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/