Ancient Greek Aesthetics: Harmony & Proportion Vitruvian Man Vitruvian Man Leonardo da Vinci , c. 1490 pen, ink and watercolour over metalpoint , 34.3 × 24.5 cm Gallerie dell'Accademia , Venice The Vitruvian Man is a famous drawing with accompanying notes by Leonardo da Vinci made around the year 1490 in one of his journals. It depicts a naked male figure in two superimposed positions with his arms apart and simultaneously inscribed in a circle and square. The drawing and text are sometimes called the Canon of Proportions . According to Leonardo's notes in the accompanying text, which are mirror writing , it was made as a study of the proportions of the (male) human body as described in a treatise by the Ancient Roman architect Vitruvius , who wrote that in the human body: a palm is the width of four fingers a foot is the width of four palms a cubit is the width of six palms a man's height is four cubits (and thus 24 palms) a pace is four cubits the length of a man's outspread arms is equal to his height the distance from the hairline to the bottom of the chin is one-tenth of a man's height the distance from the top of the head to the bottom of the chin is one-eighth of a man's height the distance from the hairline to the top of the breast is one-seventh of a man's height the distance from the top of the head to the nipples is one-fourth of a man's height the maximum width of the shoulders is one-fourth of a man's height the distance from the elbow to the tip of the hand is one-fifth of a man's height the distance from the elbow to the armpit is one-eight of a man's height the length of the hand is one-tenth of a man's height the distance from the bottom of the chin to the nose is one-third of the length of the face the distance from the hairline to the eyebrows is one-third of the length of the face the length of the ear is one-third of the length of the face Leonardo is clearly illustrating Vitruvius De Architectura 3.1.3 which reads: The navel is naturally placed in the centre of the human body, and, if in a man lying with his face upward, and his hands and feet extended, from his navel as the centre, a circle be described, it will touch his fingers and toes. It is not alone by a circle, that the human body is thus circumscribed, as may be seen by placing it within a square. For measuring from the feet to the crown of the head, and then across the arms fully extended, we find the latter measure equal to the former; so that lines at right angles to each other, enclosing the figure, will form a square. The rediscovery of the mathematical proportions of the human body in the 15th century by Leonardo and others is considered one of the great achievements leading to the Italian Renaissance . Note that Leonardo's drawing combines a careful reading of the ancient text, combined with his own observation of actual human bodies. In drawing the circle and square he correctly observes that the square cannot have the same center as the circle, the navel, but is somewhat lower in the anatomy. This adjustment is the innovative part of Leonardo's drawing and what distinguishes it from earlier illustrations. The drawing itself is often used as an implied symbol of the essential symmetry of the human body, and by extension, to the universe as a whole. It may be noticed by examining the drawing that the combination of arm and leg positions actually creates sixteen different poses. The pose with the arms straight out and the feet together is seen to be inscribed in the superimposed square. On the other hand, the &quot;spread-eagle&quot; pose is seen to be inscribed in the superimposed circle. This illustrates the principle that in the shift between the two poses, the apparent center of the figure seems to move, but in reality, the navel of the figure, which is the true center of gravity , remains motionless. From Wikipedia, the free encyclopedia.
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HARMONY & PROPORTION ANCIENT GREEK AESTHETICS
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" Man is the Measure of all Things ." Protagoras Click for Vitruvian Man interview
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Influenced by the Greeks, the Ancient Roman architect VITRUVIUS wrote that in the human body:
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Influenced by the Greeks, the Ancient Roman architect VITRUVIUS wrote that in the human body: a palm is the width of four fingers a foot is the width of four palms a cubit is the width of six palms a man's height is four cubits (and thus 24 palms) a pace is four cubits the length of a man's outspread arms is equal to his height the distance from the hairline to the bottom of the chin is one-tenth of a man's height the distance from the top of the head to the bottom of the chin is one-eighth of a man's height the distance from the hairline to the top of the breast is one-seventh of a man's height the distance from the top of the head to the nipples is one-fourth of a man's height the maximum width of the shoulders is one-fourth of a man's height the distance from the elbow to the tip of the hand is one-fifth of a man's height the distance from the elbow to the armpit is one-eight of a man's height the length of the hand is one-tenth of a man's height the distance from the bottom of the chin to the nose is one-third of the length of the face the distance from the hairline to the eyebrows is one-third of the length of the face the length of the ear is one-third of the length of the face
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What did Bodhi Tree say upon attaining enlightenment and realizing he was merely a green, fruit-bearing sphere atop a brown, bark-covered cylinder?
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Ge-om-e-try What did Bodhi Tree say upon attaining enlightenment and realizing he was merely a green, fruit-bearing sphere atop a brown, bark-covered cylinder?
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“ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats The Greeks perfected the use of Geometry
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The Greeks wished to maintain ideal proportions when building any room or like structure from the ground up.
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“ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats The Golden Rectangle
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“ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats Q: Given a flat rectangle,
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“ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats what is its most aesthetically pleasing height?
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“ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats What Does that “ MEAN” ?
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The arithmetic mean occurs when the length of the sides of a square whose PERIMETER is the same as the rectangle. ( L + W ) / 2 ( 12 + 6 ) / 2 = 9 9 exceeds 6 by 3, which is the same amount by which 12 exceeds 9. LOGIC
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The geometric mean is the nth root of the product of the variables (sides). Here n = 2. The arithmetic mean occurs when the length of the sides of a square* whose AREA is the same as the rectangle. 9 X 4 = 36 The square root of 36 is 6 * 6 x 6
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The harmonic mean is the number of variables (sides) divided by the sum of each side’s reciprocal. Here n=2 and the sides are 12 and 6. 1/12 + 1/6 = 3/12, or ¼. 2 divided by ¼ equals 8
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Leon Battista Alberti, Florentine Architect (1407-1472) "We shall therefore borrow all our Rules for the Finishing our Proportions, from the MUSICIANS, who are the greatest Masters of this Sort of Numbers, and from those Things wherein Nature shows herself most excellent and complete."
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Music, Numbers and the Universe ? "Seek truth and beauty together; you will never find them apart." Pythagoras of Samos
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“ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats Big Idea: Mathematics is the language of nature. LOGIC
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Music’s Rhythm is based on Mathematical Fractions
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J.S. Bach’s Little Fugue in G minor Bach 1685-1750 Baroque Period
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SUPERIMPOSED, RECURSIVE RHYTHMS Bach 1685-1750 Baroque Period RECURSIVE Rhythm
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SUPERIMPOSED MIRRORED MOTIFS IN ART Bach 1685-1750 Baroque Period M.C. Escher (1898 - 1972) Modern Period Recursion in Art Recursion in Art Recursion in Art
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Diatonic Scale H W W W H W W Step 256:243 9:8 9:8 9:8 256:243 9:8 9:8 Intra-Ratio A to B 243:128 G to A 27:16 E to F 4:3 F to G 3:2 B to C D to E C to D Tone 2:1 81:64 9:8 1:1 Tonic
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Filling in the Gaps The harmonic mean is the number of variables (notes) divided by the sum of each side’s reciprocal. Here n=2 and the sides are 12 and 6. TASK: Calculate the harmonic mean for the numbers 1 and 2…
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Filling in the Gaps The harmonic mean is the number of variables (notes) divided by the sum of each side’s reciprocal. Here n=2 and the sides are 12 and 6. ANSWER: For 1 and 2… 1/1 + 1/2 = 3/2 . 2 divided by 3/2 equals 4/3
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And since harmony is reducible to number, perfection is likewise reducible to number . The universe is harmonious, so it follows that the universe as a whole can be explained in terms of number . Pythagoras of Samos LOGIC
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I perfected the logical syllogism as a method for discovering TRUTH . Aristotle says, AGAIN…
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Q : Which ancient Greek philosopher took Pythagoras’ theory of the universe and “shaped” it further?
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Q : Which ancient Greek philosopher took Pythagoras’ theory of the universe and “shaped” it further? HINT : He expanded the scale of the musical scale AND the scale of the universe. PLAY
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Plato, through Timaeus, reasoned in reverse stating that the creator made the world soul out of various ingredients, and formed it into a long strip… The strip was then marked out into intervals… Plato
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“ First [the creator] took one portion from the strip (1st unit) and next a portion double the first (2nd unit) a third portion half again as much as the second (3rd unit) the fourth portion double the second (4th unit) the fifth three times the third (9th unit) the sixth eight times the first (8th unit) and the seventh 27 times the first (27th unit) ” They give the seven integers; 1, 2, 3, 4, 8, 9, 27. These contain the monad, source of all numbers, the first even and first odd, and their squares and cubes. Plato LAMBDA
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LAMBDA They give the seven integers; 1, 2, 3, 4, 8, 9, 27. These contain the monad, source of all numbers, the first even and first odd, and their squares and cubes.
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But my system wasn’t exactly perfect… Pythagoras of Samos
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But first, let’s C who knows what the word ENHARMONIC means. HINT: If you answer correctly, you must really B#.
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Allusion Trivia Following their final live performance atop a rooftop in 1969, which band’s musician quipped, “ I'd like to say thank you very much on behalf of the group and myself and I hope we passed the audition” ?
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by Lorreen Pelletier in sky apple a la mode cherry or humble shoo fly pie mulberry mud or pumpkin chocolate pecan oh ruin your appetite by Lorreen Pelletier Pie. I like a peach blueberry or banana cream and lemon meringue raspberry rhubarb mincemeat pie
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Poe, E. Near a Raven Midnights so dreary, tired and weary. Silently pondering volumes extolling all by-now obsolete lore. During my rather long nap - the weirdest tap! An ominous vibrating sound disturbing my chamber's antedoor . "This", I whispered quietly, "I ignore". Perfectly, the intellect remembers: the ghostly fires, a glittering ember . Inflamed by lightning's outbursts, windows cast penumbras upon this floor. Sorrowful, as one mistreated, unhappy thoughts I heeded : That inimitable lesson in elegance - Lenore - Is delighting, exciting...nevermore . Ominously, curtains parted (my serenity outsmarted), And fear overcame my being - the fear of "forevermore". Fearful foreboding abided, selfish sentiment confided, As I said, "Methinks mysterious traveler knocks afore. A man is visiting, of age threescore."
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Simpler Pi Poems Did I tell a witty wisecrack? (3.14159) Yes, I love a green grassland. (3.14159) Boy, I want a glass half-full of Sprite. (3.1415926) From Paul’s Page of Pi: http://www.escape.com/~paulg53/math/pi/poems.html
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Aristotle The truth can be found by analyzing nature (breaking in to parts).
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… so the hotdog guy makes the hotdog with everything on it. The monk exchanges a $20 for the hotdog and waits for his change. When no change is forthcoming, the monk asks, "Hey, where's my change?" The hotdog guy replies, "Change must come from within."
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… and the hotdog guy replies… Hey, where's my change?" The hotdog guy replies, "Change must come from within."
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“ Sorry, but you should already know… Change comes from within."
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Aristotle BIG IDEA : “Moral virtue is a state of character lying in a mean between two extremes . ” ETHICS
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Aristotle’s GOLDEN MEAN ACTION Deficiency Goodness/Mean Excess Fear cowardice courage rashness/foolhardy Drinking/Eating insensible temperate self-indulgent Truth telling modesty/ truthful self-deprecation Lending $$ stingy liberality prodigality Amusement boring ready wit buffoonery “ Moral virtue is a state of character lying in a mean between two extremes . ”
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GOLDEN SECTION Standard sized credit cards are 54mm x 86mm, creating a ratio of 0.628, less than a millimeter from a perfect golden section of 0.618. http://goldennumber.net/classic/fibonser.htm
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the Fibonacci Series 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . http://goldennumber.net/classic/fibonser.htm
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GOLDEN SECTION: PHI Starting with 0 and 1, each new number in the series is simply the sum of the two before it. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . . The ratio of each successive pair of numbers in the series approximates phi (1.618. . .) , as 5 divided by 3 is 1.666..., and 8 divided by 5 is 1.60. The ratios of the successive numbers in the Fibonacci series quickly converge on Phi or Φ. After the 40th number in the series, the ratio is accurate to 15 decimal places. 1.618033988749895 . . . http://goldennumber.net/classic/fibonser.htm
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the GOLDEN MEAN Musical scales are based on Fibonacci numbers The Fibonacci series appears in the foundation of aspects of art, beauty and life. Even music has a foundation in the series, as: There are 13 notes in the span of any note through its octave. A scale is comprised of 8 notes, of which the 5th and 3rd notes create the basic foundation of all chords, and are based on whole tone which is 2 steps from the root tone, that is the 1st note of the scale. Note too how the piano keyboard scale of C to C above of 13 keys has 8 white keys and 5 black keys, split into groups of 3 and 2. http://goldennumber.net/classic/fibonser.htm
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the GOLDEN MEAN The Fibonacci series appears in the foundation of aspects of art, beauty and life. Even music has a foundation in the series, as: There are 13 notes in the span of any note through its octave. A scale is comprised of 8 notes, of which the 5th and 3rd notes create the basic foundation of all chords, and are based on whole tone which is 2 steps from the root tone, that is the 1st note of the scale.
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"My new Hypothesis: If we're built from Spirals while living in a giant Spiral, then is it possible that everything we put our hands to is infused with the Spiral?" -- Max Cohen in the motion picture PI
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“ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats What is the difference between…
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“ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats What is the difference between… the truth and
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“ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats What is the difference between… the truth and T ruth ?
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The eternal paradox of literature: Art always tells the Truth
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John Malkovich as himself in Being John Malkovich 1999 The eternal paradox of literature: Art always tells the Truth even when it’s lying.
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The eternal paradox of literature: What’s T rue need not be true.
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The eternal paradox of literature: Although a story is fictional… Its THEME is always True.
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