SYMMETRIES, GENERALIZED NUMBERS AND HARMONIC LAWS IN MATRIX GENETICS

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1 SYMMETRIES, GENERALIZED NUMBERS AND HARMONIC LAWS IN MATRIX GENETICS J. Kappraff *, S.Petoukhov** * Mathematicia, (b. New York, NY) Address: New Jersey Istitute of Techology, USA. kappraff@verizo.et Fields of iterest: applied mathematics, mathematics ad desig Awards: The book Coectios was selected as the best book i math ad sciece professioal ad referece by the Natioal Associatio of Publishers, 99. Publicatios: The author of two books ad over 50 other publicatios. ** Biophysics, (b. Moscow, Russia) Address: Departmet of Biomechaics, Mechaical Egieerig Research Istitute Russia Academy of Scieces, d.4, Malyi Kharitoievskiy pereulok, Moscow, 0990, Russia, petoukhov@hotmail.com; Fields of Iterest: Theoretical ad Mathematical Biology (i particular, i problems of geetic codes ad bioiformatics), Biomechaics, Crystallography, Theories of Symmetry ad Self-Orgaizatio. Awards: Laureate of the State Prize of the USSR Publicatios: The author of 6 books ad more tha 50 other publicatios; (Detailed iformatio is at ) Abstract. This paper is devoted to the presetatio ad aalysis of matrix represetatios of the geetic code. Pricipal attetio is paid to a family of the geetic matrices which are costructed o the basis of Gray code orderig of their rows ad colums. This Gray code orderig reveals ew coectios of the geetic code to: 8- dimesioal bipolar algebras; Hadamard matrices; golde matrices; Pythagorea musical scale, ad a iteger triagle attributed to Nicomachus, a Syria mathematicia from secod cetury A.D. All of these mathematical etities possess symmetrical features. Some questios about silver meas ad Pythagorea triples are also described by these geetic matrices ad their geeralizatios. Keywords: geetic code, Gray code, multidimesioal algebra, Hadamard matrices, golde sectio, Pythagorea musical scale, bisymmetric matrices, silver meas, Pythagorea triples.. INTRODUCTION. The achievemets of bioiformatics have led to ew thoughts o the essece of life. "Life is a partership betwee gees ad mathematics" (Stewart, 999). But what kid of mathematics relates to the geetic code ad what kid of mathematics lies behid geetic pheomeology which accouts for the great oise-immuity of the geetic code? This questio is oe of the mai challeges i the mathematical atural scieces today. The geetic code serves to provide iformatio trasfer with a high-level of oise-immuity alog a chai of geeratios. I view of this we will look for geetic mathematics i the formalisms of the theory of discrete sigal processig, oiseimmuity codes, ad computer iformatics. This report carries forth the authors ivestigatios [Kappraff, 000a-c; Petoukhov, 00, 005, 008]. Geetic codig possesses oise-immuity. Mathematical theories of oiseimmuity codig ad discrete sigal processig are based o matrix methods for the represetatio ad aalysis of iformatio. These matrix methods, edowed with symmetry, are applied to the aalysis of esembles of molecular elemets of the geetic code. This report describes a matrix represetatio of the geetic code coected with Gray code which is widely used i the field of iformatio codig. Gray code is a system of eumeratig itegers usig 0 s ad s such that from oe iteger to the ext, a sigle bit chages from to 0 or 0 to ad that chage occurs i the least sigificat digit to result i a value ot previously obtaied. This Gray code method of presetatio, which differs sigificatly from the previous represetatio based o Kroecker families of matrices [Petoukhov, 00, 005], reveals ew coectios of the geetic code to the followig mathematical etities: 8-dimesioal bipolar algebras; Hadamard matrices; golde matrices; Pythagorea musical scale, ad the Nicomachus triagle. This mathematics is rich i symmetry. These matrices will also be show to

2 describe all of the silver meas ad Pythagorea triples based o work by Kappraff ad Adamso [Kappraff, 009].. A SPECIAL MATRIX PRESENTATION OF THE GENETIC CODE We preset the geetic alphabet ecodig the DNA/RNA bases, A (adeie), C (cytosie), G (guaie) ad U i the form of a alphabetic matrix S: 0 C A 0 U G The first colum of this matrix cotais pyrimidies, C ad U, each of which is marked by the symbol takig ito accout this mutual chemical aspect. The secod colum cotais puries, A ad G, each of which is marked by the symbol 0 takig ito accout this mutual chemical aspect. The first row of the matrix cotais C ad A, each of which is keto from the viewpoit of their physiochemical sigificace ad is marked by the symbol. The secod row of the matrix cotais U ad G, each of which is amio from the viewpoit of their physiochemical sigificace ad is marked by the symbol 0. I this way each of these itrogeous bases is expressed by o s ad s i accordace with its positio iside the matrix S: C is expressed by ; A is expressed by 0; U is expressed by 0; G is expressed by 00. This paper cosiders these biary symbols from the perspective of Gray code. I Gray code the umbers 0,,, 3, 4, 5, 6, 7 are represeted as follows: 0-000; 00; 0; 3 00; 4 0; 5 ; 6 0; Oe ca costruct a (4*4)- matrix S for the 6 geetic duplets ad a (8*8)-matrix S 3 for the 64 geetic triplets, where all rows ad colums are eumerated i ascedig order i Gray code (Figure ). Each elemet of these matrices is coded by the bits i its row ad colum, e.g., the duplet UC is represeted by 0 because U is symbolized by 0 ad C is symbolized by ). Matrix S 3 i Figure also shows what kid of amio acids (or a stop-sigal) are ecoded by each triplet of the vertebrate mitochodrial geetic code, which is the most symmetrical kow dialect of geetic codig. It should be metioed that moder sciece kows may dialects of the geetic code, as show o the NCBI s website By traditio, atural objects with the greatest symmetry are studied first after which cases i which symmetry is violated are studied. Similarly, the authors first ivestigate the vertebrate mitochodrial geetic code which is the most symmetrical. It should be oted that some authors cosider this dialect to be ot oly the most perfect ad symmetrical but also the most aciet although this opiio is subject to some debate. Let us explai a black-ad-white mosaic of the matrix S 3 i Figure. Figure shows the correspodece betwee the set of 64 triplets, sometimes referred to as codos, ad the set of 0 amio acids with stop-sigals (Stop) of protei sythesis i the vertebrate mitochodrial geetic code. The set of 64 triplets cotais 6 subfamilies of triplets, every oe of which cotais 4 triplets with the same pair of letters i the first two positios of each triplet (a example of such subsets is the case of the four triplets CAC, CAA, CAU, CAG with the same two letters CA o their first two positios). We shall refer to such subfamilies as NNtriplets. I the case of the vertebrate mitochodrial code, the set of these 6 subfamilies of NN-triplets is divided ito two equal subsets from the poit of view of degeeratio properties of the code (Figure ). The first subset cotais 8 subfamilies of so called two-positio NN-triplets, a codig value of which is idepedet of a letter i their third positio. A example of such subfamilies is the four triplets CGC, CGA, CGU, CGC, all of which ecode the same amio acid Arg, although they have differet letters i their third positio. All members of such subfamilies of NN-triplets are marked by black color i the geomatrix o Figure.

3 0 (3) () 0 () 00 (0) 00 (7) 0 (6) (5) 0 (4) 00 (3) 0 () 00 () 000 (0) 0 (3) () 0 () 00 (0) CG CU AU AG CA CC AC AA UA UC GC GA UG UU GU GG 00(7) 0(6) (5) 0(4) 00(3) 0() 00() 000(0) CGG CGU CUU CUG AUG AUU AGU AGG (Arg) (Arg) (Leu) (Leu) (Met) (Ile) (Ser) (Stop) CGA CGC CUC CUA AUA AUC AGC AGA (Arg) (Arg) (Leu) (Leu) (Met) (Ile) (Ser) (Stop) CAA CAC CCC CCA ACA ACC AAC AAA (Gl) (His) (Pro) (Pro) (Thr) (Thr) (As) (Lys) CAG CAU CCU CCG ACG ACU AAU AAG (Gl) (His) (Pro) (Pro) (Thr) (Thr) (As) (Lys) UAG UAU UCU UCG GCG GCU GAU GAG (Stop) (Tyr) (Ser) (Ser) (Ala) (Ala) (Asp) (Glu) UAA UAC UCC UCA GCA GCC GAC GAA (Stop) (Tyr) (Ser) (Ser) (Ala) (Ala) (Asp) (Glu) UGA UGC UUC UUA GUA GUC GGC GGA (Trp) (Cys) (Phe) (Leu) (Val) (Val) (Gly) (Gly) UGG UGU UUU UUG GUG GUU GGU GGG (Trp) (Cys) (Phe) (Leu) (Val) (Val) (Gly) (Gly) Figure. The matrix S for 6 duplets (o top) ad the matrix S 3 for 64 triplets with the Gray code orderig of umeratio of all rows ad colums. A correspodece betwee triplets ad amio acids is show for the case of the vertebrate mitochodrial geetic code. Names of amio acids are give by their stadard abbreviatios. The black-adwhite mosaic is explaied i the text. 8 subfamilies of the two-positio NN-triplets ad the amio acids, which are ecoded by them CCC, CCA, CCU, CCG Pro CUC, CUA, CUU, CCG Leu CGC, CGA, CGU, CGG Arg ACC, ACA, ACU, ACG Thr UCC, UCA, UCU, UCG Ser GCC, GCA, GCU, GCG Ala GUC, GUA, GUU, GUG Val GGC, GGA, GGU, GGG Gly 8 subfamilies of the three-positio NNtriplets ad the amio acids, which are ecoded by them CAC, CAA, CAU, CAG His, Gl AAC, AAA, AAU, AAG As, Lys AUC, AUA, AUU, AUG Ile, Met AGC, AGA, AGU, AGG Ser, Stop UAC, UAA, UAU, UAG Tyr, Stop UUC, UUA, UUU, UUG Phe, Leu UGC, UGA, UGU, UGG Cys, Trp GAC, GAA, GAU, GAG Asp, Glu Figure. The case of the vertebrate mitochodrial geetic code. The iitial data were take from the NCBI s web-site The secod subset cotais 8 subfamilies of three-positio NN-triplets, a codig value which depeds o a letter i their third positio. A example of such subfamilies is the four triplets CAC, CAA, CAU, CAC, two of which (CAC, CAU) ecode the amio acid His while two others, (CAA, CAG), ecode the amio acid Gl. All members of such subfamilies of NN-triplets are marked by white color i the geomatrix o Figure. So this geomatrix has 3 black triplets ad 3 white triplets. Each subfamily of four NNtriplets appears i a appropriate (x)-subquadrat of the geomatrix. The black triplets ecode the so-called high-degeeracy amio acids composed of four or more triplets, ad from Figure it ca be see that there are eight such amio acids. O the

4 other had, the white triplets ecode low-degeeracy amio acids with oly two triplets, ad it ca be see from Figure that there are twelve such amio acids [Petoukhov, 005]. The black-ad-white mosaic of the matrix S 3 (Figure ) has iversio symmetry relative to the matrix ceter. This mosaic reflects the specificity of degeeracy of the vertebrate mitochodrial geetic code. We use this mosaic matrix to explore the possible algebraic basis of the geetic code. Why it is importat to study the algebra of the geetic code? I the begiig of the 9- th cetury the followig viewpoit existed i sciece: Euclidea geometry was the sole geometry ad the arithmetic of real umbers was the sole arithmetic with which to describe the atural world. But this viewpoit was called ito questio by the discovery of o-euclidea geometries ad the quaterios of Hamilto; this algebra of quaterios preseted sciece with a ew multidimesioal arithmetic for the study of atural systems [Klie, 980]. Now sciece uderstood that differet atural systems ca possess their ow geometries ad algebras. The example of Hamilto, who spet 0 years i his attempts to describe the geometry of 3-dimesioal space by meas of 3- dimesioal algebras without success, presets a cautioary tale. Hamilto s experiece shows the importace of developig a appropriate system of algebra with which to describe a atural system. Oe ca add that geometrical ad physical-geometrical properties of various atural systems (icludig laws of coservatio, theories of oscillatios ad waves, theories of potetials ad fields, etc.) ca deped o the type of algebras which are appropriate to them. 3. CONCERNING AN ALGORITHMIC CONNECTION OF THE GENETIC (8*8)-MATRIX WITH AN 8-DIMENSIONAL ALGEBRA Is the geetic matrix S 3 (Figure ) coected with a matrix form of represetatio of a multi-dimesioal algebra? Yes, a positive aswer was received o the basis of a simple algorithm of digitizatio which uses the molecular characteristics of the itrogeous bases A, C, G, U/T from the geetic alphabet. The geomatrix S 3 (Figure ) is trasformed ito a ew umeric 8-parametric matrix Y 8 (Figure 3) by meas of this alphabetic algorithm. 00 (7) 0 (6) Y 8 = (5) 0 (4) 00 (3) 0 () 00 () 000 (0) 00(7) 0(6) (5) 0(4) 00(3) 0() 00() 000(0) X X 0 X X 3 -X 7 -X 6 -X 4 -X 5 X X 0 X X 3 -X 7 -X 6 -X 4 -X 5 -X 3 -X X 0 X X 5 X 4 -X 6 -X 7 -X 3 -X X 0 X X 5 X 4 -X 6 -X 7 -X 7 -X 6 X 4 X 5 X X 0 -X -X 3 -X 7 -X 6 X 4 X 5 X X 0 -X -X 3 -X 5 -X 4 -X 6 -X 7 X 3 X X 0 X -X 5 -X 4 -X 6 -X 7 X 3 X X 0 X Figure 3. The matrix Y 8, the black cells of which cotai coordiates with the sig + ad the white cells of which cotai coordiates with the sig -. The umeratio of the colums ad the rows is idetical to the umeratio of the colums ad the rows of the geomatrix S 3 of Figure. The cells of the matrix Y 8, which are occupied by elemets with the sig +, are marked by dark color. The cells of the matrix Y 8, which are occupied by compoets

5 with the sig -, are marked by white color. This black-ad-white mosaic of matrix Y 8 is idetical to the black-ad-white mosaic of the geomatrix i Figure. The matrix Y 8 has the 8 idepedet parameters x 0, x, x, x 3, x 4, x 5, x 6, x 7, which are iterpreted as real umbers here. It has bee discovered that Y 8 is the matrix form of a special 8- dimesioal algebra over the field of real umbers. Let us describe the alphabetic algorithm for the digitizatio of the 64 triplets [Petoukhov, 008a,b]. This algorithm is based o utilizig the two followig biaryoppositioal attributes of the geetic letters A, C, G, U/T: purie or pyrimidie ad or 3 hydroge bods. It uses also the well-kow thesis of molecular geetics that differet positios withi triplets have differet code meaigs. For example the article [Koopelcheko, & Rumer, 975] shows that the first two positios of each triplet form the root of the codo ad that they differ drastically i their fuctio from the third positio ad by its special role. I view of this alphabetic algorithm, the trasformatio of geomatrix S 3 ito matrix Y 8 is ot a abstract ad arbitrary actio at all, but such a trasformatio ca actually be utilized by a kid of bio-computer system withi orgaisms. The alphabetic algorithm of the digitizatio defies the special scheme of readig each triplet: the first two positios of the triplet are read by geetic systems from the perspective of oe molecular attribute ad the third positio of the triplet is read from the perspective of aother molecular attribute. By this alphabetic algorithm, which allows oe to recode the symbolic matrix S 3 ito the umeric matrix Y 8 (Figure 3), each triplet is read i the followig way: The two first positios of each triplet are marked by the symbol α i place of the complemetary letters C ad G o these positios ad by the symbol β i place of the complemetary letters A ad U; The third positio of each triplet is marked by the symbol γ i place of the pyrimidies (C or U) at this positio ad by the symbol δ i place of the puries (A or G); The triplets, which have the letters C or G i their first positio, receive the sig - i oly those istaces for which their secod positio is occupied by the letter A. The triplets, which have the letters A or U i their first positio, receive the sig + i those cases oly for which their secod positios are occupied by the letter C. For example, the triplet CAG receives the symbol -αβδ, because its first letter C is symbolized by α, its secod letter A is symbolized by β, ad its third letter G is symbolized by δ. This triplet possesses the sig - because its first positio has the letter C ad its secod positio has the letter A. Oe ca see that this algorithm recodes all triplets from the traditioal alphabet С, А, U, G ito the ew alphabet α, β, γ, δ. As a result, each triplet receives oe of the followig 8 expressios: ααγ = x 0, ααδ = x, αβγ = x, αβδ = x 3, βαγ = x 4, βαδ = x 5, ββγ = x 6, ββδ = x 7. We will suppose that the symbols α, β, γ, δ are real umbers. This algorithm trasforms the iitial symbolic matrix S 3 (Figure ) ito the umeric matrix Y 8 with the eight coordiates x 0, x, x, x 3, x 4, x 5, x 6, x 7 (Figure 3) for x i real, which we shall refer to as the Y-coordiates. Oe ca represet the 8-dimesioal matrix Y 8 (Figure 3) as the sum of the 8 basis matrices j k (k = 0,,, 7), each of which is coected with oe of the eight coordiates ad express it i the vector form of Equatio, rewritte explicitly i Figure 4: Y 8 = x 0 *j 0 +x *j +x *j +x 3 *j 3 +x 4 *j 4 +x 5 *j 5 +x 6 *j 6 +x 7 *j 7 ()

6 Y 8 = х 0 * х * х * х 3 * х 4 * х 5 * х 6 * х 7 * Figure 4. The represetatio of matrix Y 8 as the sum of the 8 basis matrices. The left colum shows the basis matrices, which are related to the coordiates x 0, x, x 4, x 6 with the eve idexes. The right colum shows the basic matrices, which are related to the coordiates x, x 3, x 5, x 7 with the odd idexes. The most importat ad uexpected feature of these matrices is that the set of these 8 basis matrices j 0, j, j, j 3, j 4, j 5, j 6, j 7 forms a closed set uder multiplicatio: a multiplicatio betwee ay pair of matrices from this set geerates aother matrix from this set. A multiplicatio table for these products is show i Figure 5. The result of multiplyig ay two basis elemets, which are take from the left colum ad the upper row, is show i the cell at the itersectio of its row ad colum i the multiplicatio table (for example, i accordace with this multiplicatio table j *j 5 = - j 7 ). It should be oted that this multiplicatio is ot commutative. Such a multiplicatio table defies a geetic 8-dimesioal algebra Y 8 over a field. Multiplicatio of ay two members of the octet algebra Y 8 geerates a ew member of the same algebra. Cocerig the matrix form of such multiplicatio, it meas that both factors i a multiplicatio have the idetical matrix dispositio of their 8 coordiates x 0, x,, x 7 (i the first factor) ad y 0, y,, y 7 (i the secod factor) ad the fial matrix has the same matrix dispositio of its eight coordiates, z 0, z,, z 7. This is similar to the situatio with real, complex umbers, or hypercomplex umbers where multiplicatio of ay two members of the algebraic system geerates a ew member of the same system. I other words, the expressio Y 8 =x 0 *j 0 +x *j +x *j +x 3 *j 3 +x 4 *j 4 +x 5 *j 5 +x 6 *j 6 +x 7 *j 7 ca be cosidered to be a kid of 8-dimesioal umber similar to the way i which complex umbers are -dimesioal

7 or quarterios are 4-dimesioal. J 0 J J J 3 J 4 J 5 J 6 J 7 J 0 J 0 J J J 3 J 4 J 5 J 6 J 7 J J 0 J J J 3 J 4 J 5 J 6 J 7 J J J 3 -J 0 -J -J 6 -J 7 J 4 J 5 J 3 J J 3 -J 0 -J -J 6 -J 7 J 4 J 5 J 4 J 4 J 5 J 6 J 7 J 0 J J J 3 J 5 J 4 J 5 J 6 J 7 J 0 J J J 3 J 6 J 6 J 7 -J 4 -J 5 -J -J 3 J 0 J J 7 J 6 J 7 -J 4 -J 5 -J -J 3 J 0 J Figure 5. The multiplicatio table of the basic matrices j 0, j, j, j 3, j 4, j 5, j 6, j 7 of the matrix Y8 from Figures 3 ad 4. This ew geetic algebra Y 8 belogs to a set of bipolar algebras (or Yi-Yagalgebras, or eve-odd-algebras), ad it has iterestig mathematical properties which are similar i may aspects to geetic bipolar algebras described i the works [Petoukhov, 008a,b]. Permutatios of elemets play a importat role i the theory of sigal processig. Oe ca study the ifluece that the simultaeous permutatios of the positios i all 64 geetic triplets have upo matrices S 3 ad Y 8. Six permutatios of triplets are possible : --3, -3-, 3--, -3-, --3, 3--. For example, if oe chages the iitial order, --3, i all triplets ito the ew order, -3-, the may cells of the iitial geomatrix S 3 are occupied by ew triplets. For example, the matrix cell with the triplet CAU is occupied by the triplet AUC, etc. As a result, the iitial geomatrix S 3/3 (we have icluded here a additioal idex 3 ito the previous symbol for matrix S 3 to show the appropriate juxtapositio of the --3 positios i the triplets) is recostructed ito the ew geomatrix S 3/3 with a ew black-ad-white mosaic. This ew matrix S 3/3 is trasformed by the same alphabetic algorithm of the digitizatio which was described above ito a ew iteger matrix Y 8/3 with the same eight coordiates x 0, x, x, x 3, x 4, x 5, x 6, x 7 i the ew cofiguratio. Oe ca check that this ew umeric matrix Y 8/3 is the matrix form of the represetatio of a ew 8- dimesioal bipolar algebra. The same holds for the other four matrices S 3/3, S 3/3, S 3/3 ad S 3/3, which also correspod to ew cofiguratios of triplets juxtaposed by the permutatios: 3, 3, 3, 3. These four matrices, each of which has its ow black-ad-white mosaic, are trasformed by the same alphabetic digitizatio algorithm ito ew 8-fold matrices Y 8/3, Y 8/3, Y 8/3 ad Y 8/3. Each of these four matrices also provides a matrix form of represetatio of its ow 8-dimesioal bipolar algebra. This matrix geetics approach has may coectios with priciples ad methods of symmetry (see for example [Darvas, 997, 007]). 4. A CONNECTION TO HADAMARD MATRICES. Let us cosider some coectios of the geetic matrices to matrix formalisms from the theory of discrete sigal processig. Oe of the most importat classes of matrices associated with this theory is the so called Hadamard matrices. These matrices are also used i the study of error-correctig codes such as the Reed-Muller code, i spectral aalysis, i multi-chael spectrometers with Hadamard trasformatios, i quatum computers with Hadamard gates (or logical operators), i quatum mechaics as uitary operators, etc. A huge umber of scietific publicatios are devoted to Hadamard matrices. These matrices provide for effective meas of iformatio processig. By defiitio, a Hadamard matrix of dimesio is the (*)-matrix H() with elemets + ad -. It satisfies the coditio H()*H() T = *I, where H() T is the trasposed matrix ad I is the (*)-idetity matrix. The Hadamard matrices of dimesio k are give by the recursive formula H( k ) = H() (k) = H() H( k- ) for k N, where deotes the Kroecker (or tesor) product, (k) deotes Kroecker expoetiatio, k ad N are itegers, H() is illustrated i Figure 6.

8 - - H() = - ; H(4) = H( K- ) H( K- ) H( K ) = -H( K- ) H( K- ) Figure 6. The family of Hadamard matrices H( k ) based o the Kroecker product Is there ay atural coectio betwee the geetic matrices S 3/3, S 3/3, S 3/3, S 3/3, S 3/3 ad S 3/3 ad Hadamard matrices? A positive aswer to this questio could lead to ew thoughts about the structure of the geetic code. The aswer to this questio is positive: such a algorithmic coectio does exist. It is associated with fudametal ad somewhat eigmatic features of the geetic code, amely, ) the mutual replacemet of the letters U ad T i RNA ad DNA ad, ) the differece of these letters from other letters A, C, G due to the absece of amids (amio-groups) withi them. This algorithm is amed the U-algorithm, ad it was used i the study of geomatrices of the Kroecker type [Petoukhov, 008a,b]. Let us demostrate its applicatio to the geomatrix S 3/3 (Figure ). CGG (Arg) CGA (Arg) CAA (Gl) CAG (Gl) UAG (Stop) UAA (Stop) UGA (Trp) UGG (Trp) CGU (Arg) CGC (Arg) CAC (His) CAU (His) UAU (Tyr) UAC (Tyr) UGC (Cys) UGU (Cys) CUU (Leu) CUC (Leu) CCC (Pro) CCU (Pro) UCU (Ser) UCC (Ser) UUC (Phe) UUU (Phe) CUG (Leu) CUA (Leu) CCA (Pro) CCG (Pro) UCG (Ser) UCA (Ser) UUA (Leu) UUG (Leu) AUG (Met) AUA (Met) ACA (Thr) ACG (Thr) GCG (Ala) GCA (Ala) GUA (Val) GUG (Val) AUU (Ile) AUC (Ile) ACC (Thr) ACU (Thr) GCU (Ala) GCC (Ala) GUC (Val) GUU (Val) AGU (Ser) AGC (Ser) AAC (As) AAU (As) GAU (Asp) GAC (Asp) GGC (Gly) GGU (Gly) AGG (Stop) AGA (Stop) AAA (Lys) AAG (Lys) GAG (Glu) GAA (Glu) GGA (Gly) GGG (Gly) Figure 7. Top: the result of the trasformatio of matrix S 3/3 ito a ew mosaic geomatrix. Bottom: a Hadamard (8*8)-matrix with the similar mosaic. Accordig to the U-algorithm we ivert the sigs i cells of the matrix S 3/3 every time the letter U occupies the first or the third positios of a triplet. For example, by this U-algorithm the cells with the triplets UCA ad GAU chage their sig oce, while the cell with the triplet UAU chages its sig twice which meas that the sig of this cell is uchaged. As a result of such U-algorithmic sig chages, a ew mosaic geomatrix appears (Figure 7, top). Its mosaic is idetical to the correspodig mosaic of a Hadamard matrix. Actually, if each black triplet (white triplet) i this geomatrix is replaced by umber + ( - ) a umeric matrix is formed (Figure 7, bottom). Oe ca

9 easily check that this matrix satisfies the defiitio of Hadamard (8*8)-matrices: H(8)*H(8) T = 8*I. The five geomatrices S 3/3, S 3/3, S 3/3, S 3/3 ad S 3/3 are also coected with Hadamard matrices because they are trasformed ito their ow Hadamard matrices by meas of the same U-algorithm. Oe ca suppose that this U-algorithm (of ivertig the sigs every time the letter U or T appears i a odd positio of triplets) is coected with the biological mechaism of mutual replacemet of the letters U ad T at trasitio from RNA to DNA ad vice versa. 5. GENOMATRICES WITH NUMBERS OF HYDROGEN BONDS Each itrogeous base has a particular umber of hydroge bods: the complemetary bases, C ad G, have 3 hydroge bods, while the complemetary bases, A ad U, have hydroge bods. Ivestigatios of the Kroecker family of geomatrices reveals iterestig mathematical properties of these matrices i the case for which a symbol of each triplet is replaced by the umber of ways i which the hydroge bods of its bases ca iteract give by the product of the umber of hydroge bods of its bases. For example the triplet CAG is replaced by a iteger 3**3=8, etc. I what follows, we study the iteger based geomatrices formed by these iteger product replacemets as ordered by Gray code. As a result, the symbolic geomatrices S, S 3, etc. (Figure ) are trasformed ito a family of iteger matrices M, M 3, etc. (Figure 8). These matrices are bisymmetric that is symmetric relative to both diagoals. M = 3 3 ; M = ; M 3 = Figure 8. Iteger matrices with elemets represeted by iteger products of hydroge bods i mooplets, duplets ad triplets Notice that i M the atural umbers,3 appear, i M the umbers 4,6,9 appear while i M 3 the umbers 8,,8,7 appear, with each row ad colum expressig the same species of positive itegers with o iteger appearig adjacet to itself i a row or colum. These sets of itegers come from a triagle of positive itegers attributed to the d cetury AD Syria mathematicia Nicomachus (Kappraff, 000) ad represets sequeces of musical fifths. The Nicomachus Triagle, T(,k), is reproduced i Table k k where the itegers i the -th row are { 3,0 k }; 0. Here if the cetral iteger 6 is thought to be the legth of a strig represetig a fudametal toe, the 4 ad 9 of row 3 represet the strig legths correspodig to the risig ad fallig musical fifths, :3 ad 3:. Also the fifth row represets the strig legths that give rise to a petatoic scale with fudametal strig legth of 36 uits while the itegers i row 7 represet strig legths of a heptatoic scale with 6 as the strig legth of the fudametal. The Triagle T(,k) i Table has the property that every row, colum, diagoal, ad lie joiig ay two elemets cotais a geometric progressio. Table. The Nicomachus Triagle, T(,k) Table. Pascal s Triagle etc etc.

10 T(,k) is the triagle of coefficiets i the expasio of ( + 3x) ; give by the geeratig fuctio. For example, 8,,8,7 are geerated by y ( + 3x) 3 3 ( + 3x) = x + 3 8x + 7x where we see that there is oe 8, oe 7, three s ad three 8 s i each row or colum of matrix M 3. Furthermore if we set x = we fid that the sum of the elemets i each row or colum of M equals 5, e.g., for M 3 the sum = 5. I other words, successive itegers from a row of the Nicomachus triagle are multiplied by successive itegers from rows of Pascal s Triagle, give i Table, e.g., (, 3, 3, ) (8,, 8, 7) where deotes dot product i order to sum the row ad colum elemets of M 3. Petoukhov [00, 005] has show that, / τ / τ P = M = () / τ τ + 5 where τ = is the golde mea. Associatig τ with ad 00, ad / τ with 0 ad 0, we obtai matrices of the square roots of each of the higher M matrices deoted by P. For example, τ / τ / P = M = τ / τ (3) / τ τ / τ τ 00 correspods to τ τ = τ, 00 correspods to / τ / τ = / τ, ad 0 correspods to τ / τ =. It ca be show that the elemets of P = are all powers of the golde mea [Kapraff, 009]. 6. GENERALIZED BISYMMETRIC MATRICES / M Bisymmetric matrices have bee ivestigated by Petoukhov [00, 005] i the study of matrix geetics. Here we cosider geeral forms of bisymmetric matrices. We start with the matrix M : M = a b The higher order matrices M are determied i a similar maer as was doe for the matrix with a = 3 ad b=. They cotai colums ad rows with itegers from each row of the geeralized Nicomachus Triagle i Table 3 with multiplicities give by Pascal s Triagle. For example, usig elemets of row 3 of Table 3, b a a ab b ab M = ab a ab b (5) b ab a ab ab b ab a Table 3. Geeralized Nicomachus Triagle b a (6) b ab a b 3 ab a b a 3 b 4 ab 3 a b a 3 b a 4 (4)

11 The elemets of M are geerated by (b + ax) with each row ad colum of M summig to (a+b). O the other had we have show [Kappraff, 009] that the square root of M ad M have elemets α, β that are geeralizatios of the golde mea. These square root matrices P ad P ca be expressed as, α αβ β αβ α β P = ad P = αβ α αβ β (7a,b) β α β αβ α αβ αβ β αβ α where, α = ad b β a a b β =. (8a,b) Therefore α ad β are real for a>b ad complex for a<b. Sice P = M that, ad from this it follows that, a = α + β, = αβ α + β =, it follows b, (9a,b) a + b, ad α β + β α Also, α ad β are roots of the fourth degree polyomial, a = b (0a,b) 4 x ( α + β ) x + α β = 0 () Makig use of Equatios 8a,b, Equatio is rewritte, 4 b x ax + = 0 () 4 where usig Equatio 8a ad 9a, a (3) = b 4β + β Equatio ca also be rewritte as, 4 b x = ax, (4) 4 ad if we cosider the geometric sequece, 4 6 8, α, α, α, α, (5) where, b 4 α = aα α, (6) 4 this correspods to a geeralized Fiboacci sequece, {c } i which, b c = 0, c = ad c = ac c. (7a) 4

12 The ratio of successive terms, α is complex, the c c c + approaches α from below. + approaches c α i the limit where, i the case that α deotes the square of the absolute value. If we let g = c ck ck, the g ca be show to satisfy the recursio, 4 ( b a b g = a g + ) g. (7b) 6 4 Settig b = ad lettig a = N ±, Equatio 3 reduces to, Also. It follows that, + β = N ± β β m β = N ad α = / β. (8a,b) (9a) Sice α / β =, Equatio 9a is rewritte, α m = N (9b) α We refer to solutios of the equatios, x - /x = N ad x + /x = N (0) as the N-th silver meas of the first ad secod kid respectively ad deote them as SM ( N ) ad SM ( N ) [Kappraff, 000b]. Whe N =, x = SM () = τ, the golde mea. Therefore, i Equatio 9b, α = SM ( N ) or α = SM ( N ). As a result of Equatio 0, α satisfies oe of the equatios, Therefore, the sequece x = Nx ± (), α, α, 3 α, 4 α, () is a geeralized Pell sequece [Kappraff, 000b] ad satisfies the recursio, as does the sequece, { c k } where, α = Nα ± α (3) where c c (4a) k = Nck ± ck + approaches α i the limit. We also fid that whe b =, Equatio 7b has c the special solutio: g = k, i.e., c ck ck = k for all (4b)

13 This meas that if k = 0, the sequece {c } is a geometric sequece. Otherwise it is a approximate geometric sequece. We cosider eight examples: Example : a = 3, b =, N=. α = SM ( ) = τ ad β = / τ, row ad colum elemets are geerated by( + 3x), row ad colum sum = 5, Sequece 7 yields {0,,3,8,, }, eve idexed Fiboacci terms with ratio of successive terms approachig τ. I Equatio 4b, we fid that k =. The golde mea has foud may applicatios. LeCorbusier made it the basis of his Modulor system of architectural desig (Kappraff, 000c). Example : a = 6, b =, N =. Replacig this ito Equatio 8 yields α = SM () = +, β =, row ad colum elemets are geerated + by ( + 6x), row ad colum sum = 8 Sequece 7a yields: {0,,6,35,04, }, approachig β. I Equatio 4b, we fid that k =. The proportio, + is commoly kow as the silver mea ad was the basis of the system of proportios used i the Roma empire (Kappraff, 000c). Example 3: a = 5, b = 4, α =, β =, row ad colum elemets geerated by (4 + 5x), row ad colum sum = 9. Sequece 7, i.e., 4 c = 5c 4c, yields : {0,,5,,85,34,365, } = { c = } as the 3 ratio of successive terms teds to 4. Example 4: a = 5, b = 3, α = 3/, β = /, row ad colum elemets geerated by ( 5 + 3x), row ad colum sum = 8. The geeralized k k Nicomachus Triagle i Table 4 is geerated from { 3 5 }(0 k ). Table 4. Geeralized Nicomachus Triagle Geerated by (3,5) Each colum of this Triagle represets a sequece of musical fifths ad recreates the aciet Pythagorea scale, whereas ay three successive colums geerate the toes of the aciet Just scale [Kappraff, 000, 009], [McClai, 976] Example 5: a = 4, b = 3, α =, β = elemets geerated by ( 4 + 3x), row ad colum sum = 4 7, row ad colum 7. Example 6: a = 7, b =, N = 3, α = SM ( 3) = τ, β = / τ, row ad colum elemets geerated by( + 7x), row ad colum sum = 9. Example 7: a =, b =, α = /, β = /. Row ad colum elemets are geerated by ( + x).

14 All elemets of the geeralized Nicomachus Triagle (see Table 3) are oes but takig ito accout multiplicity yields Pascal s Triagle (see Table ) whose! (,k)-th elemet is equal to. k!( k)! The ratio of successive terms i Sequece 7, i.e., c = 0, c = ad c = c c, yields:, ¾, /3, 5/8, 3/5, 7/, 4/7, approachig the value of ½. 4 These ratios are the fudametal, musical fourth, fifth, mior sixth, major sixth of the aciet Just scale, ad two approximatios to the major ad mior seveths, all approachig the octave value of ½. If the modulus M of a pair of successive approximatig fractios a/b ad c/d is defied as M = (ad bc) the all moduli of the approximatig sequece have the value, e.g., (x4 x3) =, (3x3 4x) =, etc. As a result, the approximatig fractios appear as elemets of successive rows of the Farey Table to the right of ½ [Kappraff, 000b]. iπ / 6 π / 6 Example 8: a =, b =, N = i, α = SM ( i) = e, β = e i, row ad colum terms are geerated by ( + x), row ad colum sum = 3. The geeralized Nicomachus Triagle yields, Table 5. Geeralized Nicomachus Triagle Geerated by (,) Multiplyig the elemets of Table 5 by the elemets of Pascal s Triagle to accout for multiplicity yields the square of Pascal s Triagle, a triagle whose (i,j)-th etry is (i,j)x i-j where (i,j) is the elemet of the i-th row ad j-th colum of Pascal s Triagle. Table 6. Square of Pascal s Triagle The rows of Table 6 give the umber of vertices, edges, faces, cells, etc of hypercubes, of icreasig dimesio, e.g., H 0 (poit) V = ; H (lie segmet) V =, E = ; H (square) V = 4, E = 4, F=; H 3 (cube) V = 8, E =, F = 6, C = ; H 4 (tesseract) V = 6, E = 3, F = 4, C = 8, hypercube =. Sequece 7 geerates the sequece: 0,,,0,-,-,0,,,, ad the ratio of successive terms should approach α =. I fact, the ratio of a subsequece approaches idetically as it should. I Eq. 4b, we fid oce agai that k =. If a =, b =, the the geeralized Nicomachus Triagle is idetical to the oe for a =, b = but the colums are i reverse order. O the other had = α ad β. = 7. PYTHAGOREAN TRIPLES AND THE SQUARE OF A x BISYMMETRIC MATRIX. For x,y real umbers with x>y, ad

15 x y M = y x The square of this matrix is, (5) M = x y y x + y xy x = xy x + y. (6) Notice that M has values that are the hypoteuse ad altitude of a right triagle whose base is x y. As a result, if x ad y are itegers, the { x + y, xy, x y } is a Pythagorea triple, i.e., three iteger sides of a right triagle. Compare this with the complex umber x + iy ad its square, ( x + iy) = x y + ixy. Here the argumet of x+iy is doubled while its xy modulus is squared, i.e., if ta θ = y / x the ta θ = while the modulus x y squares from x + triagle is x + y so that it is idetical with the triagle i Figure 3. y to x + y as show i Figure 3. The hypoteuse of this Figure 9. Pythagorea triples x + y xy [x + y ] / θ y θ x x -y We ow idetify x y y x ta θ = y / x with the ordered pair (x,y) or equivaletly with the complex umber x + iy so that =c. It ca be show that the ordered pair, (c,), correspods to a triagle with the radius r of the iscribed circle, ad area A give by, r = c, A = ( c )( c)( c + ) It follows that the radius r of the iscribed circle ad the area of triagle (a,b) is, r = b ( a / b ) = ab b ad 4 A = b ( a / b )( a / b)( a / b + ) = ab( a b ) (7a,b) It also follows from Equatios 7 that, r = area semiperimeter where this equatio holds for triagles that are ot right triagles.

16 7. THE SQUARE ROOT OF A X BISYMMETRIC MATRIX It follows from Equatio 6 that, x + y xy x xy + y / x y = y x We ow pose the problem to fid x ad y such that, a b b a / = x y y x. (8) where a is the hypoteuse ad b is the altitude of a right triagle with vertex agle θ whose base is a b, ta = b / a b usig stadard trigoometric idetities, ta θ / = y / θ ad x cosθ taθ / = where cosθ = + cosθ After some algebra, a b a. As a result, a taθ / = a b b which implies that, x = kb ad y = k( a a b ). (9) But, sice the hypoteuse of the right triagle with vertex θ / is a, x + y = a (30) Replacig Equatios 9 ito 30 ad solvig for k it follows after some algebra that, b x = ad y which agrees with Equatios 8a ad b. y = a a b 7.3 PYTHAGOREAN TRIPLES AND POWERS OF x BISYMMETRIC MATRICES For a,b atural umbers with a>b, eve powers of a arbitrary x bisymmetric matrix, a b b a result i a sequece of pairs of whole umbers that are hypoteuse ad side of Pythagorea triples for all values of. The third side will be powers of a b. If a b = c for c a atural umber, i.e., if {a,b,c} is a Pythagorea triple, the all

17 powers of the bisymmetric matrix results i hypoteuse ad side of Pythagorea triples with the third side beig powers of c. It should be oted that the first umber of these Pythagorea triples, a, represets the legth of the hypoteuse ulike the first umber of (a,b) which was the legth of a side. Example : (3,) = 3 Therefore the Pythagorea triple is {3,,5} 3 r = = ad A = (3 )(3 ) = = 3 33 with {33,3,5} = = with {783,78,5} Example : a=5, b=4 where {5,4,3} is a triple = 40 4 with {4,40,9} r ad A = (5 4)(5 4 ) = 80 where = = = = with {365,364,7} 8. Coclusio Petoukhov s geomic matrices have led to ew geomic algebras, geeralizatio of the golde mea, geeralized Fiboacci sequeces, geeralized Nicomachus Triagles, ad to a algorithm for geeratig Pythagorea triples. Pascal s Triagle plays a importat role. Although Petoukhov s matrices reproduce the sequeces of musical fifths foud i the rows of the Nicomachus Triagle, there is o obvious coectio betwee the geomic matrices ad the musical scale. Refereces Darvas, G. (007) Symmetry. Basel: Birkhäuser book. Darvas G., Petoukhov S.V. (005) Symmetries ad tesors of geetic codes. A bioiformatic approach to the geometrizatio of biology. Proceedigs of FIS005, The Third Coferece o the Foudatios of Iformatio Sciece, Paris, July 4-7, 005. Editor: Michel Petitjea, Olie Editio, ISBN , Published i Basel, Switzerlad He, M., Petoukhov, S.V., Ricci,P., 004. Geetic Code, Hammig Distace ad Stochastic Matrices. Bulleti for Mathematical Biology. v.66, p Kappraff, J., Oct. 000a. The Arithmetic of Nicomachus of Gerasa ad its Applicatios to Systems of Proportio. Nexus Network Joural (a electroic joural, Vol. 4, No. 3. Kappraff, J., 000b. Beyod Measure: A Guided Tour through Nature, Myth, ad Number, Sigapore: World Scietific. Kappraff, J., 000c. Coectios: The Geometric Bridge betwee Art ad Sciece. Sigapore: World Scietific. Kappraff, J., G.W. Adamso Geeralized DNA Matrices, Silver Meas, ad Pythagorea Triples. Forma, v.4, #. Kappraff, J Aciet Harmoic Law. Symmetry Festival 009. McClai,.E.G., 976. The Myth of Ivariace, York Beach, Me.: Nicolas-Hays, Ic. Petoukhov, S.V., 00. Geetic Codes : Biary Sub-Alphabets, Bi-Symmetric Matrices ad Golde Sectio ad Geetic Codes II: Numeric Rules of Degeeracy ad a Chroocyclic Theory - Symmetry i Geetic Iformatio, ed. Petoukhov S.V., ISBN , special double issue of the joural Symmetry: Culture ad Sciece, Budapest: Iteratioal Symmetry Foudatio, p. Petoukhov, S.V., 005. The rules of degeeracy ad segregatios i geetic codes. The chroocyclic

18 coceptio ad parallels with Medel s law s - Advaces i Bioiformatics ad its Applicatios (editors M.He, G.Narasimha, S.Petoukhov), Proceedigs of the Iteratioal Coferece (Florida, USA, 6-9 December 004), Series i Mathematical Biology ad Medicie, v.8, pp.5-53, New Jersey-Lodo-Sigapore-Beijig, World Scietific. Petoukhov, S.V., 008a. The degeeracy of the geetic code ad Hadamard matrices, Website of the Los-Alamos Laboratory, ArXiv; Petoukhov, S.V., 008b. Matrix Geetics, Algebras of the Geetic Code, ad Noise Immuity, 36 pages, Moscow: RCD Petoukhov S.V., & He, M., 009. Symmetrical Aalysis Techiques for Geetic Systems ad Bioiformatics: Advaced Patters ad Applicatios. Hershey, USA: IGI Global.