the emergence of the new technologies, we can see these multi-dimensional forms in motion as animated sequences.

  An animation showing multi-dimensions

And yet, even with the animated view, it is hard to imagine the forth spatial dimension. We are in three dimensional equivalent of the fictional world called "Flatland: A Romance of Many Dimensions" by English schoolmaster Edwin Abbott Abbott.

A house in a Flatland

Here the author is describing "a two-dimensional world referred to as Flatland which is occupied by geometric figures. Women are simple line-segments, while men are regular polygons with various numbers of sides. The narrator is a humble square, a member of the social caste of gentlemen and professionals in a society of geometric figures, who guides us through some of the implications of life in two dimensions. The square has a dream about a visit to a one-dimensional world (Lineland) which is inhabited by "lustrous points."

The Lineland

He attempts to convince the realm's ignorant monarch of a second dimension but finds that it is essentially impossible to make him see outside of his eternally straight line. He is then visited by a three-dimensional sphere, which he cannot comprehend until he sees Spaceland for himself."

(http://www.geom.uiuc.edu/~banchoff/Flatland/).

Similarly, we find it difficult to comprehend anything that is simultaneously perpendicular to all our known x, y, z axis. Having said that, I find myself equally incapable of portraying the Lineland or the Flatland as well.

Did anyone ever try to build a tesseract house? There is a science fiction story "And he built a crooked house" by Robert Heinlein, in which an ambitious architect Quintus Teal, attempts to build a four dimensional house, and once it is built, very strange things start happening to the occupants, as the house does what seems to be a teleportation from one point of a universe to another, or a movement in four dimensional space. Perhaps this story inspired another artist and architect Paul Laffoley, to put together detailed plans of a tesseract house. (http://paullaffoley.net/paullaffoley/)

Tesseract house plans by P. Laffoley

This house has actually been built by Seifert Surface, but only in a virtual realm of the "Second life".

Heinlein inspired tesseract house in Second Life

Leaving the room we are in, walking straight through four doors and ending up back where he began. And so a graduate student in Stanford’s math department has managed to create, more than 60 years after Heinlein first conceived it, a home that seems to exist in more than three dimensions.  http://nwn.blogs.com/nwn/2011/06/tesseract-crooked-house-in-second-life.htmlThere are however other more complex multidimensional object.

Multidimensional objects

Although mathematicians can work with the fourth dimension abstractly by adding a fourth coordinate to the three that we use to describe a point in space, a fourth spatial dimension remains difficult to visualise.

Sculpture of a shadow of a four dimensional object

Above is a sculpture designed by Penn State professor of mathematics Adrian Ocneanu (http://science.psu.edu/news-and-events/2005-news/math10-2005.htm/), which measures about six feet in every direction, presents the three-dimensional "shadow" of a four-dimensional solid object. Ocneanu's sculpture similarly maps four-dimensional solid into a space perceptible to the human observer. His process, radial stereography, presents a new way of making this projection. He explained the process by analogy to mapping a globe of the Earth onto a flat surface.
"Four-dimensional models are useful for thinking about and finding new relationships and phenomena," said Ocneanu. "The process is actually quite simple- think in one dimension less." To explain this concept, he points to the two-dimensional map of the three-dimensional world. The interview with the professor and the overview of the sculpture can be found here (http://www.youtube.com/watch?v=viKTj78ge-0)

Re(Calibrator) Phase1

This project is a venture into translation between the gesture-software-hardware, into the realm of the possibilities of hybrids, the collision of physical and digital tools and techniques.
It is also a quest of a unique framework, which will accommodate both digital creativity and traditional craftsmanship.
At the early stages of my research I started looking into art inspired by geometry and mysticism, and geometry inspired by imagination and the unknown. My fascination was about the dimensionality of our world and the efforts of scientists to discover other dimensions, a multi-dimensional unified theory of everything that is beyond the perception of our three dimensional world.

Some artists, portrayed reality in surprising and surreal ways, and perhaps one of the most recognisable person of these genre is M. C. Escher, who was not only a skillful master of presenting surreal worlds with much conviction, but also a proficient researcher of mathematics, geometry, tessellation and infinity.

Reptiles by Escher

What interests me the most in Escher's work, is this particular work on 2D reptiles, that step out of their flat world, gaining the Z depth and walking in a circle, marching through the three dimensional world, reaching the top and looking down into their flat existence, only to then return back to their flat homeland again. This translation between 2D and 3D really intrigues me and what happens in between, the transitional two reptiles are half 2D and half 3D, stuck in between two dimensions.
This is a world, an imaginary scenario, which arguably does not exist, this is an example of an illusion, a mystery, a parallel world. And why do we seek to be mystified? Albert Einstein thought that "The most beautiful thing we can experience is the mysterious. It is the source of all true art and science" (preface, Mazes for the Mind) I tend to agree with this, the mysterious seems to draw us in a tempting, inspiring and sometimes overwhelming ways.
The source of the mystery varies.
Gaudi often found inspiration for his mysterious and bizarre architecture, in nature and animal world.

Casa Batllo by Gaudi

The balconies of Casa Batllo are derived from fish skull structure (page 56, Twenty buildings every architect should understand). The scale and the placing of this idea makes the building appear mysterious, as if we have stepped into some kind of a dream world of exaggeration and distortion, or an entirely different world.
In his "Diary of a Genius" (http://www.picassomio.com/diary-genius-book-0971457832.html) Salvador Dali mentions, that many of his painting are based on his dreams.

Corpus Hypercubus by Dali 

In Corpus Hypercubus (Crucifix) Dali depicts Jesus crucified on a four dimensional cube (tesseract). The hypercube, in this painting, is unfolded, in such a way, that we can see all 8 cubes simultaneously.

Hypercube unfolded

Did Dali dream of the extra dimension? His critics argue that his use of this mathematical symbol is a "visual opportunism" and that the artist knew nothing of the meanings and mathematical principles behind them (http://www.theartistsalvadordali.com/salvador-dali-painting-poster-print-corpus-hypercubus.htm). There is no evidence to support or disprove this, but one things is certain, Dali was mystified by the unfamiliar, odd geometry.
Mysterious geometry is an elegant representation of an impossible structure. 

The impossible cube structure

These objects are also called optical illusions and Escher used such structures in most of his graphic works.
Can there be extra dimensions? And if so, how does this relate to my search for the manual-digital hybrid? I would like to think that there is a way to practically implement the notions of mysterious extra dimensions and the idea of in between or "trapped" dimensions in this voyage.