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Article The Two Supreme of ’s —the One and the Indefinite Dyad—the of a Straight Line into Extreme and Mean , and Pingala’s Matr¯ ameru¯

Maria Antonietta Salamone Department of and Society, Faculty of Philosophy, Complutense University of Madrid, 28040 Madrid, Spain; salamonema@filos.ucm.es

 Received: 6 December 2018; Accepted: 8 January 2019; Published: 16 January 2019 

Abstract: The objective of this paper is to propose a mathematical interpretation of the continuous geometric proportion (, 32a) with which Plato accomplishes the goal to unify, harmonically and symmetrically, the Two Opposite Elements of Timaeus Cosmos— and —through the Mean Ratio. As we know, from the algebraic point of view, it is possible to compose a continuous geometric proportion just starting from two different quantities a (Fire) and b (Earth); their sum would be the third term, so that we would obtain the continuous geometric proportion par excellence, which carries out the agreement of opposites most perfectly: (a + b)/a = a/b. This equal proportion, applied to linear , corresponds to what called the Division into Extreme and Mean Ratio (DEMR) or The Golden Proportion. In fact, according to my mathematical interpretation, in the Timaeus 32b and in the 991 a–b, Plato uses Pingala’s Matr¯ ameru¯ or The First Analogy of the Double to mould the body of the Cosmos as a whole, to the pint of identifying the two supreme principles of the Cosmos—the One (1) and the Indefinite Dyad (Φ and1/Φ)—with the DEMR. In effect, Fire and Earth are joined not by a single Mean Ratio but by two (namely, and ). Moreover, using the Platonic approach to analyse the geometric properties of the shape of the Cosmos as a whole, I think that Timaeus constructed the 12 pentagonal faces of by means of elementary Golden (a/b = Φ) and the Matr¯ ameru¯ sequence. And, this would prove that my mathematical interpretation of the platonic texts is at least plausible.

Keywords: Plato; ; Timaeus; Epinomis; Justice; Equality; Division in Extreme and Mean Ratio; Golden Proportion; Pingala’s Matr¯ ameru;¯ the One and the Indefinite Dyad

1. Introduction: The of Justice and the Paradigm of the Line of the Horizon In my recent research [Salamone, 2017: 27–30] I proposed a cosmological interpretation of the term díke¯ in , in agreement with , which establishes the synonymy between justice and equality, through the use of the paradigm of the Divided Line. I analysed the etymological origin of the Greek word díke¯ which derives from the Sanskrit root word * di´s-(dik)whose refers to the «cosmological of the Line of the Horizon, that is to say, the apparent boundary line that divides the Cosmos into two equal parts: the Earth and the Sky». Furthermore, I tried to demonstrate the philosophical relationship between the idea of cosmic justice and the paradigm of the Line of the Horizon from the point of view of the founders of Greek scientific . In my opinion, the «original conception representing justice as a divine division of the Cosmos into two equal parts, or cosmic dasmós, turns out to be both spatial (the Sky is separated from the Earth) and temporal (Day is split from Night), and has its roots in ancient not only of Greek origin, but also Indo-Iranian, Hindu, Old Persian, Egyptian, Babylonian and Chinese ( ... ). The representation is as

Symmetry 2019, 11, 98; doi:10.3390/sym11010098 www.mdpi.com/journal/symmetry Symmetry 2019, 11, x 2 of 13 Symmetry 2019, 11, 98 2 of 13 but also Indo-Iranian, Hindu, Old Persian, Egyptian, Babylonian and Chinese (…). The follows: the Cosmosrepresentation began as is an as undifferentiated follows: the Cosmos mass, began without as anyan undifferentiated internal boundaries mass, or without limits any internal (’sboundaries boundless or ). limits (Anaximander’s This mass separated boundless into univ two parts,erse). This Earth mass and separated Sky, which into were two parts, Earth opposed or contrary,and Sky, male which and were female. opposed Finally, or contrary, the male male and theand female female. were Finally, united the bymale , and the the female were contraries wereunited combined, by Eros, and the gave contraries birth to were the individual combined, and gave of birth , to orthe of individual things» [1 existence]. It is of Gods, or probable that Platoof things» refers [1]. to theIt is paradigmprobable that of the PlatoLine refers of the to Horizon the paradigmin his of the Line when of the he Horizon speaks in his Republic of the Dividedwhen Line tohe explain speaks his of cosmologicalthe Divided Line doctrine to explain of his [2 ].cosmological In this way, doctri politicalne andof ideas moral [2]. In this way, questions intertwinepolitical with and cosmological moral questions and metaphysicalintertwine with ones, cosmological as prescribed and at metaphysical the beginning ones, of the as prescribed at Book of the Timaeusthe beginningwith reference of the toBookThe of Republic the Timaeus. with reference to The Republic.

2. Plato’s Cosmology2. Plato’s and Cosmology the Division and into the ExtremeDivision and into Mean Extreme Ratio and (DEMR) Mean Ratio (DEMR) Against the innumerableAgainst the coexistinginnumerable coexisting of theuniverses Atomists, of the and Atomists, the succession and the of singlesuccession of single universes whichuniverses had figured which in had some figured Ionian in some Ionian and in syst ,ems and in Plato Empedocles, contemplated Plato thecontemplated the Cosmos as a singleCosmos and as unique a single living and creature:unique living « creature: has come «Heaven to be has and come is and to shallbe and be is hereafter and shall be hereafter one and unique»one [3 and]. According unique» [3]. to the According ’s to the views,philosopher’s the creator views, made the onlycreator one made copy only in order one copy in order that the Cosmosthat should the Cosmos resemble should its model resemble in respect its model of its in uniqueness. respect of its Uniqueness uniqueness. is ,Uniqueness and is perfection, and the Cosmos isthe marvellous Cosmos foris marvellous possessing for it. Inpossessing fact, if it it. were In fact, not unique,if it were there not wouldunique, be there body would left be body left outside it, whoseoutside strong it, powerswhose strong might powers impair itsmight andimpair even its destroylife and it. even That destroy is why it. the That Cosmos is why the Cosmos must be (1) wholemust and be complete,(1) whole consistingand complete, of parts consisting each of whichof parts is each whole of and which complete; is whole (2) and single complete; or (2) single unique (not oneor of unique many coexistent(not one of ); many (3)coexistent everlasting worlds); (not destroyed (3) everlasting and superseded (not destroyed by another and superseded by ), which itanother could hardlyworld), be which if it were it could exposed hardly to assaultsbe if it were from exposed outside. to Also, assaults Aristotle from explained outside. Also, Aristotle that there is noexplained body, place, that or there void is or no body, outside place, the or Cosmos void or [time4]. outside the Cosmos [4]. Furthermore, PlatoFurthermore, reduced thePlato four reduced material the Elements four material of Empedocles Elements of to Empedocles two from start—as to two from did start—as did , whoParmenides, postulated who Fire postulated and Earth—and Fire and made Earth— “theand middle” made a “the blend middle” of these. a blend of these. In fact, for Empedocles,In fact, for who Empedocles, postulated who four postulated elements: four elements:

Fire and Earth, moreover,Fire and Earth, are extremes moreover, and are purest: extremes Water and and purest: Air, onWa theter contrary,and Air, on are the intermediates contrary, are intermediates and more like blends.and more The like same blends. course The is same followed course by is those followed who by advocate those who three advocate (we may three compare (we may compare what what Plato doesPlato in the does Divisions: in the forDivisions: he makes for “the he middle”makes “the a blend). middle” Indeed a blend). there is Indeed practically there no is practically no difference betweendifference those who between postulate those two who and postulate those who two postulate and those three, who exceptpostulate that three, the former except split that the former split the middle elementthe intomiddle two, element while theinto latter two, treatwhile it the as latter only onetreat[De it as Generatione only one [De et Generatione Corruptione, et Corruptione, 330b] [5]. 330b] [5].

Plato deducesPlato the need deduces of two the Primary need of Elementstwo Primary by theElements following by the argument. following (1) argument. There must (1) There must be be two (not onetwo primary (not one form primary of , form asof thematter, Ionian as the monists Ionian had monists held) had because held) Firebecause is needed Fire is to needed to make make the world’sthe world’s body visible, body visible, and Earth and to Earth make toit make tangible. it tangible. Fire and Fire Earth and Earth had been had been commonly commonly regarded regarded as theas two the extreme two extreme and contrary and contrary elements elements (or end elements), (or end elements), since Fire belongssince Fire to the belongs Sky in to the Sky in opposition to theopposition Earth; (2) to But, the twoEarth; cannot (2) But, hold two together cannot withouthold together a middle without element a middle which element serves as which serves as bond, becausebond, the Cosmos because is athe three-dimensional Cosmos is a three-dimensional space. Also, the spac threee. Also, terms the must three be terms in proportion, must be in proportion, and the most perfectand the bond most is perfect that proportion bond is that which proportion makes the which most makes perfect the unity most between perfect unity the middle between the middle and extreme elements.and extreme Actually, elements. according Actually, to Euclid, according in order to Euclid, to construct in order aproportion, to construct at a leastproportion, three at least three terms are necessary;terms (3) are The necessary; most perfect (3) The type most of proportion perfect type is theof proportion geometrical is continued the geometrical proportion continued [6], proportion which Plato next[6], proceeds which Plato to define, next proceeds as follows: to define, as follows: Hence the , whenHence he the began god, when to put he together began to the put body together of the the Universe, body of the set Universe, about making set about it on making Fire it on Fire and and Earth. ButEarth. two things But two alone things cannot alone be cannot satisfactorily be satisfactorily united withoutunited without a third; a forthird; there for mustthere bemust be some bond some bond betweenbetween them them drawing drawing them them together. together. And And of allof all bonds bonds the th beste best is is that that which which makes makes itself itself and the terms it and the terms itconnects a unity in the fullestfullest sense; and itit isis thethe naturenature ofof aa continuedcontinued proportionproportion toto effect this most effect this most perfectlyperfectly (ττοῦτοoῦτo δὲδε` πέφυκεν πεϕυκεν´ ἀναλογίαναλoγία κάλλιστα καλλιστα´ ἀποτελεῖνπoτελεῖν)) [Timaeus,[Timaeus, 31b]. 31b].

In fact, accordingIn fact, to Huffman: according «Atto Huffman: Timaeus «At 31c–32c Timaeu it iss 31c–32c clear that it is the clear term thatἀναλ theo termγία means ἀναλογία means the the continuouscontinuous geometric ἀgeometricναλoγία whatἀναλογία we would what callwe awould continuous call a geometriccontinuous proportion, geometric orproportion, just or just a a continuous proportioncontinuous (Cornford, proportion 1937: (Cornford, 45ff). When 1937: Plato45ff). discussesWhen Plato the arithmeticdiscusses the and arithmetic harmonic and harmonic means at Timaeus 36a ff, he does not use the term ἀναλογία. Similarly at EN 1131 a31, Aristotle

Symmetry 2019, 11, 98 3 of 13 meansSymmetry at 2019 Timaeus, 11, x 36a ff, he does not use the term ἀναλoγία. Similarly at EN 1131 a31, Aristotle3 of 13 defines ἀναλoγία as “equality of ratio”, and, a few lines later, he tells us that “the call thisdefines sort ἀναλογία of ἀναλoγ ίasα geometrical“equality ofἀ ratio”,ναλoγ and,ία” (K aαλ fewoῦ σιlines δὲ τlater,ὴν τ oheια ύtellsτην usἀναλ thatoγ “theίαν γεωµετρικmathematiciansὴν oἱ µαθηµατικcall this sortoί, 1131of b13)»ἀναλογία [7]. geometrical ἀναλογία” (Καλοῦσι δὲ τὴν τοιαύτην ἀναλογίαν γεωμετρικὴνTo tell the οἱ , μαθηματικοί here, as,in 1131 many b13)» other [7]. places, Plato is compressing his statement of technical mattersTo totell such the truth, a point here, that as only in many expert other readers places could, Plato fully is appreciate compressing its meaning.his statement Therefore, of technical I am tryingmatters to to explain such a mypoint interpretative that only expert hypothesis readers in could accordance fully appreciate with the its newer meaning Tübingen. Therefore, paradigm, I am whichtrying givesto explain greatest my importance interpretative to Plato’s hypothesis so-called in Agraphaaccordance Dogmata with (Unwrittenthe newer Tübingen Lectures) orparadigm, Lectures Onwhich the Goodgives[ 8greatest]. The question, importance actually, to Plato’s is as follows: so-called what Agrapha kind of Dogmata continuous (Unwritten geometric Lectures) proportion or canLectures accomplish On the theGood goal [8]. to The unify question, harmonically actually, the is twoas follows: opposite what Elements, kind of Fire continuous and Earth? geometric As we know,proportion from can the accomplish algebraic point the goal of view, to unify it is harmon possibleically to compose the two a opposite continuous Elements, geometric Fire proportionand Earth? justAs we starting know, from from two the different algebraic quantities point of A (Fire)view, andit is B possible (Earth); theirto compose sum would a continuous be the third geometric term A + Bproportion (Fire + Earth), just starting so that wefrom would two different obtain the quantiti continuouses A (Fire) geometric and B proportion(Earth); theirpar sum excellence would, whichbe the carriesthird term out theA + agreement B (Fire + Earth), of opposites so that most we perfectly:would obtain the continuous geometric proportion par excellence, which carries out the agreement of opposites most perfectly: (A(A + + B):AB):A == A:BA:B This equal proportion, applied to linear geometry, correspondscorresponds toto whatwhat EuclidEuclid calledcalled the Division inin ExtremeExtreme andand MeanMean RatioRatio (DEMR)(DEMR) oror thethe Golden ProportionProportion (Figure(Figure1 1):): «A «A straight straight line line is is said said to to have have beenbeen divided in extreme extreme and and mean mean ratio, ratio, when when the the ra ratiotio of ofthe the whole whole line line to the to the larger larger segment segment is the is thesame same as the as theratio ratio of ofthe the larger larger segment segment to to the the sm smalleraller segment» segment» [9]. [9]. The The Golden Proportion, alsoalso knownknown as thethe DivineDivine Proportion,Proportion, is thethe irrationalirrational number 1.618 whichwhich is represented by the Greek LetterLetter PhiPhi ((ΦΦ).

Figure 1. The Division of a straight Line into Extreme andand MeanMean Ratio.Ratio.

InIn agreementagreement withwith manymany scholarsscholars like Brunes [[1100],], Neufert [[11],11], Michel [[12],12], Herz-Fischler and, Timaeus moremore recently,recently, Gaiser Gaiser [13 [13],], Krämer Krämer [14 ][14] and and Olsen Olsen [15], [15], in in Timaeus32a, Plato 32a, may Plato allude may to allude the Division to the ofDivision a line intoof a Extreme line into and Extreme Mean Ratioand Mean to explain Ratio the to unity explain and the harmony unity ofand opposite harmony Elements of opposite (Fire andElements Earth), (Fire obtained and Earth), thanks obtained to “the middlethanks one”:to “the the middle Golden one”: Ratio. the Golden Ratio. This isis mymy mathematicalmathematical interpretationinterpretation ofof Plato’sPlato’s well-knownwell-known fragment:fragment: For whenever,whenever, ofof threethree numbers,numbers, thethe middlemiddle oneone [1][1] betweenbetween anyany twotwo [1.618[1.618 andand 0.618]0.618] thatthat areare eithereither solidssolids oror planesplanes ()(squares) isis suchsuch thatthat asas thethe firstfirst isis toto it,it, soso isis itit toto the last [1.618:1 = 1:0.618] 1:0.618],, andand converselyconversely asas thethe lastlast isis toto thethe middle,middle, soso isis thethe middlemiddle toto thethe firstfirst [0.618:1[0.618:1 == 1:1.618],1:1.618], thenthen sincesince thethe middlemiddle becomesbecomes firstfirst andand lastlast [1:1.618=0.618:1],[1:1.618=0.618:1], and again the last and firstfirst becomebecome middlemiddle [1:0.618[1:0.618 == 1.618:1] , in that way all will necessarily come to play the same part towards one another, andand byby soso doingdoing theythey willwill allall makemake aa unityunity[Timaeus, [Timaeus, 32a].32a]. InIn fact,fact, thethe intelligibleintelligible formform of thethe GoldenGolden Proportion is that all termsterms are interchangeableinterchangeable and, beingbeing interchangeable,interchangeable, all formform aa harmoniousharmonious unity.unity. Moreover, the numbersnumbers ofof thisthis geometricalgeometrical proportion «are either solids or planes», which means that they apply not only to linear geometry but also to plane geometry and solid geometry, as we shall see in a moment. Golden Proportion: 1.618:1 = 1:0.618 = Φ

Symmetry 2019, 11, 98 4 of 13 proportion «are either solids or planes», which means that they apply not only to linear geometry but also to plane geometry and solid geometry, as we shall see in a moment.

Golden Proportion: 1.618:1 = 1:0.618 = Φ

Conversely: 0.618:1 = 1:1.618 = 1/Φ

Alternately: 1:0.618 = 1.618:1 = Φ

Alternately: 1:1.618 = 0.618:1 = 1/Φ

In cosmological terms, this indicates that the Cosmos is a multiplicity deriving from a unity of opposites (Fire and Earth), according to the old cosmogony. Actually, and in accordance with the Pythagorean of Croton—who said that «Harmony in every way arises out of opposites. For harmony is the unification of what is a mixture of many ingredients and the agreement of the disagreeing» [16], the cause of harmony is the agreement of the Elements, and the for this agreement is their equal proportion (isomoiria) and the fact that no one of them is more powerful than any other (isonomia). Plato’s Cosmos is characterized precisely by this superior harmony which derives from the conjunction/unification of opposite Elements through «the middle intelligible one» [17]. In my opinion, it is also impossible to solve this geometric of Plato if we do not consider Aristotle’s arguments, which explain not only that the divisibility and continuity of magnitudes depend on the number of dimensions—one sort continuous in one dimension (longitude/linear geometry), another in two (longitude and altitude/plan geometry), and another in three or all dimensions (longitude, altitude and depth/solid geometry)—but also that the triad is the divine number of the whole Cosmos. The Cosmos, in fact, is completely perfect because it is a three-dimensional space, which is divisible into all dimensions.

The Matr¯ ameru¯ Sequence and the Construction of the Body of the Timaeus Cosmos If the Cosmos is completely perfect due to the fact that it is a three-dimensional space, it consequently follows that (as Parmenides said) beyond the Cosmos, there is no other magnitude, since the three dimensions are all there is, and what is divisible into three dimensions (longitude, altitude and depth) is divisible into all. Cosmologically, this implies that the Timaeus Cosmos is one and unique, and not one of the innumerable coexisting worlds of the Atomists. Once again, the Cosmos is perfect because of its uniqueness, that is because it is based on a unique model (the Form of living being). Moreover, this is the reason why Plato says that the shape of the Cosmos must be solid in its form, that is, Fire and Earth are joined not by a single geometric mean but by two (namely, Air and Water):

Now, if it had been required that the body of the Universe should be a plane surface with no depth, a single mean would have been enough to connect its companions and itself; but in fact the world was to be solid in form, and solids are always conjoined, not by one mean, but by two. Accordingly the god set water and air between fire and earth, and made them, so far as was possible, proportional to one another, so that as fire [A] is to air [B], so is air [B] to water [C] and as air [B] is to water [C], so is water [C] to earth [D], and thus he bound together the frame of a world visible and tangible [Timaeus, 32b].

According to Bury, “two mean terms are required for a continuous proportion of ‘solid’ (or cubic) numbers” [18]. Mathematically, four quantities A, B, C and D are said to be divided into Golden Proportion, if the ratio of the first term and the second term is equal to the ratio of the second term and the third term, which is equal to the ratio of the third term and the fourth term (A/B = B/C = C/D = Φ). For example, in Pingala’s Matr¯ ameru,¯ which was discovered around 400–300 B.C. (?) [19], the consecutive numbers 1, 2, 3, 5, 8, 13 are in Golden Proportion because 1/1, 2/1, 3/2, 5/3, 8/5, 13/8 = approximately equal to Φ (as well 1/1, 1/2 2/3, 3/5, 5/8, 8/13 = approximately equal to 1/Φ). In fact, the ratio of any two sequential Matr¯ ameru¯ numbers approximates to the of Φ. The larger Symmetry 2019, 11, x 2 of 13 but also Indo-Iranian, Hindu, Old Persian, Egyptian, Babylonian and Chinese (…). The representation is as follows: the Cosmos began as an undifferentiated mass, without any internal boundaries or limits (Anaximander’s boundless universe). This mass separated into two parts, Earth and Sky, which were opposed or contrary, male and female. Finally, the male and the female were united by Eros, the contraries were combined, and gave birth to the individual existence of Gods, or of things» [1]. It is probable that Plato refers to the paradigm of the Line of the Horizon in his Republic Symmetry 2019, 11, 98 5 of 13 when he speaks of the Divided Line to explain his cosmological doctrine of ideas [2]. In this way, politicalSymmetry and 2019 moral, 11, x questions intertwine with cosmological and metaphysical ones, as prescribed5 ofat 13 the beginning of the Book oftheSymmetry the consecutive Timaeus 2019, 11 with, x numbers reference in the to sequence,The Republic the. more accurate the approximation of Φ (Table1). Let5 of me 13 try a few: Table 1. Pingala’s Mātrāmeru. 2. Plato’s Cosmology and the Division into Extreme and MeanTable Ratio 1. (DEMR)Pingala’s Mātrāmeru. A B B/A TableA 1.BPingala’s M B/Aatr¯ ameru.¯ Against the innumerable coexisting1 universes1 of the 1 Atomists, and the succession of single 1 1 1 universes which had figured in some Ionian1 2syst ems and 2 in AEmpedocles, B Plato contemplated B/A the 1 2 2 Cosmos as a single and unique living creature:2 3 «Heaven has 1.5 come1 to be 1and is and shall 1 be hereafter 2 3 1.5 one and unique» [3]. According to the philosopher’s3 5 1.666666666 views, the1 …creator made 2 only one 2 copy in order 3 5 1.666666666 … that the Cosmos should resemble its model5 in resp8 ect of its 1.6 uniqueness. 2 Uniqueness 3 is 1.5 perfection, and the Cosmos is marvellous for possessing it. In fact, if it were 3not5 unique,8 5 there 1.666666666 would 1.6 be . .body . left 8 13 1.625 5 8 1.6 outside it, whose strong powers might impair its life and even destroy8 13 it. That 1.625is why the Cosmos … … … 8 13 1.625 … … … must be (1) whole and complete, consisting144 of233 parts 1.618055556 each of which...... … is whole and complete; (2) single 144 233 1.618055556 … or unique (not one of many coexistent233 worlds); 377 (3)1.618025751 everlasting144 … (not destroyed 233 1.618055556 and superseded . . . by another world), which it could hardly be if it were exposed to233 assaults233 377 377 from 1.618025751 outside. 1.618025751 Also, … . .Aristotle . explainedThe that Mā trthereāmeru, is no or body,the place, or sequence, void or time is itself outside simple the to Cosmos follow. [4].In this sequence (1, 1, 2, 3, 5, The Mātrāmeru, or the Fibonacci sequence, is itself simple to follow. In this sequence (1, 1, 2, 3, 5, 8, 13,Furthermore, 21, 34, 55, 89, Plato 144, reduced …, ∞),The each the M four numberatr¯ ameru,¯ material is the or Elem thesum Fibonaccients of the of twoEmpedocles sequence, preceding isto itselfnumbers two from simple (for start—as to instance, follow. did In1 + this sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …, ∞), each number is the sum of the two preceding numbers (for instance, 1 + Parmenides,1 = 2, 2 + 3 =who 5, 3 postulated + 5 = 8). 8,As 13,Fire you 21, and go 34, farther Earth— 55, 89, andand 144, fart made...her, ∞ to“the), eachthe middle” right number in athis blend is sequence, the of sum these. of the the ratio two of preceding a term numbers (for instance, 1 = 2, 2 + 3 = 5, 3 + 5 = 8). As you go farther and farther to the right in this sequence, the ratio of a term to Inthe fact, one forbefore Empedocles, it will get1 +who closer 1 = postulated 2, and 2 + 3closer = 5, four 3to + Φelements: 5. =Furthering 8). As you this go observation, farther and shapes farther can to the be rightcreated in this sequence, the ratio to the one before it will get closer and closer to Φ. Furthering this observation, shapes can be created basedFire uponand Earth, length moreover, measurementsof are a term extremes toof the theand one Mpurest:ā beforetrāmeru Wa itter willnumbers andget Air, closer onin thesequence. andcontrary, closer For are to example,intermediatesΦ. Furthering creating this a observation, shapes can based upon length measurements of the Mātrāmeru numbers in sequence. For example, creating a rectangleand more with like the blends. values Thebe of same createdany course of the based istwo followed uponsuccessive by length those M ā measurementswhotrāmeru advocate numbers three of (we theforms may Matr¯ what compareameru¯ is known what numbers as the in sequence. For example, with the values of any of the two successive Mātrāmeru numbers forms what is known as the “GoldenPlato does Rectangle”. in the Divisions: Youcreating can forstart ahe rectanglewith makes a “the with middle” theand values add a blend). a ofsquare any Indeed of of the thethere two same is successive practically size to form M noatr¯ aameru¯ new numbers forms what is “”. You can start with a square and add a square of the same size to form a new rectangledifference (Figure between 2). those Continue knownwho postulate adding as the two “Goldensquares and those whos Rectangle”. whoe sidespostulate Youare three,the can length startexcept with thatof the athe square longerformer and splitside add of athe square of the same size to rectangle (Figure 2). Continue adding squares whose sides are the length of the longer side of the rectangle;the middle the element longer into side formtwo, will while a newalways the rectangle latter be treata (Figuresuccessive it as only2). Continue Moneā tr[Deāmeru Generatione adding number. squares etEventually, Corruptione, whose sides the are large the length of the longer side rectangle; the longer side will always be a successive Mātrāmeru number. Eventually, the large rectangle330b] [5]. formed will lookof like the a rectangle; Golden Rectangle— the longer sidethe longer will always you continue, be a successive the closer Matr ¯it willameru¯ be. number. Eventually, the large rectangle formed will look look like like a a Golden Golden Rectangle— Rectangle—thethe longer longer you you continue, continue, the the closer closer it itwill will be. be. Plato deduces the need of two Primary Elements by the following argument. (1) There must be two (not one primary form of matter, as the Ionian monists had held) because Fire is needed to make the world’s body visible, and Earth to make it tangible. Fire and Earth had been commonly regarded as the two extreme and contrary elements (or end elements), since Fire belongs to the Sky in opposition to the Earth; (2) But, two cannot hold together without a middle element which serves as bond, because the Cosmos is a three-dimensional space. Also, the three terms must be in proportion, and the most perfect bond is that proportion which makes the most perfect unity between the middle and extreme elements. Actually, according to Euclid, in order to construct a proportion, at least three terms are necessary; (3) The most perfect type of proportion is the geometrical continued proportion Figure 2. Mātrāmeru Squares.Figure 2. Matr¯ ameru¯ Squares. [6], which Plato next proceeds to define, as follows: Figure 2. Mātrāmeru Squares. HenceIn my the god,opinion, when ithe isbegan possible Into put my that,together opinion, in the it body book is possible of Epinomisthe Universe, that,, Plato inset the aboutrefers book making preciselyEpinomis it on Fireto, Platothe and Golden refers precisely to the Golden In my opinion, it is possible that, in the book Epinomis, Plato refers precisely to the Golden ProportionEarth. But andtwo thingsto M aloneātrProportionāmeru cannot numbers, be and satisfactorily to M whenatr¯ ameru¯ united he numbers,withoutexplains a third; whenthe differencefor he there explains must between thebe some difference bondgeometric, between geometric, arithmetic Proportion and to Mātrāmeru numbers, when he explains the difference between geometric, arithmeticbetween them and drawingharmonic themand sequen together. harmonicces And of sequencesnumbers. of all bonds ofIn th numbers.fact,e best in is thatThe In whichFirst fact, inAnalogymakesThe Firstitself of Analogytheand Doublethe terms of, the that itDouble is, in, that is, in the Matr¯ ameru¯ arithmetic and harmonic sequences of numbers. In fact, in The First Analogy of the Double, that is, in theconnects Mātrāmeru a unity sequence in the fullestsequence in which sense; in andthe which itnew is the the number new number isof thea continued isprevious the previous proportion two twoadded to added effect together this together most (ἡ μὲν (ἡ µ εν`δὴ δὴ πρώτη τoῦ διπλασίoυ the Mātrāmeru sequence in which the new number is the previous two added together (ἡ μὲν δὴ πρώτηperfectly τοῦ ( τοῦτοδιπλασίου δὲ πέφυκεν κατ᾿κατ ᾿ἀριθμὸν ἀναλογίαριθµὸν ἓν νκάλλιστα πρπρὸςὸς δδύούo ἀποτελεῖνκατ κατὰα` λ óλόγονγo) ν[Timaeus, ϕερ φερομένηoµενη´ 31b].), the), the first first two two sequential sequential numbers are 1 and 1, and the πρώτη τοῦ διπλασίου κατ᾿ ἀριθμὸν ἓν πρὸς δύο κατὰ λόγον φερομένη), the first two sequential numbers are 1 and 1, andscale the proceedsscale proceeds up to numberup to nu 8.mber This 8. is This my mathematicalis my mathematical interpretation interpretation of Plato’s problematic fragment: In fact, according to Huffman:numbers «At are Timaeu1 and 1,s and31c–32c the scaleit is clear proceeds that the up termto nu ἀναλογίαmber 8. This means is my the mathematical interpretation of Plato’s problematic fragment: continuous geometric ἀναλογίαof Plato’sAnd what thereforeproblematic we would there fragment: willcall bea continuous need of studies: geometric the most proportion, important and or firstjust is,a in fact, of numbers in continuousAnd thereforeproportion there (Cornford, will bethemselves; need 1937: of studies: 45ff). not of Whthe those mosten whichPlato important arediscusses corporeal, and firstthe but is,arithmetic of in the fact, whole of and numbers origin harmonic of thein odd and the even, and the And therefore there will be need of studies: the most important and first is, in fact, of numbers in means themselves;at Timaeus not 36a of ff,those he whichdoesgreatness arenot corporeal, use of their the influencetermbut of ἀναλογίαthe on whole the nature origin. Similarly ofof .the oddat ENWhen and 1131 the he even, hasa31, learnt andAristotle the these things, there comes next themselves; not of those which are corporeal, but of the whole origin of the odd and the even, and the greatness of their influenceafter on the these nature what of they realit cally. byWhen the veryhe has ridiculous learnt these name things, of geometry, there comes when it proves to be a manifest greatness of their influence on the nature of reality. When he has learnt these things, there comes next after these what theylikening call by ofthe numbers very ridiculous not like onename another of geometry, by nature when by referenceit proves toto thebe provincea of planes; and this next after these what they call by the very ridiculous name of geometry, when it proves to be a manifest likening of numberswill not be clearlylike one seen another by him by nature who is by able reference to understand to the province it to be a of marvel planes; not of human, but of divine manifest likening of numbers not like one another by nature by reference to the province of planes; and this will be clearly seenorigin. by him And who then,is able after to un that,derstand the numbers it to be a thricemarvel increased not of human, and like but to of the solid nature, and those and this will be clearly seen by him who is able to understand it to be a marvel not of human, but of divine origin. And then, afteragain that, which the havenumbers been thri madece increased unlike, he and likens like by to another the solid art, nature, namely, and that which its adepts called divine origin. And then, after that, the numbers thrice increased and like to the solid nature, and those again which have beenstereometry; made unlike, and he a divine likens and by marvellousanother art, thing namely, it is tothat those which who its envisage adepts it and reflect how the whole those again which have been made unlike, he likens by another art, namely, that which its adepts called stereometry; and a divine and marvellous thing it is to those who envisage it and reflect how called stereometry; and a divine and marvellous thing it is to those who envisage it and reflect how the whole of nature moulds off species and class, as power [the One: 1] and its opposite [the the whole of nature moulds off species and class, as power [the One: 1] and its opposite [the Indefinite Dyad: Φ and1/Φ] continually turn upon the double according to each analogy. Thus the Indefinite Dyad: Φ and1/Φ] continually turn upon the double according to each analogy. Thus the

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of nature moulds off species and class, as power [the One: 1] and its opposite [the Indefinite Dyad: Φ and1/Φ] continually turn upon the double according to each analogy. Thus the first analogy is of the double [in point of number: 1, 1], passing by numerical in the proportion of one to two [1 + 1 = 2], and that which is according to power is double; that which passes to the solid and tangible is likewise again double having proceeded from one to eight [1,1,2,3,5,8]. But, that passing to a mean of the double, as much more than the less as it is less than the greater, [in fact, in the geometric sequence of Matr¯ ameru,¯ between 3 and 8 the geometric mean is 5, which is 2 more than 3 and 3 less than 8] while the other mean exceeds and is exceeded by the same portion of the extremes themselves—between six (6) and twelve (12) comes the whole-and-a-half [in the arithmetic sequence, between 6 and 12 the arithmetic mean is 9, which is 3 more than 6 and 3 less than 12] and whole-and-a-third [in the harmonic sequence, between 6 and 12 the harmonic mean is 8, which is 6 plus 1/3 of 6 and 12 minus 1/3 of 12]—turning between these very two, to one side or the other, this analogy assigned to men an accordant and proportioned use for the purpose of rhythm and harmony in their pastimes, and has been bestowed by the blessed dance of the Muses [Epinomis, 990e-991b] [20].

In order to better understand the implication of Matr¯ ameru¯ consecutive numbers in Platonic cosmology, we have to consider Plato’s , which is the body of the Cosmos as existing before the Cosmos was made or the designed the geometrical figures of the Primary Bodies [21]. In fact, at this point, Plato clarifies the existence of the Receptacle of all Becoming (Timaeus, 48e-49) which is compared to a Mother (Timaeus, 50d) of what has come to be visible and otherwise sensible (Timaeus, 51b). The Receptacle is an intermediate order among the world of Being which «always is real and has no becoming» and the world of Becoming, which «becomes and passes away, but never has real being». The point is that all that becomes «must needs become by the agency of some cause: for without a cause nothing can come to be» (Timaeus, 28a-9). Plato finally identifies the Receptacle with Space «which is everlasting, not admitting destruction; providing a situation for all things that come into being, but itself apprehended without the senses by a sort of bastard reasoning, and hardly an of belief» (Timaeus, 51–52b). In accordance with Aristotle, Space is, for Plato, the place, the medium or substratum—formless and unshapen—in which the Elements are conceived as a material for composing things, or in which all becoming takes place. Before the creation of the Cosmos, the elements are present in indefinite form; their is disorderly. There is no substance yet, but a flux of shifting qualities—Fire is hot, Air is cold, Water is wet and Earth is dry—appearing and vanishing in permanent Space.

3. The Two Material Elements of Timaeus Cosmos, and the Shapes of the Four Primary Bodies of the Universe Plato rejected the old Milesian doctrine of a single fundamental form of matter (the ), which was to serve both as the original Element of the cosmos and as the permanent ground underlying change. As we know, Thales of had held Water to be a component of all things. of Colophon had posited that all things came to be from Earth. of had considered Fire as the most fundamental Element, and Anaximenes of Miletus had believed that Air was underlying all other things. Plato only partially accepted the belief of the Pythagorean pluralists, like Parmenides of Elea, who had postulated that the material Elements of the cosmos were two: Fire and Earth; or Empedocles of Akragas, who had suggested that in the world was created from the four Elements: Fire, Earth, Air and Water. However, and in contrast to them, Plato admitted the possibility of the transmutation of Elements. In fact, Parmenides’ cosmic space consisted of being, which did not change, did not become, did not move, always remained the same and was everlasting; whereas Empedocles had assumed that although the four Elements could be mixed together in various proportions, and the Elements themselves were unbreakable and could never be changed. In contrast, Plato did not hold material Elements to be really primary, permanent and with a nature, Symmetry 2019, 11, 98 7 of 13

and worked hard to prove it. In fact, Plato’s theory allowed primary Elements to be transmuted and Symmetry 2019, 11, x 7 of 13 recombined into different types of bodies:

generation,We must, but in fact, we speak consider as if in men itself knew the naturewhat fire of fireand andeach water, of the airothers and is, earth, positing before them the generationas original of principles,the Heaven elements and their (as it conditions were, letters) before of the the univ Heavenerse; was.whereas For one to this who day has noever one so haslittle explained their shouldgeneration, not rank but them we speak in this as analogy if men knew even whatso low fire as andsyllables each of[Timaeus, the others 48 is, b3]. positing them as original Theprinciples, unexplained elements existence (as it were, of letters) the four of the Elements universe; whereashad been one whotaken has as ever the so starting little intelligence point for cosmogony,should their not rank properties them in thisand analogybehaviour even had so low been as syllablesassumed[Timaeus, as if men 48 really b3]. knew what Fire and Earth were.The unexplained Plato now existencedenies them of the the four status Elements of «Pri hadmary been Elements» taken as the(from starting this pointmoment, for cosmogony,Timaeus willtheir call properties them the «Primary and behaviour Bodies had of the been Cosmos») assumed and as iftries men to reallyconsider knew their what intelligible Fire and form. Earth How were. doPlato Primary now Bodies denies themcome the into status being? of «PrimaryWhat are Elements»they made (from of? What this moment, are their Timaeus original willforms? call themPlato’s the answer«Primary is that Bodies Primary of the Bodies Cosmos») actually and derive tries to fr considerom the two their Elementary intelligible Triangles form. How that do are Primary now taken Bodies ascome the two into irreducible being? What Elements are they for made the of? geometrica What arel theirconstruction original forms?of the Four Plato’s Primary answer Bodies/Solids is that Primary ofBodies the Cosmos actually (Timaeus derive, from53c─d). the That two Elementaryis to say, the Triangles faces of that the areFour now Solids taken are as decomposed the two irreducible into “fundamental”Elements for the Triangles geometrical (we must construction remember of theAristo Fourtle’s Primary statement Bodies/Solids about the triad of the being Cosmos the (divineTimaeus , number53c–d). of That the iswhole to say, Cosmos) the faces which of the FourPlato Solids consider decomposed as the real Primary into “fundamental” Elements («as Triangles it were, (we letters»)must remember of the Universe Aristotle’s (Figure statement 3). The abouttwo fundamental the triad being Triangles the divine are numberconsidered of thein such whole a way Cosmos) as towhich explain Plato how considers the change as theand real the Primary transmutation Elements of («asElements it were, is possible. letters») ofThe the first Universe Element (Figure is the3 ). scaleneThe two fundamental (30°/60°/90°)—«the Triangles are half-equilateral»—by considered in such a which way as the to explainDemiurge how generated the change the and first the threetransmutation Primary Bodies of Elements of the isUniverse, possible. the The particles first Element of Fire, is theAir scaleneand Water. triangle The (30 half-equilateral◦/60◦/90◦)—«the is obtainedhalf-equilateral»—by by dropping whicha perpendicular the Demiurge from generated any the of first the three equilateral Primary to Bodies the opposite of the Universe, side. The the sidesparticles of the of half-equilateral Fire, Air and Water. have Theleng half-equilateralths corresponding is obtained to the numbers by dropping 1, 2, a√ perpendicular3. This is accordingly from any describedangle of thebelow equilateral (54b) as to «having the opposite the greater side. The side sides (of of the the two half-equilateral containing havethe right lengths angle) corresponding triple in √ squareto the of numbers the lesser». 1, 2, 3.The This second is accordingly Element describedis the isosceles below (54b)triangle as «having (45°/45°/90°), the greater whose side sides (of the correspondtwo containing to the thenumbers right angle)1, 1, √2—«the triple inhalf-square»—by square of the lesser». which the The Demiurge second Element generated is the the isoscelesfourth √ Primarytriangle Body, (45◦/45 the◦ /90particle◦), whose of Earth. sides correspond to the numbers 1, 1, 2—«the half-square»—by which the Demiurge generated the fourth Primary Body, the particle of Earth.

Figure 3. The Two Material Elements of the Timaeus Cosmos. Figure 3. The Two Material Elements of the Timaeus Cosmos. In this sense, we have to pick up Plato’ statement: In this sense, we have to pick up Plato’ statement: Here one of the two elements, having generated these bodies, had done it. But the Herewent one on of to the generate two elements, the fourth having body, generated being together these bodies, in sets had of four, done with it. But their the right isosceles triangle meeting wentat the on centre,to generate thus the forming fourth abody, single being equilateral together quadrangle. in sets of four, Six with such their quadrangles, right angles joined meeting together, at theproduced centre, thus eight forming solid angles, a single each equilateral composed quad by arangle. set of three Six such plane quadrangles, right angles. joined The shape together, of the producedresulting eight body solid was cubical,angles, each having composed six quadrangular by a set of equilateral three plane planes right as angles. its faces The[Timaeus, shape of 55b].the resulting body was cubical, having six quadrangular equilateral planes as its faces [Timaeus, 55b]. Consequently, we will be able to understand Platonic if we know the proprieties of the Consequently, we will be able to understand Platonic physics if we know the proprieties of the two Primary Elements by which the Demiurge first constructed geometrically, and then harmoniously two Primary Elements by which the Demiurge first constructed geometrically, and then joined the four Primary Bodies of the Cosmos (Figure4) through the Golden Proportion [ 22]. harmoniously joined the four Primary Bodies of the Cosmos (Figure 4) through the Golden Specifically, the Primary Body of Fire is a or a Tetrahedron (“the simplest and smallest Proportion [22]. Specifically, the Primary Body of Fire is a Pyramid or a Tetrahedron (“the simplest and smallest figure”) made of 4 equilateral triangles consisting of 24 scalene right triangles altogether; the Primary Body of Air is an made of 8 equilateral triangles consisting of 48 scalene right triangles altogether; the Primary Body of Water is an made of 20 equilateral

Symmetry 2019, 11, 98 8 of 13

figure”) made of 4 equilateral triangles consisting of 24 scalene right triangles altogether; the Primary BodySymmetry of 2019 Air, is11, an x Octahedron made of 8 equilateral triangles consisting of 48 scalene right triangles8 of 13 altogether; the Primary Body of Water is an Icosahedron made of 20 equilateral triangles consisting oftriangles 120 scalene consisting right trianglesof 120 scalene altogether; right triangles and the Primaryaltogether; Body and of the Earth Primary is a Cube Bodymade of Earth of 6is squares a Cube consistingmade of 6 squares of 24 isosceles consisting right of triangles 24 isosceles altogether. right triangles altogether.

Figure 4. The Four Primary Bodies of the Timaeus Cosmos. Figure 4. The Four Primary Bodies of the Timaeus Cosmos. The platonic belief is that Tetrahedron, Octahedron and Icosahedron are composed of elementary The platonic belief is that Tetrahedron, Octahedron and Icosahedron are composed of scalene triangles, which means that transformations and transmutations into different kinds of solids elementary scalene triangles, which means that transformations and transmutations into different are possible only among the particles of Fire, Air and Water; while the Cube is composed of elementary kinds of solids are possible only among the particles of Fire, Air and Water; while the Cube is isosceles triangles, which means that particles of Earth cannot be transformed by any of others (Timaeus, composed of elementary isosceles triangles, which means that particles of Earth cannot be 54b–55c). The exclusion of Earth from the cycle of transformation was severely criticized by Aristotle. transformed by any of others (Timaeus, 54b─55c). The exclusion of Earth from the cycle of From a modern scientific , of course, Plato’s mapping from mathematical ideals to transformation was severely criticized by Aristotle. physical reality looks hopelessly wrong. Nevertheless, Plato’s cosmology anticipated the spirit of From a modern scientific perspective, of course, Plato’s mapping from mathematical ideals to modern theoretical physics. His program of describing the material world by analysing («reducing») physical reality looks hopelessly wrong. Nevertheless, Plato’s cosmology anticipated the spirit of it to a few atomic substances (the two irreducible elements), each with simple properties, existing in modern theoretical physics. His program of describing the material world by analysing («reducing») great numbers of identical copies, coincides with modern understanding. it to a few atomic substances (the two irreducible elements), each with simple properties, existing in Thegreat Rational numbers Element of identical of the Timaeus copies, Cosmos,coincides and with the Shapemodern of theunderstanding. Universe as a Whole: The Dodecahedron

The RationalFor a deeper Element understanding of the Timaeus of Cosmos, the cosmic and the body Shape generation, of the Universe transformation as a Whole: andThe unification,Dodecahedron we have finally to consider the ultimate principles of Plato’s Cosmos with reference to the Unwritten Doctrines,For a deeper which understanding are «very hard of tothe apprehend». cosmic body generation, It means that—in transformation accordance and unification, to Cornford’s we suggestion—thehave finally to consider supreme the principles ultimate (theprinciples intelligible of Plato’s forms) Cosmos known with to mathematicians reference to the areUnwritten «Lines» andDoctrines, «Numbers»: which indeed, are «very if Primary hard Bodiesto apprehend». are made up It of means Planes, that—in and Planes accordance are made upto ofCornford’s Triangles, Trianglessuggestion—the themselves supreme can be principles constructed (the with intelligible Lines and forms) Lines known can be expressedto mathematicians as Numbers. are And,«Lines» all ofand this «Numbers»: is consistent indeed, with my if interpretation.Primary Bodies Essentially, are made forup me, of Planes, the myth and of thePlanes Demiurge are made moulding up of allTriangles, sorts of trianglesTriangles andthemselves figures outcan of be constructed (Timaeus, 50a–c)with Lines explains, and inLines clearer can terms, be expressed the dualism as betweenNumbers. «the And, Power all of and this its is Opposite», consistent thatwith is my between interpretation. the permanent Essentially, nature for of theme, rational the myth of the ofDemiurge the Cosmos moulding (The One)—which all sorts of triangles Plato now and calls figures «Matrix», out of almostgold (Timaeus certainly, 50a in─ referencec) explains, to in a specific clearer sequenceterms, the ofdualism numbers between and structures «the Power («By and nature its Oppo it issite», there that as ais matrixbetween for the everything»)—and permanent nature the of shiftingthe rational qualities principle of the of fourthe Cosmos Primary (The Bodies One)—whi of the Cosmosch Plato (The now Indefinite calls «Matrix», Dyad). almost The resultcertainly is the in constructionreference to a of specific the Dodecahedron, sequence of thenumbers fifth regular and stru solid,ctures which («By is nature the shape it is ofthere the cosmicas a matrix body for as aeverything»)—and whole: the shifting qualities of the four Primary Bodies of the Cosmos (The Indefinite Dyad). The result is the construction of the Dodecahedron, the fifth regular solid, which is the shape of theThere cosmic still body remained as a onewhole: construction, the fifth, and the god used it for the whole, making a of animal figures thereon [Timaeus, 55c]. There still remained one construction, the fifth, and the god used it for the whole, making a pattern Butof animal what figures about thereon the Dodecahedron [Timaeus, 55c]. and its decomposition into triangles? Taylor observes that «Timaeus does not describe its construction (Is it just a touch of Pythagorean ‘reserve’?») [23]. But what about the Dodecahedron and its decomposition√ into triangles? Taylor observes that Along the same lines, Olsen states that «Plato gives us the 2 triangle for the construction of the «Timaeus does√ not describe its construction (Is it just a touch of Pythagorean ‘reserve’?») [23]. Along Cube, and the 3 triangle for the construction of the Tetrahedron, Octahedron and Icosahedron. But, the same lines, Olsen states that «Plato gives us the √2 triangle for the construction of the Cube, and the √3 triangle for the construction of the Tetrahedron, Octahedron and Icosahedron. But, the triangle (or the root numbers embedded in it) necessary for the construction of the Dodecahedron is most conspicuously absent. Regarding the √2 and √3 primitive triangles, however, Plato states cryptically (and yet very revealingly for the astute student)»:

Symmetry 2019, 11, 98 9 of 13 the triangle (or the root numbers embedded in it) necessary for the construction of the Dodecahedron √ √ Symmetryis most 2019 conspicuously, 11, x absent. Regarding the 2 and 3 primitive triangles, however, Plato9 states of 13 cryptically (and yet very revealingly for the astute student)»: Now all triangles are derived from two, each having one right angle and the other angle acute. Of theseNow triangles, all triangles one are has derived on either from side two, the each half having of a right one rightangle, angle the division and the otherof which angle is acute.determined Of these by equaltriangles, sides one (the has right-angled on either side isosceles); the half the of other a right ha angle,s unequal the divisionparts of a of right which angle is determined allotted to unequal by equal sidessides (the right-angled isosceles);scalene). This the other we assume has unequal as the parts first ofbeginning a right angle of fire allotted and the to unequalother bodies, sides following(the right-angled the account scalene). which This combines we assume likelihood as the firstwith beginningnecessity; ofthe fire principles and the otheryet more bodies, remote following than thesethe account are known which to combinesHeaven and likelihood such men with as necessity; Heaven thefavours. principles (…) Now, yet more of the remote two thantriangles, these the are isoscelesknown to is Heaven of one type and only; such menthe scalene, as Heaven of an favours. endless ( number.... ) Now, Of ofthis the unlimited two triangles, multitude the isosceles we must is chooseof one typethe best, only; if thewe scalene,are to make of an a beginning endless number. on our Of own this principles. unlimited Accordingly, multitude we if mustanyone choose can tell the usbest, of ifa better we are kind to make that a he beginning has chosen on ourfor the own cons principles.truction Accordingly, of these bodies, if anyone his will can be tellthe usvictory, of a better not ofkind an thatenemy, he hasbut chosenof a friend for. theFor construction ourselves, however, of these we bodies, postulate his will as the be thebest victory, of these not many of antriangles enemy, onebut kind, of afriend. passing For over ourselves, all the rest; however, that, namely, we postulate a pair of as which the best compose of these the many equilateral triangles triangle. one kind, The reasonpassing is over too long all the a rest;story; that, but namely,if anyone a pairshould of whichput matter compose to the the test equilateral and discover triangle. that The it is reason not so, is thetoo prize long ais story; his with but all if anyonegood will. should [Timaeus, put matter 53c–54b]. to the test and discover that it is not so, the prize is his with all good will [Timaeus, 53c–54b]. However, in line with Heath, who considers that Timaeus is an exposition of Pythagorean ideas («ThereHowever, seems to in be line therefore with Heath, no room who for considers doubt that that theTimaeus constructionis an exposition of a of Pythagorean by means of ideas an isosceles(«There seemstriangle to behaving therefore each noof roomits base for angles doubt thatdouble the of construction the vertical of angle a pentagon was due by meansto the Pythagoreans»)of an isosceles triangle[24], I think having that each the ofTimaeus its base Dode anglescahedron double is of composed the vertical by angle 12 pentagonal was due tofaces the consistingPythagoreans») of 60 Golden [24], I thinkTriangles that (5 the Golden Timaeus Triangle per face) having is composed each of by its 12 base pentagonal angles double faces ofconsisting the vertical of 60 angle Golden (36°/72/72°). Triangles That (5 Golden is, using Triangles the Platonic per face) approach having also each to of analyse its base the angles geometric double ◦ ◦ propertiesof the vertical of the angle shape (36 of/72/72 the cosmic). That body is, using as a thewhole, Platonic I discover approach that alsothe toPrimary analyse Element the geometric of the Dodecahedronproperties of the is shapethe Golden of the cosmicTriangle, body which as a co whole,uld confirm I discover the that validity the Primary of my Elementhermeneutical of the hypothesisDodecahedron (Figure is the 5). Golden In fact, Triangle, the Golden which Triangle could confirm is an isosceles the validity triangle of my «such hermeneutical that the ratio hypothesis of the (Figure5). In (a) fact, to thebase Golden (b) is equal Triangle to the is an Golden isosceles Ratio, triangle a/b = «suchΦ» [25]. that the ratio of the hypotenuse (a) to base (b) is equal to the Golden Ratio, a/b = Φ»[25].

Figure 5. The Rational Element of the Timaeus Cosmos. Figure 5. The Rational Element of the Timaeus Cosmos. For this reason, the Pentagon naturally contains the Divisions into Extreme and Mean Ratio. The PentagonFor this reason, could the actually Pentagon be defined naturally as contains a regular the polygon Divisions of five into sides Extreme whose and Mean Ratio. are The in a Pentagon could actually be defined as a of five sides whose diagonals are in a Golden Proportion with its sides. Furthermore, each Pentagon contains a , also called the five- point star, pentacle or pentangle, (obtained by connecting all the vertices of the Pentagon through the diagonals), that is a star polygon in which we find a second Pentagon, and so on. The fact of being

Symmetry 2019, 11, 98 10 of 13

GoldenSymmetry Proportion2019, 11, x with its sides. Furthermore, each Pentagon contains a Pentagram, also called10 of the 13 five-point star, pentacle or pentangle, (obtained by connecting all the vertices of the Pentagon through theable diagonals), to generate that an isindefinite a star polygon series inof whichPentagon wes find and a second Pentagon, inserted and into so on. one The another fact of can being be ableused toto generateshow that an the indefinite series and the of side of the and Pentagon Pentagrams are incommensurable, inserted into one that another is, the can ratio be usedof their to showlengths, that Φ, the cannot diagonal be expressed and the sideas a ofratio the between Pentagon integers. are incommensurable, Also, since the ratio that is,of thea pair ratio of ofconsecutive their lengths, MātrΦā,meru cannot numbers be expressed is appr as aoximately ratio between equal integers. to Φ, we Also, can since obtain the an ratio approximate of a pair of consecutivePentagon and M atr¯Pentagramameru¯ numbers using M isā approximatelytrāmeru numbers equal as tolineΦ ,lengths. we can obtain an approximate Pentagon and PentagramIn the figure using below Matr¯ (Figureameru¯ 6), numbers there are as lineMātr lengths.āmeru numbers 5, 8, 13. The ratio of these three pairsIn of the consecutive figure below M (Figureātrāmeru6), therenumbers are M isatr¯ approximatelyameru¯ numbers equal 5, 8, 13.to TheΦ, ratioso that of thesethe Pentagram three pairs ofsymbolism consecutive can M beatr¯ directlyameru¯ numbersrelated to is the approximately Golden Proportion. equal to AsΦ, we so thatknow, the the Pentagram Pentagram symbolism became one can beof the directly symbols related of the to thePythagorean Golden Proportion. School and As was we ca know,lled “five the Pentagram alpha” because became it represented one of the symbols the five ofBodies the Pythagorean of the Cosmos: School Air, andWater, was Earth, called Fire “five (already alpha” validated because it by represented Empedocles) the and five Spirit Bodies (added of the Cosmos:by ). Air, Water, And, Earth, this Firewould (already mathematically validated by prove Empedocles) that my and research Spirit (addedhypothesis by Pythagoras). is at least And,plausible. this would mathematically prove that my research hypothesis is at least plausible.

Figure 6. ¯ ¯ Figure 6.The The construction construction of theof pentagonalthe pentagonal faces fa ofces Dodecahedron of Dodecahedron by means by of means Matrameru of M sequence.ātrāmeru sequence. But what does the Dodecahedron represent? The Dodecahedron is the fifth Body of the Cosmos, Ether (the ), and represents the shape of the Universe as a whole, which is realized when the four But what does the Dodecahedron represent? The Dodecahedron is the fifth Body of the Cosmos, Primary Bodies of the Cosmos unite, harmoniously and symmetrically, through the matrix for everything, Ether (the Soul), and represents the shape of the Universe as a whole, which is realized when the four that is the rational and permanent principle of the Cosmos—The One or the Golden Ratio—and The Primary Bodies of the Cosmos unite, harmoniously and symmetrically, through the matrix for First Analogy of the Double or the Pingala’s Matr¯ ameru¯ (Figure7). everything, that is the rational and permanent principle of the Cosmos—The One or the Golden Ratio— and TheAth. First Athenian. Analogy On of the Double most likely or the account Pingala’s there M areātr toāmeru be reckoned (Figure five 7). solid bodies, from which oneAth. might Athenian. fashion On things the most fairest likely and account best; butthere all are the to rest be reckoned of creation five has solid a single bodies, shape, from for which there one is nothingmight fashion that could things come fairest to beand without best; but a body all the and rest never of creation possessing has a any single colour shape, at all, for exceptthere is only nothing that reallythat could most come divine to creature,be without the a soul. body And and thisnever alone, posses onesing may any say, colour has the at businessall, except of only fashioning that really and manufacturing,most divine creature, whereas the the soul. body, And as wethis call alone, it, has one that may of beingsay, has fashioned the business and created of fashioning and seen. Butand themanufacturing, other—let us whereas repeat it, the for body, not once as we only call be it, it has said—has that of being the properties fashioned of and being created unseen, and of seen. knowing But andthe other—let being thought, us repeat and ofit, beingfor not endowed once only with be memoryit said—has and the reckoning properties by alternationsof being unseen, of odd of andknowing even. Theand bodies,being thought, then, being and five, of being we must endowed name themwith asme fire,mory water, and andreckoning thirdly by air, alternations earth fourth, of and odd ether and fifth;even. and by predominance of these are each of the many varieties of creatures perfected [Epinomis, 981The b–d].bodies, then, being five, we must name them as fire, water, and thirdly air, earth fourth, and ether fifth; and by predominance of these are each of the many varieties of creatures perfected [Epinomis, 981 b─d].

SymmetrySymmetry2019 2019,,11 11,, 98 x 1111 of 1313

Figure 7. The Shape of the Timaeus Cosmos as a whole. Figure 7. The Shape of the Timaeus Cosmos as a whole. 4. Conclusions 4. ConclusionsAs a result, the Two Ontological, Epistemological, and Ethical Principles of Plato’s Cosmos, the OneAs anda result, the Indefinitethe Two Ontological, Dyad, in my Epistemological, view, refer to the and Division Ethical of Principles a straight Lineof Plato’s into ExtremeCosmos, and the MeanOne and Ratio the (Φ Indefinite:1 = 1:1/Φ Dyad,): in my view, refer to the Division of a straight Line into Extreme and (1)MeanThe Ratio Rational (Φ:1 = Principle1:1/Φ): of the Universe: The Being (Soul) (1) The GoldenRational Ratio Principle (1)—The of the Good Universe: The Being (Soul) The Golden Ratio (1)—The Good Ontological Principle: The One (1) OntologicalEpistemological Principle: Principle: The One (1) EpistemologicalEthical Principle: Principle: Justice Identity Ethical Principle: Justice (2) The Two Material Principles of the Universe: The Becoming (Fire and Earth) (2) The GreaterTwo Material and the Principles Lesser (Φ ofand the 1/Universe:Φ)—The The Evil Becoming (Fire and Earth) The Greater and the Lesser (Φ and 1/Φ)—The Evil OntologicalOntological Principles: Principles: The Indefinite The Indefinite Dyad Dyad (1.618 (1.618 and 0.618) and 0.618) EpistemologicalEpistemological Principles: Principles: Sameness Sameness and Difference and Difference EthicalEthical Principles: Principles: Excess Excess and Deficiency and Deficiency

In point of fact, and in agreement with Aristotle: In point of fact, and in agreement with Aristotle: InIn realityreality thisthis doctrinedoctrine thatthat ‘the‘the one’one’ andand ‘excess‘excess andand defect’defect’ areare thethe principlesprinciples ofof allall thatthat isis turnsturns outout toto havehave beenbeen inin possessionpossession ofof men’smen’s mindsminds fromfrom ofof old;old; butbut notnot alwaysalways inin thethe samesame way,way, forfor thethe earlierearlier thinkersthinkers mademade thethe ‘one’‘one’ thethe receptivereceptive subjectsubject andand thethe contrastedcontrasted ‘two’‘two’ thethe activeactive agents,agents, whereaswhereas moremore recentlyrecently (Plato)(Plato) thethe ‘two’‘two’ havehave sometimessometimes beenbeen regardedregarded asas thethe subjectsubject passivelypassively actedacted uponupon andand thethe ‘one’‘one’ as as the the agent agent[The [The Physics, Physics, 189 189 b b 10–20] 10─20] [ 26[26].].

Funding:Funding: ThisThis researchresearch receivedreceived nono externalexternal funding.funding. Acknowledgments:Acknowledgments:I I would would like like to thankto thank two anonymoustwo anonymous reviewers reviewers for their for valuable their andvaluable stimulating and stimulating comments, and my dear friend Masi Ribaudo for having meticulously corrected the text in English, greatly improving the comments, and my dear friend Masi Ribaudo for having meticulously corrected the text in English, greatly

Symmetry 2019, 11, 98 12 of 13 mathematical interpretation of the Platonic quotations contained therein. Without his critical support and his linguistic advice, this work would probably not have been completed. Conflicts of Interest: The author declares no conflict of interest.

References

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