The Elegant Mathematics of Vitruvian Man, Leonardo da Vinci’s Most Famous Drawing: An Animated Introduction

Near­ly 500 years after his death, we still admire Leonar­do da Vin­ci’s many and var­ied accom­plish­ments in paint­ing, sculp­ture, archi­tec­ture, sci­ence, and quite a few oth­er fields besides, most of which would have begun with his putting down some part of the for­mi­da­ble con­tents of his head on to a piece of paper. (As we’ve seen, some­times he need­ed to draw up a to-do list first.) Some of those works remained on paper, and even became famous in that hum­ble form. If you’ve only seen one of Leonar­do’s draw­ings, for instance, it’s almost cer­tain­ly Vit­ru­vian Man.

Leonar­do’s cir­ca-1490 study of the pro­por­tions of the human body — or to put it in more com­mon terms, the pic­ture of the naked fel­low stand­ing inside a square and a cir­cle — stands at an inter­sec­tion of art and math­e­mat­ics, one at which Leonar­do spent a great deal of time through­out his life. The Ted-ED les­son above, writ­ten by edu­ca­tor James Ear­le, gets into “the geo­met­ric, reli­gious and philo­soph­i­cal sig­nif­i­cance of this decep­tive­ly sim­ple draw­ing” whose title ref­er­ences the first-cen­tu­ry BCE Roman archi­tect and civ­il engi­neer Mar­cus Vit­ru­vius Pol­lio, who claimed that “the navel is the cen­ter of the human body, and that if one takes a com­pass and places the fixed point on the navel, a cir­cle can be drawn per­fect­ly around the body.”

Vit­ru­vius also real­ized that “arm span and height have a near­ly per­fect cor­re­spon­dence in the human body, thus plac­ing the body per­fect­ly inside a square as well.” Both he and Leonar­do saw real impli­ca­tions in this align­ment between anato­my and geog­ra­phy, begin­ning with the notion that build­ings and oth­er works of man should also take on these “per­fect” pro­por­tions. All of this ties in with the prob­lem, first pro­posed by ancient geome­ters, of “squar­ing the cir­cle,” that is, find­ing a pro­ce­dure to hand-draw a square and a cir­cle both of equal area. Leonar­do used Vit­ru­vian Man to point toward one pos­si­ble solu­tion using the human body.

You can learn more about the impor­tance and lega­cy of the draw­ing in the BBC doc­u­men­tary The Beau­ty of Dia­grams, avail­able on Youtube (part one, part two). “Although the dia­gram does­n’t rep­re­sent some huge sci­en­tif­ic break­through,” says its host, math­e­mati­cian Mar­cus du Sautoy, “it cap­tures an idea: that math­e­mat­ics under­pins both nature and the man­made world. It rep­re­sents a syn­the­sis of archi­tec­ture, anato­my, and geom­e­try. But it’s the per­fec­tion and ele­gance of Leonar­do’s solu­tion to this rid­dle of the square and the cir­cle in Vit­ru­vius which gives the dia­gram its pow­er and its beau­ty.” And judg­ing by the unabat­ed pop­u­lar­i­ty of Vit­ru­vian Man par­o­dies, it looks to have at least anoth­er half-mil­len­ni­um of rel­e­vance ahead.

Relat­ed Con­tent:

Down­load the Sub­lime Anato­my Draw­ings of Leonar­do da Vin­ci: Avail­able Online, or in a Great iPad App

Leonar­do da Vinci’s Bizarre Car­i­ca­tures & Mon­ster Draw­ings

How to Build Leonar­do da Vinci’s Inge­nious Self-Sup­port­ing Bridge: Renais­sance Inno­va­tions You Can Still Enjoy Today

Leonar­do da Vinci’s Vision­ary Note­books Now Online: Browse 570 Dig­i­tized Pages

Ralph Steadman’s Wild­ly Illus­trat­ed Biog­ra­phy of Leonar­do da Vin­ci (1983)

Based in Seoul, Col­in Mar­shall writes and broad­casts on cities and cul­ture. His projects include the book The State­less City: a Walk through 21st-Cen­tu­ry Los Ange­les and the video series The City in Cin­e­ma. Fol­low him on Twit­ter at @colinmarshall or on Face­book.

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  • David Bradley says:

    …except, of course, that Leonar­do aban­doned this avenue of rea­son­ing when he realised the per­fect pro­por­tions he assumed Vit­ru­vius had found were wrong. The dia­gram, although famous, shows noth­ing about human anato­my. It’s 500-year old #Fak­e­News. Just ask the cura­tor of the Queen’s col­lec­tion of Leonar­do da Vin­ci works…I para­phrase what he told me 5 years ago:

    Roman archi­tect Mar­cus Vit­ru­vius Pol­lio who sug­gest­ed that some­how the per­fect man could with arms and legs akim­bo tran­sect a per­fect cir­cle and a per­fect square. In his work, Leonar­do mea­sure lengths, ratios and angles but could not find the per­fect ratios sug­gest­ed by Vit­ru­vius 1500 years before. Instead, he obtained odd frac­tions 5/11’s, 7/17th’s none of which seemed to point to the per­fect cir­cle or the per­fect man and Leonar­do turned back from this dead-end.

  • anthony g cila says:

    thank you, a lot of infor­ma­tion made avail­able in one place, let me add to the dis­course (for your con­sid­er­a­tion):

    Hav­ing actu­al­ly seen Di Vin­ci’s Vit­ru­vian Man on dis­play in an Ital­ian muse­um (Venice) decades ago I real­ized copies/drawings of it do not reveal the hid­den genius of Di Vin­ci (it was carved into the thick paper not mere­ly drawn on it). It is a mas­ter­ful­ly drawn “rebus” of sorts, caus­ing many a learned man/woman has gazed up on its like­ness think­ing it was a 3D man inside a 2D square and 2D cir­cle (men­tal gym­nas­tics ensued … about round­ing the square or squar­ing the cir­cle. All miss­ing what was actu­al­ly there. The 3D man is drawn in a 3D cube and 3D sphere of equal vol­umes (that is but one of their rela­tion­ships); proved by the math:

    …Di Vin­ci’s nod to Archimedes AND Vit­ru­vian, if not to Tris­megis­tus too (ternary hides the qua­ter­nary):

    the cube vol­ume (L*W*H) 6*6*6= 216 and
    the sphere vol­ume (radius cubed*pi*4/3) 3.722(cubed)*3.142*(4/3)= 216 (all round­ed num­bers from the hun­dredths!)

    …round­ing the square/squaring the cir­cle (non­sense):
    cir­cum­fer­ence of the square (6*4) would require the cir­cum­fer­ence of the cir­cle to have a diam­e­ter of 7.64 (24/3.142),
    Di Vin­ci’s cir­cle has a diam­e­ter of 7.444 it’s a non-starter.

    .. along those lines, the gold­en ratio is also encod­ed (in plain view, not just encod­ed in the pyra­mid math)
    Di Vin­ci’s Vit­ru­vian Man (from the cen­ter of the circle/sphere down = 61.8% of the square!

    I did pen online (decades ago) my obser­va­tion of Di Vin­ci’ ‘Vit­ru­vian Man’ to be in a 3D cube and sphere of equal vol­umes but was ignored by the aca­d­e­m­ic dog­ma cham­pi­ons (dogged­ly hang­ing on to their “alleged con­sen­sus”) squar­ing the cir­cle (its side is 24.075% longer that the square’s side) or round­ing the square (a myopic view to be sure) … much like Strat­for­dians insist­ing Shake­speare is mag­i­cal­ly the work of a lone minor actor/wool mer­chant (Shak spar) attribut­ing him alone the pseu­do­nym/pen-name and there can be no oth­er (???). Like the sur­geon respond­ing to Pres­i­dent Rea­gan’s ques­tion (as he lay wound­ed at hos­pi­tal after an assas­si­na­tion attempt), “are you a repub­li­can?” … the response was, “we are all repub­li­cans now” … WE ARE ALL OXFORDIANS NOW TOO (those of us that can smell what the Rock is cook­ing, in the ver­nac­u­lar of the day).

    …draw­ing a square and cir­cle of equal areas, not a prob­lem… clear­ly not what he did. Like­wise, draw­ing a cube and sphere of equal vol­umes math is easy if you know the math. The RELATIONSHIP BETWEEN THEM HAVING BEEN REVEALED in DaVin­ci’s Vit­ru­vian Man is NO ACCIDENT/MERE COINCIDENCE, oth­ers assume he was round­ing the square etc. (all the while ignor­ing the obvi­ous). It is an illu­sion to think the 3D man is in mere­ly in a 2D square and a 2D cir­cle… he is in a 3D cube and 3D sphere of equal vol­ume val­ues 6*6*6=216=radius cubed*pi*(4/3). KISS (keep it sim­ple) … hid­den in plain sight…genius.

    DID YOU NOTICE: the sphere’s radius cubed (round­ed at thou­sandths) equals the degrees and min­utes of GIZA­’s pyra­mid slope?!?!
    …there are TOO MANY “COINCIDENCES” in the Vit­ru­vian man to be mere coin­ci­dences…
    …there is a lot “more” going on here than just a guy in a cou­ple rudi­men­ta­ry geo­met­ric shapes…
    …in addi­tion to di Vin­ci’s “round­ing to in his head” accu­ra­cy:

    anoth­er proof (con­cealed con­cept?) about vol­ume:
    SINCE: L*W*H= vol­ume of cube and radius cubed*pi*(4/3)= vol­ume of sphere
    THEN: the diam­e­ter sphere that equals any side of cube (it fits into) a sphere will always be 52.359% of the cube … pi/6=.52359

    radius cubed*pi*(4/3)= vol­ume
    (3.722 cubed*pi*(4/3))= vol­ume
    51.562*3.142*1.333=215.956 (who would­n’t round­ed that to 216)
    (note: pyra­mid angle 51.84 … is 3.72867 cubed maybe Da Vin­ci was off by a few hun­dredths from his radius)

    using his diam­e­ter for the vol­ume of the imag­i­nary square that sphere occu­pies 7.445*7.445*7.445= 412.661646125
    …imbed­ded pat­tern coin­ci­dence: I doubt it to be a par­i­ty check but worth not­ing.

    sequence of 4 #‘s fol­low) (1+2=) 3. (skip 6’s) 1 (skip 6) 4 (skip 6) 2 (round­ed up form 5) “3.142”
    (4–1) 3. (instruc­tion: skip next two (66)) 1 (skip one) 4 (skip one) 2 (round­ed up from 5) “3.142” is what he used … coin­ci­dences do hap­pen it appears (be aware when they are not).

    3 sep­a­rate pas­sages refer to the same peri­od of time “dif­fer­ent­ly””
    ONCE GENERICALLY (42 MONTHS … a 30-day win­dow), and
    …con­ceal­ing pi mas­ter­ful­ly…

    num­bers used in the ternary (3) con­ceal­ing the qua­ter­nary: 01223456 and a dec­i­mal some­where
    … zero thru six with two twos ??? per­haps some­thing is squared or root­ed, but what?
    … BEYOND ODDLY SPECIFIC THERE IS NO 7,8 or 9 (delib­er­ate­ly out of sight out of mind!!!)
    … rev 13:18 (of 666 fame) 13x18=234
    … like Da Vin­ci’s back­wards writ­ing (he did­n’t do it all the time, so I doubt it was to avoid smear­ing as he wrote
    … 234 becomes 432, add them togeth­er vio­la “666”
    …a par­i­ty check like rev 12:6 con­tain­ing 1260
    …now you know the real rea­son for the addi­tion of chap­ter and verse num­bers to the bible in the 1500’s
    … reversed the miss­ing 789 to get 987
    … 9.87 with the dec­i­mal … AND IT IS THE SQUARE OF 3.142 … PI

    BACK TO DI VINCI…squares in cir­cles in squares, etc. (aka flower of live in a cir­cle):

    vol­ume of a sphere (that has the diam­e­ter of 7.445)/volume of a cube (7.445 cubed) = % of cube’s vol­ume occu­pied by sphere

    216.02486/412.5785=.52359 (con­verse­ly, vol­ume of sphere*.52359=volume of cube that con­tains it)
    ((pi/6= .52359 …))

    Archimedes used a cube occu­pied by a sphere cut in half to deter­mine the rela­tion­ship between them
    (hav­ing a flat side (slice) to work with):

    vol­ume of sphere with same diam­e­ter as a 6x6x6 cube: 113.083722 (who would­n’t round that to 113)
    …113 is the 30th prime num­ber (per­haps he used prime #‘s as we used log­a­rithms … short cuts???)
    (5th cen­tu­ry AD Chi­nese math­e­mati­cian cal­cu­lat­ed pi to sev­en places)
    (per­haps Da Vin­ci was obliv­i­ous to Chi­nese advance­ments)
    (though Da Vin­ci 15th cen­tu­ry AD/CE pi looks bet­ter than Archimedes 4th cen­tu­ry BC/BCE approx­i­ma­tions)
    (being between 22/7 (3.1428) and 223/71 (3.1408))
    (was Da Vin­ci mock­ing Archimedes with his improved sphere vol­ume accu­ra­cy … Archimedes’ excelled in vol­ume)
    (by the 15th cent AD/CE 3.1415926 was known in Europe, some­thing already known in Chi­na since 3rd cent AD/CE)

    Look, sev­er­al hun­dreds of years and none of our math­e­mati­cians have stum­bled on Da Vin­ci’ truths in The Vit­ru­vian Man (sure­ly, I am not the 1st, mere decades ago). Archimedes was ham strung with his pi (22/7, 222/7); Da Vin­ci was not mere­ly regur­gi­tat­ing some­thing oth­ers built upon, he found a way to encode it in plain view.

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