The on-line transmission from the flight2017-10-17 16:27:00
The Fifth Perspective
This name was given by a prominent Polish mathematician Karol Borsuk to the curve in a plane which reminded him of a broken bike wheel after a ride on Warsaw streets paved with cobblestones.
The curve differs from a regular circle by infinitely many osciliations with the same amplitudes which condensate at the breaking point. Topologically they are different, which means that one curve may not be continuously trasformed onto another one. However according to the theory of shape, developed by Karol Borsuk, the regular circle and Varsowian Circle have the same shape.
In particular they share the property of spliting the plane into inner and outer parts.
Karol Borsuk was a world leading mathematician working in topology. His whole life was closely connected with Warsaw. During the second world war he authored and produced toys-logical puzzles, which helped him and his family to survive the war. In recent years, one of them "Animals Breeding" (Hodowla Zwierzątek) regained its popularity and it is on sell as "Superfarmer" game.
One of Warsaw streets, in Ursynów district, is named after him. But its surfice is quite smooth!
Streets of Warsaw
The higher we rise above the ground, the wider are our horizons; the horizons of perception and understanding. Still, this is when little, everyday things escape our attention, and it is the little things, like atoms, that form our reality. Even the widest perspective needs a pair of down-to-earth lenses to fully grasp our surrondings.
The First Perspective
OUR FRIENDLY SHADOW
On a sunny day we are never alone. We are permanently accompanied on our stroll around Warsaw by a shadow. It is an ever agreeable friend.
Our brain and mental processes have been shaped by evolution through image information based on the perception of light. Although the phenomenon is generally undervalued, light seems to be one of the most important factors shaping biological evolution. It is not surprising then that luminous space is indicative of the biological front, while dark space, not carrying any information, remains behind us. Also for this reason, although a shadow may be physically both in front and behind us, we always perceive it as being from behind, even when the sun shines in the back and the shadow precedes our steps. We also seem, usually, not to pay much attention to our own shadow.
Our shadows meet the shadows of other people, they tip their hats, shake hands, greet each other and bid each other farewell, blend into the shadow of nearby buildings, only to emerge on the other side. Shadows do not behave exactly as do the objects whose shadows they are – they are considerably more dynamic than the objects. Even when the object is still, the shadow lengthens, shortens, and circles the object as the sun moves. The city pulsates with the life of shadows.
The existence of shadows is related to many interesting and philosophically fascinating questions. What is the role of the human being in this relationship: that of author, proprietor, partner, or merely the look-alike, of the shadow? Beware! the shadow's owner may be taxed. Can predator-like shadows seize power over... exactly, over whom, just as in Andersen’s tale? There truly have been such cases.
We should perhaps take a look at our companion. Can we use the shadow to define an outline seen from the ground-based, horizontal perspective? Let's try.
The Second Perspective
The original artefact is not only a naturally formed artistic object, a collection of algae, lichens and mosses of the outskirts of Warsaw, but also a bio-indicator of the degree of environmental pollution. Specimen composition of lichen present on an object depend not only on the ground but also is an indicator of long-term air quality (scale lichens).
The measurements of air pollution conducted from the balloon present momentary results, captured in one brief moment. The lichen scale is a unique synthetic image of air quality in the city over the last months.
The author added that unusual shoe to the latest summer collection as a little artistic happening and psychological experiment in one. The results proved that very few clients of the shoe shop noticed its peculiarity.
The Third Perspective
Art of Impression: An afternoon on the grass is a part of Saski Garden. The photographer was mesmerized by a pair of shoes noticed from one-kilometre altitude.
The Fourth Perspective
THE SIERPIŃSKI CARPET
Quantum, molecular, biological, and presumably also social structure of nature is based on hidden simplicity of the common expression, although its external appearance reveals a tremendous variety and the complexity of structures or behaviours. One of the unusual descriptions of the natural world are fractals - patterns exhibiting recurrent self-similarity.
Graphs of multiple complex polynomials, at first sight extremely complicated and completely chaotic, actually show amazing simplicity of their basic geometrical forms which repeat with great regularity and order. Every randomly chosen fragment of the pattern, when magnified looks like the whole, and its components also resemble the whole object. The graphic representation of polynomials with minor changes of the basic formula mimic practically all shapes found in nature. Therefore the fractal geometry is called the geometry of the nature and casts new light on our universal knowledge about the world of nature.
One of the first described fractals was the Sierpiński carpet. But the world fame as their discoverer and the founder of this discipline, and also a precursor of the chaos theory is associated with the name of Benoît B. Mandelbrot. Both Wacław Sierpiński and Benoit Mandelbrot were Varsovians. Mandelbrot was born in Warsaw and soon had to emigrate. Wacław Sierpiński, the co-founder of the Polish mathematical school, remained associated with the city and with the University of Warsaw till the end of his life.
The notion of the chaos has many meanings and forms. In the chaotic urban pattern of today’s Warsaw it is difficult to find any unifying rule. When seen from above, the city especially clearly exhibits this lack of order. And perhaps this reflects the lack of social need for such order, though already for more than a hundred years airplanes fly over Warsaw?
I propose, to design a selected space, for example the planned scientific-technological centre within the Siekierki Arc or its fragment, as a fractal pattern. This would be not only an important event in the history of the town-planning, but also a homage to the world-famous Polish mathematicians connected with Warsaw. Geometrical fractal patterns are among the few mathematical concepts best suitable to be implemented into the real space. Warsaw would gain a modern symbol of the public city space recognizable worldwide of considerably greater rank than the Palace of the Culture or the fake palm-tree on one of the roundabouts. This would also be an essential element of the spatial identity of the capital.
The idea is all the more tempting that the bend of Vistula itself already is an element of the fractal geometry there.
On the Photo - historical fractal called the Sierpiński carpet. Remarkably, the surface area of the Sierpiński carpet equals 0. Will the null surface area allow an exemption from land tax?
The Sixth Perspective
For good measure: a classic snapshot of a street in Warsaw, taken from the ground level - the photo was randomly selected from thousands of others.