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Jun
27th
Mon
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Ludwig Wittgenstein: “Logic takes care of itself; all we have to do is to look and see how it does it”

Ludwig Wittgenstein, Tractatus logico-philosophicus (1921):


1 — The world is everything that is the case.


2 — What is the case, the fact, is the existence of atomic facts


2.01 — An atomic fact is a combination of objects (entities, things)


2.013 — Every thing is, as it were, in a space of possible atomic facts. I can think of this space as empty, but not of the thing without the space.




2.032 — The way in which objects hang together in the atomic fact is the structure of the atomic fact.

Ludwig Wittgenstein, Austrian philosopher, mathematician, logician, who held the professorship in philosophy at the University of Cambridge (1889-1951), the first quote: Journal entry (13 Oct 1914), also in § 5.47 of Tractatus logico-philosophicus (1921), (transl. by C.K. Ogden) Drawings by Jorinde Voigtsource

Feb
26th
Sat
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Left Brain - Right Brain, Paint by Mercedes Benz (tnx endlessforms)
Left brain: I am the left brain. I am a scientist. A mathematician. I love the familiar. I categorize. I am accurate. Linear. Analytical. Strategic. I am practical. Always in control. A master of words and language. Realistic. I calculate equations and play with numbers. I am order. I am logic. I know exactly who I am.
Right brain: I am the right brain. I am creativity. A free spirit. I am passion. Yearning. Sensuality. I am the sound of roaring laughter. I am taste. The feeling of sand beneath bare feat. I am movement. Vivid colors. I am the urge to paint on an empty canvas. I am boundless imagination. Art. Poetry. I sense. I feel. I am everything I wanted to be.

Left Brain - Right Brain, Paint by Mercedes Benz (tnx endlessforms)

Left brain: I am the left brain. I am a scientist. A mathematician. I love the familiar. I categorize. I am accurate. Linear. Analytical. Strategic. I am practical. Always in control. A master of words and language. Realistic. I calculate equations and play with numbers. I am order. I am logic. I know exactly who I am.

Right brain: I am the right brain. I am creativity. A free spirit. I am passion. Yearning. Sensuality. I am the sound of roaring laughter. I am taste. The feeling of sand beneath bare feat. I am movement. Vivid colors. I am the urge to paint on an empty canvas. I am boundless imagination. Art. Poetry. I sense. I feel. I am everything I wanted to be.

Dec
20th
Mon
permalink

Leonardo da Vinci, Geometrical solids animation, Video credit: Victoria and Albert Museum

“This animation shows how Leonardo composed illustrations of a dodecahedron: a solid composed of twelve pentagons – in other words, twelve identical, five-sided faces. He split it into segments in order to calculate its volume, sliced off its corners and built pyramids on the resulting faces.”

(Source: amiquote)

Jun
1st
Tue
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The Mathematical Art of M.C. Escher

"For me it remains an open question whether [this work] pertains to the realm of mathematics or to that of art." — M.C. Escher



— BBC-4, 2005

Douglas R. Hofstadter on M. C. Escher’s drawings

"To my mind, the most beautiful and powerful visual realizations of this notion of Strange Loops exist in the work of the Dutch graphic artist M. C. Escher, who lived from 1902 to 1972. Escher was the creator of some of the most intellectually stimulating drawings of all time. Many of them have their origin in paradox, illusion, or double-meaning.

Mathematicians were among the first admirers of Escher’s drawings, and this is understandable because they often are based on mathematical principles of symmetry or pattern… But there is much more to a typical Escher drawing than just symmetry or pattern; there is often an underlying idea, realized in artistic form. And in particular, the Strange Loop is one of the most recurrent themes in Escher’s work. Look, for example, at the lithograph Waterfall, 1961 (Fig. 5), and compare its six-step endlessly falling loop with the six-step endlessly rising loop of the J.S. Bach's "Canon per Tonos". The similarity of vision is remarkable. Bach and Escher are playing one single theme in two different “keys”: music and art.

                                               Figure 5, M. C. Escher, Waterwall, 1961

Escher realized Strange Loops in several different ways, and they can be arranged according to the tightness of the loop. The lithograph Ascending and Descending, 1960 (Fig. 6), in which monks trudge forever in loops, is the loosest version, since it involves so many steps before the starting point is regained.

                          Figure 6, M. C. Escher, Ascending and Descending, 1960

A tighter loop is contained in Waterfall, which, as we already observed, involves only six discrete steps. You may be thinking that there is some ambiguity in the notion of a single “step”-for instance, couldn’t Ascending and Descending be seen just as easily as having four levels (staircases) as forty-five levels (stairs)% It is indeed true that there is an inherent haziness in level-counting, not only in Escher pictures, but in hierarchical, many-level systems. We will sharpen our understanding of this haziness later on.

But let us not get too distracted now’ As we tighten our loop, we come to the remarkable Drawing Hands (Fig. 135), in which each of two hands draws the other: a two-step Strange Loop.

                                                 Figure 135, M. C. Escher, Drawing Hands

  Figure 136, Abstract diagram of M.C. Escher’s Drawing Hands. On to, a seeming paradox. Below, it’s relsolution.

"And yet when I say "strange loop", I have something else in mind — a less concrete, more elusive notion. What I mean by “strange loop” is — here goes a first stab, anyway — not a physical circuit but an abstract loop in which, in the series of stages that constitute the cycling-around, there is a shift from one level of abstraction (or structure) to another, which feels like an upwards movement in a hierarchy, and yet somehow the successive “upward” shifts turn out to give rise to a closed cycle. That is, despite one’s sense of departing ever further from one’s origin, one winds up, to one’s shock, exactly where one had started out. In short, a strange loop is a paradoxical level-crossing feedback loop.” — (D. Hofstadter,  I Am a Strange Loop, p. 101-102)

And finally, the tightest of all Strange Loops is realized in Print Gallery (Fig. 142): a picture of a picture which contains itself. Or is it a picture of a gallery which contains itself? Or of a town which contains itself? Or a young man who contains himself’? (…)

                                              Figure 142, M. C. Escher, Print Gallery

Implicit in the concept of Strange Loops is the concept of infinity, since what else is a loop but a way of representing an endless process in a finite way? And infinity plays a large role in many of Escher’s drawings. Copies of one single theme often fit into each’ other, forming visual analogues to the canons of Bach. Several such patterns can be seen in Escher’s famous print Metamorphosis (Fig. 8). It is a little like the “Endlessly Rising Canon”: wandering further and further from its starting point, it suddenly is back. In the tiled planes of Metamorphosis and other pictures, there are already suggestions of infinity.

                                   Figure 8, M. C. Escher, Metamorphosis II

But wilder visions of infinity appear in other drawings by Escher. In some of his drawings, one single theme can appear on different levels of reality. For instance, one level in a drawing might clearly be recognizable as representing fantasy or imagination; another level would be recognizable as reality. These two levels might be the only explicitly portrayed levels. But the mere presence of these two levels invites the viewer to look upon himself as part of yet another level; and by taking that step, the viewer cannot help getting caught up in Escher’s implied chain of levels, in which, for any one level, there is always another level above it of greater “reality”, and likewise, there is always a level below, “more imaginary” than it is.

This can be mind-boggling in itself. However, what happens if the chain of levels is not linear, but forms a loop? What is real, then, and what is fantasy? The genius of Escher was that he could not only concoct, but actually portray, dozens of half-real, half-mythical worlds, worlds filled with Strange Loops, which he seems to be inviting his viewers to enter.”

Douglas R. Hofstadter, Gödel Escher Bach: an Eternal Golden Braid, Penguin Books, 1999, p. 18-23.

Inspirations: A Short Film Celebrating the Mathematical Art of M.C. Escher

Cristóbal Vila, Inspirations, February 2012

See also:

The Official M.C. Escher Website
The Strange Worlds of M C Escher, Escape Into Life
Mathematical Art of M.C. Escher, Platonic Realms MiniText
Nature by Numbers - a short movie inspired on numbers, geometry and nature by Cristóbal Vila

May
30th
Sun
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William Blake, “I Want! I Want!” 1793, Engraving, Butlin 201 40

“By the 1790s, even the most romantically inclined philosophers no longer believed there was a man in the moon. Yet William Blake, with his characteristic combination of unfettered imagination and irrepressible logic, simply surmised that he must have had a very long ladder.

I Want! I Want! is the title of a small etching Blake produced at the height of the French revolutionary terror. A tiny, anonymous figure stands poised on the first rung of a celestial ladder, lodged like a giant toothpick in the smiling crescent moon. It is a devastatingly simple metaphor for the destructive ambition that grows out of humankind’s capacity to dream. The etching is used here as a striking frontispiece for an exhibition in which nine contemporary artists construct their own utopias.”

— Alfred Hickling, I Want! I Want! Northern Gallery, Sunderland, The Guardian, 27 August 2003
May
9th
Sun
permalink
The Road to Success, published 1913
"Motivational Poster (click image twice to blow up size), adapted to musical education from an original drawing issued by the National Cash Register Co. to point the road to business success. It originally appeared in THE ETUDE music magazine, published October 1913 and includes some of the unique language and jargons of the time. Great inspirational and motivational tool/gift for anyone, especially young people starting out, new businesses and anyone seeking success (aren’t we all?)..  Just follow the ‘Road to Success,’ making sure not to fall prey to ‘bohemianism,’ shiftlessness’ or one of the many other pitfalls along the way. Don’t lose time at the ‘Hotel Know It All’ or ‘Mutual Admiration Society’ waiting for compliments such as ‘Caruso can’t touch you’ or ‘You’ll set the world on fire.’ Once you reach the ‘Gate of Ideals’ you’ve nearly reached the big zither(?) of success.
‘This (literally) lyrical prize is achieved by first entering the Gate of Opportunity. People are running through, but some have already settled in to the sit-down life of ease and comfort in what looks like the Beer Garden of Bohemianism.
Some manage to pass by those delights to check in to the Hotel Know It All, because they hold to mottoes such as Nobody can tell me, or I don’t need to practice, or I’m a born genius, or yet: I don’t need system.
Similarly misguided cries are heard on the patio of the Mutual Admiration Society: You’re the Hit of the Age, You’ll Set the World on Fire, You’re a Wonder My Boy, or (my favourite): Caruso Can’t Touch You.
Those who avoid those three establishments of ill repute might still fall victim to the deep, dark well of Illiteracy, or the spinning, disorienting wheel of Conceit. A select few manage to board the train called Right System at the Railroad Station.
That doesn’t stop some from running along the rail track towards Success, only to succumb to the ugly hand of Vices, the spinning fan of Bad Habits (blowing its victims towards Oblivion), or the pitfall of Bad Reputation. Others fall prey to Charlatanism, or get tangled up in the webs of Jealousy and Do It Tomorrow.
Those who overcome all these perils will enter the gates of System. But while the train crosses a bridge across the river Failure, those on foot are threatened by the Cauldron of Misrepresentation, and tempted by Short Cuts.
Some do manage to wade across the river to the other side, but there must overcome Bad Temper, Carelessness, Shiftlessness and Bad Memory. Then there’s Lack of Preparation, a giant rock which the train can tunnel through effortlessly, while the surviving pedestrians must trek across it.
Sprees, Laziness and Bad Business Methods then still threaten them, until at last they come before two gates, the one for Weak Morals remaining forever closed, the Gate of Ideals open to the train (and some on foot).
Conclusion: you can be successful without adopting the Right System, but your chances are far smaller. And you’ll have to make a lot bigger effort to get there.” (Source: OldiesCountry)

The Road to Success, published 1913

"Motivational Poster (click image twice to blow up size), adapted to musical education from an original drawing issued by the National Cash Register Co. to point the road to business success. It originally appeared in THE ETUDE music magazine, published October 1913 and includes some of the unique language and jargons of the time. Great inspirational and motivational tool/gift for anyone, especially young people starting out, new businesses and anyone seeking success (aren’t we all?)..  Just follow the ‘Road to Success,’ making sure not to fall prey to ‘bohemianism,’ shiftlessness’ or one of the many other pitfalls along the way. Don’t lose time at the ‘Hotel Know It All’ or ‘Mutual Admiration Society’ waiting for compliments such as ‘Caruso can’t touch you’ or ‘You’ll set the world on fire.’ Once you reach the ‘Gate of Ideals’ you’ve nearly reached the big zither(?) of success.

‘This (literally) lyrical prize is achieved by first entering the Gate of Opportunity. People are running through, but some have already settled in to the sit-down life of ease and comfort in what looks like the Beer Garden of Bohemianism.

Some manage to pass by those delights to check in to the Hotel Know It All, because they hold to mottoes such as Nobody can tell me, or I don’t need to practice, or I’m a born genius, or yet: I don’t need system.

Similarly misguided cries are heard on the patio of the Mutual Admiration Society: You’re the Hit of the Age, You’ll Set the World on Fire, You’re a Wonder My Boy, or (my favourite): Caruso Can’t Touch You.

Those who avoid those three establishments of ill repute might still fall victim to the deep, dark well of Illiteracy, or the spinning, disorienting wheel of Conceit. A select few manage to board the train called Right System at the Railroad Station.

That doesn’t stop some from running along the rail track towards Success, only to succumb to the ugly hand of Vices, the spinning fan of Bad Habits (blowing its victims towards Oblivion), or the pitfall of Bad Reputation. Others fall prey to Charlatanism, or get tangled up in the webs of Jealousy and Do It Tomorrow.

Those who overcome all these perils will enter the gates of System. But while the train crosses a bridge across the river Failure, those on foot are threatened by the Cauldron of Misrepresentation, and tempted by Short Cuts.

Some do manage to wade across the river to the other side, but there must overcome Bad Temper, Carelessness, Shiftlessness and Bad Memory. Then there’s Lack of Preparation, a giant rock which the train can tunnel through effortlessly, while the surviving pedestrians must trek across it.

Sprees, Laziness and Bad Business Methods then still threaten them, until at last they come before two gates, the one for Weak Morals remaining forever closed, the Gate of Ideals open to the train (and some on foot).

Conclusion: you can be successful without adopting the Right System, but your chances are far smaller. And you’ll have to make a lot bigger effort to get there.” (Source: OldiesCountry)