Fractal approximations to some famous constants
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The document provides fractal approximations of famous mathematical constants by representing them as fractions. The fractions become more complex at higher levels of accuracy. The nominator and denominator of each fraction are further decomposed into prime numbers to demonstrate the transcendental nature of the constants. Approximations are given for constants such as pi, e, the golden ratio, Feigenbaum constants, and more. The author developed C++ programs to calculate the fractal approximations and prime decompositions.
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Fractal approximations to some famous constants
- 1. Fractal approximations to some famous constants Chris De Corte chrisdecorte@yahoo.com KAIZY BVBA Beekveldstraat 22 9300 Aalst, Belgium December 20, 2013 1
- 2. The goal of this document is to share with the mathematical community the fractal approximations of some famous constants. The fractals have also been decomposed into prime products. I have used the constants that I found on Wikipedia as a reference. 2
- 3. CONTENTS CONTENTS Contents 1 Key-Words 4 2 Introduction 4 3 Method and techniques 4 4 Famous constants 4 5 Results 7 6 Discussions 7 7 Acknowledgments 7 8 References 7 3
- 4. 4 1 FAMOUS CONSTANTS Key-Words Mathematical constant, fractal, prime decomposition, Archimedes, Napier, Pythagoras, Theodorus, Euler–Mascheroni, Golden ratio, Plastic constant, Embree–Trefethen, Feigenbaum, Twin prime, Meissel–Mertens, Brun, Catalan, Landau–Ramanujan, Viswanath, Ramanujan–Soldner, Erd˜s–Borwein, Bernstein, Gauss–Kuzmin–Wirsing, Hafner–Sarnak–McCurley, o Golomb–Dickman, Cahen, Laplace, Alladi–Grinstead, Lengyel, L´vy, Ap´ry, Mills, Backe e house, Porter, Lieb, Niven, Sierpi˜ski, Khinchin, Frans´n-Robinson, Landau, Universal paran e bolic, Omega, MRB, Reciprocal Fibonacci. 2 Introduction The following document originated out of my interest for primes, prime decomposition and transcendental numbers. As I had never seen a document that summarizes the famous constants as fractals, I thought this might be of interest to some people. Furthermore, I continued in splitting the nominators and denominators into their prime products to show that indeed the constants are transcendental. 3 Method and techniques I developed a C++ program to do the calculation. The program starts off with a simple fraction (ex. 1/3) as a first estimate and then gradually increases the nominator and/or denominator each time a higher accuracy can be achieved up to a final fraction with a more complicated nominator and denominator. The program eventually stops when the demanded accuracy has been reached. The program only increases the nominator and/or denominator when a higher accuracy can be achieved. As such, the proposed fractions are the simplest for their accuracy. Of coarse many other (simple and complex) fractions can be proposed depending on the demanded accuracy. Using another C++ program, I decomposed the resulting nominators and denominators into their prime products. 4 Famous constants In next table, I summarize the approximation of famous constants as a fractal. As can be calculated, they are accurate up to the displayed number of digits. 4
- 5. 4 FAMOUS CONSTANTS Other constants can be calculated on demand. constant constant name value fraction in primes 69305155 22060516 41826352 15387055 54608393 38613965 50843527 29354524 31251629 54142032 39088169 24157817 9018811 6808099 35129 50000 52743222 11295983 36551513 14603619 50865831 77050550 19671359 75225884 17896708 9408621 41689151 47886179 45180024 49325023 29038748 37997709 14149853 12500000 54256793 37383177 79092334 49226721 15281111 54542379 6694823 22046884 5·163·85037 22 ·97·56857 24 ·29·109·827 5·19·161969 72 ·239·4663 5·132 ·45697 72 ·337·3079 22 ·41·71·2521 31251629 24 ·3·7·161137 37·113·9349 73·149·2221 41·219971 192 ·18859 35129 24 ·55 2 ·7·418597 2·3 11295983 17·29·151·491 3·4867873 32 ·19·109·2729 2·52 ·179·8609 19671359 22 ·17·29·37·1031 22 ·19·235483 3·3136207 72 ·17·50047 11·4353289 23 ·3·19·99079 11·587·7639 22 ·227·31981 3·71·178393 23·191·3221 25 ·58 23·2358991 3·13·958543 2·443·89269 3·2459·6673 19·37·21737 3·193·94201 6694823 22 ·59·93419 π Archimedes 3.14159265358979 e √ 2 √ 3 Euler 2.71828182845905 Pythagoras 1.414213562373095 Theodorus 1.732050807568877 γ Euler-Mascheroni 0.577215664901532 φ Golden ratio 1.618033988749894 ρ Plastic 1.324717957244746 β∗ Embree-Trefethen 0.70258 δ Feigenbaum 4.66920160910299 α Feigenbaum 2.502907875095892 C2 Twin prime 0.660161815846869 M1 Meissel–Mertens 0.261497212847642 B2 Brun twin 1.9021605823 B4 Brun quadruplet 0.8705883800 K Catalan 0.915965594177219 K Landau–Ramanujan 0.764223653589220 K Viswanath 1.13198824 µ Ramanujan–Soldner 1.451369234883381 EB Erd˜s–Borwein o 1.606695152415291 β Bernstein 0.280169499023869 λ Gauss–Kuzmin–Wirsing 0.303663002898732 5
- 6. 4 FAMOUS CONSTANTS constant constant name value fraction in primes σ Hafner–Sarnak–McCurley 0.353236371854995 λ Golomb–Dickman 0.624329988543550 − Cahen 0.6434105463 − Laplace limit 0.662743419349181 − Alladi–Grinstead 0.8093940205 Λ Lengyel 1.0986858055 − L´vy e 3.275822918721811 ζ(3) Ap´ry e 1.202056903159594 θ Mills 1.306377883863080 − Backhouse 1.456074948582689 − Porter 1.4670780794 − Lieb’s square ice 1.5396007178 − Niven 1.705211140105367 K Sierpi˜ski n 2.584981759579253 − Khinchin 2.685452001065306 F Frans´n-Robinson e 2.807770242028519 L Landau 0.5 P2 Universal parabolic 2.295587149392638 Ω Omega 0.567143290409783 CM RB MRB 0.187859 ψ Reciprocal Fibonacci 9089869 25733106 26035979 41702272 23797887 36987095 25429634 38370255 38795875 47932001 16763347 15257635 126753231 38693554 61264192 50966133 55816841 42726413 64207145 44096044 47738351 32539748 7115023 4621343 128685955 75466288 32824543 12698172 68778274 25611433 64723195 23051457 1 2 115551430 50336329 52802900 93103279 187859 1000000 32338241 9624804 9152544 17509351 1429·6361 2·33 ·7·19·3583 47·251·2207 27 ·73·4463 3·7932629 5·43·71·2423 2·12714817 3·5·7·11·139·239 53 ·149·2083 13·67·113·487 16763347 5·1091·2797 3·11·37·103811 2·107·180811 26 ·11·17·5119 3·107·179·887 2459·22699 61·700433 5·23·347·1609 22 ·911·12101 47738351 22 ·1171·6947 443·16061 4621343 5·19·1354589 24 ·317·14879 53·619331 22 ·32 ·29·12163 2·31·1109327 563·45491 5·31·211·1979 32 ·11·13·17911 1 2 2·5·11555143 1619·31091 22 ·52 ·19·27791 93103279 7·47·571 26 ·56 13·2487557 22 ·3·7·149·769 25 ·3·95339 17509351 4/π − 1 Chris’ constant 1 3.359885666243177 0.522723200877063 1 drawing a line with this slope can be used to square a circle. One vertical side of the square will go through the interception of the line with the circle 6
- 7. 8 5 REFERENCES Results One can easily test the proposed fractions on their accuracy. If a different accuracy or constant is needed, It can be calculated on demand. 6 Discussions As one can see, all the fractions calculated have different primes in the nominator then in the denominator. This would mean that all these constants are transcendental numbers. 7 Acknowledgments I would like to thank this publisher, his professional staff and his volunteers for all the effort they take in reading all the papers coming to them and especially I would like to thank this reader for reading my paper till the end. I would like to thank my wife for keeping the faith in my work during the countless hours I spend behind my desk. 8 References • http://en.wikipedia.org/wiki/Mathematical constant 7