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    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 "spread-eagle" 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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    Greek Power Point1.4 Segment 3 - Presentation Transcript

    1. HARMONY & PROPORTION ANCIENT GREEK AESTHETICS
    2. " Man is the Measure of all Things ." Protagoras Click for Vitruvian Man interview
    3. Influenced by the Greeks, the Ancient Roman architect VITRUVIUS wrote that in the human body:
    4. 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
    5. 6:1 Proportion
    6. One Foot Six Feet
    7. Six Feet Twelve Feet Two Feet
    8. Doric Order
    9. 8:1 Proportion
    10.  
    11. Corinthian Order
    12. Corinthian Style: Temple to the goddess Nike
    13. Temple to the Corporate god, Nike Post-Modern Consumer Style
    14. Temple of Apollo at Corinth Earliest Surviving Doric Structure
    15. The Oracle at Delphi Northeast of Corinth
    16. What did Bodhi Tree say upon attaining enlightenment and realizing he was merely a green, fruit-bearing sphere atop a brown, bark-covered cylinder?
    17. 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?
    18. Ge-om-e-try
    19. GEE, I’m a tree!
    20. “ 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
    21. The Greeks wished to maintain ideal proportions when building any room or like structure from the ground up.
    22. “ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats The Golden Rectangle
    23. “ 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,
    24. “ 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?
    25. “ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats What Does that “ MEAN” ?
    26. 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
    27. 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
    28. 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
    29. LOGIC Physical Harmony
    30. 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."
    31. Music, Numbers and the Universe ? "Seek truth and beauty together; you will never find them apart." Pythagoras of Samos
    32. “ 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
    33. Music’s Rhythm is based on Mathematical Fractions
    34. SUBDIVISIONS
    35. SUBDIVISIONS OF THE WHOLE
    36. FRACTIONS of FRACTIONS
    37. J.S. Bach’s Little Fugue in G minor Bach 1685-1750 Baroque Period
    38. SUPERIMPOSED, RECURSIVE RHYTHMS Bach 1685-1750 Baroque Period RECURSIVE Rhythm
    39. 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
    40. THEME & VARIATION IN MUSIC LOGIC AESTHETICS
    41. Fifth Symphony Ludwig van Beethoven 1770-1827 Classical Period
    42. Fifth Symphony Ludwig van Beethoven 1770-1827 Classical Period
    43. Fifth Symphony Ludwig van Beethoven 1770-1827 Classical Period
    44. Fifth Symphony Ludwig Van Beethoven 1770-1827 Classical Period
    45. Fifth Symphony Ludwig Van Beethoven 1770-1827 Classical Period
    46. Fifth Symphony Ludwig Van Beethoven 1770-1827 Classical Period
    47. Fifth Symphony Ludwig van Beethoven 1770-1827 Classical Period
    48. Mathematics is ALSO the language of HARMONY & PITCH
    49.  
    50. Diatonic Scale (Ionian Mode)
    51. Diatonic Scale (Ionian Mode)
    52.  
    53.  
    54. Q: What is the scientific symbol for Wavelength ?
    55. Hint: It’s the 11th letter of the Greek alphabet … Daaah.
    56. lamb-da
    57.  
    58.  
    59.  
    60. Concert A=440 means that the universally agreed upon standard for the tone of “A” is 440 Hertz, or 440 cycles (vibrations) per second .
    61. Each tone can be subdivided into 100ths to ensure an almost perfect match.
    62. Q: Why is it so easy to understand how a Chromatic tuner works? Hint
    63. BECAUSE IT MAKES CENTS !
    64. Q: Why might a chromatic tuner also be useful in diagnosing pain?
    65. BECAUSE IT KNOWS ALL YOUR HERTZ !
    66. Semitone (Half-step or Minor Second)
    67. Wholetone (Whole-step or Major Second)
    68. Diapason (Octave)
    69. Diapason (Octave)
    70.  
    71. 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
    72. 1:2 RATIO Octave
    73. 2:3 RATIO Perfect Fifth
    74. 2:3 RATIO Perfect Fifth
    75. 3:4 RATIO Perfect Fourth
    76. 3:4 RATIO Perfect Fourth
    77. SACRED CONSONANCE
    78. But what about all of those black notes in between?
    79. Filling in the Gaps
    80. 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…
    81. 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
    82. Filling in the Gaps
    83.  
    84. HARMONY Is PERFECTION… Pythagoras of Samos
    85. 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
    86. I perfected the logical syllogism as a method for discovering TRUTH . Aristotle says, AGAIN…
    87. A P O L L O God of Truth and Music
    88. Goddess of Harmony and Concord www.aris-titanium.com/.../ hirez/harmonia.jpg
    89. The Role of Greek Music
    90. Department of Classics, University of Cincinnati Sacred Fragments
    91.  
    92.  
    93. Q : Which ancient Greek philosopher took Pythagoras’ theory of the universe and “shaped” it further?
    94. 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
    95. PLATO’S CONTRIBUTION PYTHAGORAS’ CONTRIBUTION
    96. 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
    97. “ 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
    98. Plato METAPHYSICS
    99. The Universe is One Big …
    100. Beauty Truth Truth Beauty LETTER ?
    101. Beauty Truth Truth Beauty CHIASMUS
    102. Plato
    103. Armillary Sphere Celestial Circles Plato
    104. Beauty Truth Truth Beauty Tangible Beauty (in the circle of the different) is a Representation of Intangible Truth (in the circle of the Same).
    105.  
    106. If you know BEAUTY , you know TRUTH . If you know TRUTH , you can begin to understand life’s profound mysteries .
    107. And THAT’S Why I’m So Darned Important in the Grand Scheme of Things! Art Says…
    108. Music of the Spheres
    109. 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.
    110. But my system wasn’t exactly perfect… Pythagoras of Samos
    111. But first, let’s C who knows what the word ENHARMONIC means. HINT: If you answer correctly, you must really B#.
    112. And the award goes to… THE WITH B Sharps
    113. 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” ?
    114. It was THE JOHN LENNON BEATLES
    115. ENHARMONIC TONES
    116.  
    117.  
    118. Minute shift
    119. Superimposed shift
    120. Order in Chaos Or Chaos in Order ?
    121. STYLE ALERT! Order Chaos Chaos Order
    122. CHIASMUS Order Chaos Chaos Order
    123. by Lorreen Pelletier
    124. by Lorreen Pelletier Not to be confused with…
    125. by Lorreen Pelletier A more primal equation…
    126. by Lorreen Pelletier
    127. 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
    128. 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."
    129. 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
    130. Poetry of Your Own Okay, Now Try Some Pi…
    131. 3. 1415926535897932384626433832795028841971693993751 05820974944 592307816406286208998628034825342117067982148086513282306647 093844609550582231725359408128481117450284102701938521105559 644622948954930381964428810975665933446128475648233786783165 271201909145648566923460348610454326648213393607260249141273 724587006606315588174881520920962829254091715364367892590360 011330530548820466521384146951941511609433057270365759591953 092186117381932611793105118548074462379962749567351885752724 891227938183011949129833673362440656643086021394946395224737 190702179860943702770539217176293176752384674818467669405132 000568127145263560827785771342757789609173637178721468440901 224953430146549585371050792279689258923542019956112129021960 864034418159813629774771309960518707211349999998372978049951 059731732816096318595024459455346908302642522308253344685035 261931188171010003137838752886587533208381420617177669147303 598253490428755468731159562863882353787593751957781857780532 171226806613001927876611195909216420198938095257201065485863 278865936153381827968230301952035301852968995773622599413891 249721775283479131515574857242454150695950829533116861727855 889075098381754637464939319255060400927701671139009848824012 858361603563707660104710181942955596198946767837449448255379
    132. Robert Fludd: Utriusque Cosmi Maioris scilicet et Minoris Metaphysica Physica Atque Technica Historia, pub.1617.
    133.  
    134. LAMBDA
    135. A P O L L O God of Truth and Music
    136. A THOUSAND YEARS of Greek Arete
    137. A THOUSAND YEARS of Greek Arete
    138. A THOUSAND YEARS of Greek Arete
    139.  
    140. David’s Death of Socrates: Neoclassical Period (1787)
    141. The only true wisdom consists in knowing that you know nothing. SOCRATES
    142. Socrates
    143. So-crates, dude The only true wisdom consists in knowing that you know nothing. Bill & Ted
    144. Socrates
    145. Socrates
    146. Plato Socrates
    147. Plato Socrates
    148. Aristotle Plato Socrates
    149. Aristotle Plato
    150. Plato
    151. Plato “ Let No Man Destitute of Geometry Enter My Doors.”
    152. POINT LINE PLANE SOLID SQUARES CUBES 1 st even & odd
    153. Celestial Circles Armillary Sphere Turner, Gerard. Antique Scientific Instruments. Dorset: Blandford
    154. Plato All sense experience is faulty & transient.
    155. Plato Thus, the truth can be found by looking beyond nature. EPISTEMOLOGY
    156. Plato Horse vs. Horseness: Theory of Forms
    157. Plato
    158. Diagram of Plato’s The Allegory of the Cave
    159. Aristotle No matter without form; no form without matter.
    160. Aristotle’s Poetics
    161. Aristotle The truth can be found by analyzing nature (breaking in to parts).
    162. … 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."
    163. "Hey, where's my change?"
    164. … and the hotdog guy replies… Hey, where's my change?" The hotdog guy replies, "Change must come from within."
    165. “ Sorry, but you should already know… Change comes from within."
    166. Aristotle BIG IDEA : “Moral virtue is a state of character lying in a mean between two extremes . ” ETHICS
    167. 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 . ”
    168. the GOLDEN MEAN & PHI
    169. the GOLDEN RECTANGLE
    170. 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
    171. the Fibonacci Series 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . http://goldennumber.net/classic/fibonser.htm
    172. 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
    173. 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
    174. the GOLDEN MEAN
    175. 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.
    176. the GOLDEN MEAN AESTHETICS
    177. the GOLDEN MEAN Later applications of
    178. GOLDEN RECTANGLE The
    179. the GOLDEN MEAN Later applications of
    180. the GOLDEN RECTANGLE
    181. The Golden Rectangle
    182. The Golden Rectangle
    183. The Golden Rectangle
    184. The Golden Rectangle
    185. The Golden Rectangle
    186. The Golden Rectangle
    187. The Golden Rectangle
    188. The Golden Rectangle “ Moral virtue is a state of character lying in a mean between two extremes . ”
    189. Inherent in Perfection is the UNATTAINABLE “ Moral virtue is a state of character lying in a mean between two extremes . ”
    190. The Spiral of Fifths
    191.  
    192.  
    193.  
    194. "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  
    195.  
    196. “ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats The Golden Rectangle
    197. “ Beauty is Truth and Truth Beauty, That is All Ye Know on Earth and All Ye Need to Know.” Keats The Golden Rectangle
    198. Now here’s real a “ PARADOX ”…
    199. But first, a real “ Pair o’ Docks”…
    200. “ 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…
    201. “ 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
    202. “ 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 ?
    203. The eternal paradox of literature: Art always tells the Truth
    204. John Malkovich as himself in Being John Malkovich 1999 The eternal paradox of literature: Art always tells the Truth even when it’s lying.
    205. The eternal paradox of literature: What’s T rue need not be true.
    206. The eternal paradox of literature: Although a story is fictional… Its THEME is always True.
    207. THEME equals Truth EPISTEMOLOGY
    208. When reading literature… Art Says…
    209. Truth doesn’t come easily. Art Says…
    210. Using your powers of inference… Art Says…
    211. … you have to “urn” it. Art Says…
    212. What we obtain too cheap, we esteem too lightly. Thomas Paine Says… (1737-1809) Writer and Revolutionary
    213. Authors rarely come right out and TELL you the story’s THEME.
    214. Art Says… By imagining, struggling, sympathizing, as we read, we make the life lesson more personally meaningful .
    215. AND more memorable. Art Says…

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