Galleries of Visualizations about Golden Ratio, Silver Ratio, Bronze Ratio,
Original Metallic Ratios, Similar Metallic Ratios,
and These Equiangular Spirals (Logarithmic Spirals)
based on Pythagorean Theorem, Binomial Theorem, Addition theorem, Addition theorem,
and knight's moving types or skipped Generalized Fibonacci or Lucas Sequences
with Pell or Jacobsthal Sequence

Shingo Nakanishi (Osaka Institute of Technology), "Visualizations and Geometric Characterizations of Equiangular Spirals about Similar Metallic Ratios Using Generalized Fibonacci Sequences, Pythagorean Theorem, and Binomial Theorem", Personal research note in Japanese, (2021.12.03)

Stairways to heaven using divine proportion, golden ratio

Extended version of continued fractions and nested radicals about 2 times of the silver ratio.

Extended version of continued fractions and nested radicals about n times of the metallic ratio.

The symmetries about second continued fractions which do not follow current rules, these related nested radicals, and these equiangular spirals (logarithmic spirals).

The symmetries about twelfth continued fractions which do not follow current rules, these related nested radicals, and these equiangular spirals (logarithmic spirals).

There might be two roots about metallic ratios.

How beautiful about nested radicals and continued frations are?

Fig.1 Concepts about Kepler triangles using golden ratio and Pythagorean theorem

Fig. 2 Golden ratio using Kepler triangle and cumulative distribution function of standard normal distribution

Fig. 3 Mathematical and geometric characterizations about similar metric ratios and these Pythagorean theorem

Fig.4 Geometric definitions about similar metallic ratios based on generalized Fibonacci sequences and Pythagorean theorem

Fig. 5 Concept of similar metallic ratios using simultaneous probabilities on the joint standard normal distribution

Fig .6 First similar metallic ratio related to the golden ratio using Pythagorean theorem, Kepler triangles, and Fibonacci sequence

Fig .7 Second similar metallic ratio related to the silver ratio using Pythagorean theorem, right-angled isosceles triangle, and Jacobsthal sequence

Fig .8 Third similar metallic ratio related to the bronze ratio using Pythagorean theorem and generalized Fibonacci sequence

Fig .9 Twelfth similar metallic ratio related to the platinum ratio using Pythagorean theorem and generalized Fibonacci sequence

Fig .10 Fractals and equiangular spirals using Pythagorean theorem and generalized Fibonacci sequence about first ratio called the golden ratio and second ratio related to the silver ratio

Fig. 11 Several illustrated cases of series about similar metallic ratios

Fig .12 Archimedean spirals of similar metallic ratios

Fig .13 Geometric characterizations about similar metallic ratios related to the ellipses with the same focuses (±1,0)

Fig .14 General forms and matrices of generalized Fibonacci sequences using similar metallic ratios

Fig .15 Geometric relations about similar metallic ratios using these right triangles based on geometric means related to these eigen values

Fig .16 Visualizations of Fibonacci sequences and Jacobsthal sequences using Pascal's triangles

Fig .17 Weighted digit shifts of Fibonacci sequences and Jacobsthal sequences using Pascal's triangles

Fig .18 Visualizations of generalized Fibonacci sequences using Pascal's triangles

Fig .19 Relations about continued fractions of similar metallic ratios and applied Pascal's triangles

Fig .20 Visualizations of negative generalized Fibonacci sequences using Pascal's triangles

Fig .21 Concepts of equiangular spirals of similar metallic ratios using binomial theorem and Gaussian plane

Fig .22 Visualizations and Definitions of equiangular spirals of similar metallic ratios on the Gaussian plane using Kepler triangles or right-angled isosceles triangles

Fig .23 Visualizations and Definitions of equiangular spirals of similar metallic ratios on the Gaussian plane using binomial theorem and De Moivre's formula

Fig .24 Concepts of similar metallic ratios using Pythagorean theorem, ellipses with the same focuses, and equiangular spirals on the Gaussian plane (Case n=12)

Fig .25 Concepts of similar metallic ratios using Pythagorean theorem, ellipses with the same focuses, and equiangular spirals on the Gaussian plane (Case n=1,2,3,6,9,and 12)

Fig .26 Concepts of similar metallic ratios using Kepler triangles and right-angled isosceles triangles for equiangular spirals on the Gaussian plane (Case n=1 or 2)

Fig .27 Harmonies of similar metallic ratios using Pythagorean theorem, Kepler triangles, and right-angled isosceles triangles

Fig .28 Relations of Archimedean spirals and these areas about squares (Part 1)

Fig .29 Relations of Archimedean spirals and these areas about squares (Part 2)

Fig .30 Relations of Archimedean spirals and these areas about squares (Part 3)

Fig .31 Concepts about Pythagorean theorem related to the similar metallic ratios

Fig .32 Harmonies of similar metallic ratios using Kepler triangles and right-angled isosceles triangles for Archimedean and equiangular spirals (Case n=1 and 2)

Fig. 33 Mt. Ikoma and Yodo river from Omiya campus of Osaka Institute of Technology, Vitruvian man, and others

Fig. 34-1 Harmonies and symmetries about Newton's binomial theorem, applied Pascal's triangle, and generalized Fibonacci sequences

Fig. 34-2 These beauty and attractiveness using nested radicals, continued fractions, and equiangular spirals

Fig. 35 The concept of this research note about similar metallic ratios with standard normal distribution

Fig. 36 The new symmetry of golden ratio using continued fractions and nested radicals (Part 1)

Fig. 37 The new symmetry of golden ratio using continued fractions and nested radicals (Part 2)

Fig. 38 Relations of Kepler triangles, generalized Fibonacci sequence, and equiangular spirals

Fig. 39 Relations of right-angled isosceles triangles, generalized Fibonacci sequence, and equiangular spirals

Fig. 40 Relations of right triangles and generalized Fibonacci sequence

Fig. 41 The concept of equiangular spirals using circles, points, lines, and parabolas

Fig. 42 The geometric concepts of golden ratio and similar metallic ratios

Fig. 43 Symmetries of equiangular spirals using equilateral triangle and 12 (Part 1)

Fig. 44 Symmetries of equiangular spirals using equilateral triangle and 12 (Part 2)

Fig. 45 Symmetries of equiangular spirals using equilateral triangle and 12 (Part 3)

Fig. 46 Symmetries of equiangular spirals using equilateral triangle and 12 (Part 4)

Fig. 47 Symmetries of equiangular spirals using equilateral triangle and 12 (Part 5)

Fig. 48 Design patterns of some illustrative numerations based on metallic ratios such as Zeckendorf representations

Fig. 49 Design patterns of some illustrative numerations based on similar metallic ratios such as Zeckendorf representations

Fig. 50 Concept of two illustrative equiangular spirals based on similar metallic ratios (Part 1)

Fig. 51 Concept of two illustrative equiangular spirals based on similar metallic ratios (Part 2)

Fig. 52 Concept of cuboids based on similar metallic ratios using Kepler cuboid and cube

Fig. 53 Concept of Kepler cuboid and golden ratio

Fig. 54 Concept of cube and second ratio

Fig. 55 Development views of cuboids using similar metallic ratio from n=1 to 3

Fig. 56 Concept of cuboids using similar metallic ratio from n=1 to 3

Fig. 57 Parabola, spiral of Theodorus, and 12 cuboids using similar metallic ratio from n=1 to 12

Fig. 58 Spiral staircases using similar metallic ratio from n=1 to 3

Fig. 59 Geometric symmetries of two types of equiangular spirals using similar metallic ratio from n=1 to 12 (Part 1)

Fig. 60 Geometric symmetries of two types of equiangular spirals using similar metallic ratio from n=1 to 12 (Part 2)

Fig. 61 Concept of parabola and cardioid of equiangular spirals using similar metallic ratio from n=1 to 12

Fig. 62 Concept of radical roots, sinusoidal spiral, and equiangular spirals using similar metallic ratio from n=1 to 12 (Part 1 Lemniscate of Bernoulli, Hyperbola, Humbert cubic, Kiepert cubic)

Fig. 63 Concept of radical roots, sinusoidal spiral, and equiangular spirals using similar metallic ratio from n=1 to 12 (Part 2 Lemniscate of Bernoulli, Hyperbola, Humbert cubic, Kiepert cubic)

Fig. 64 Concept of radical roots, sinusoidal spiral, and equiangular spirals using similar metallic ratio from n=1 to 12 (Part 3 Humbert cubic, Kiepert cubic)

Fig. 65 Concept of radical roots, sinusoidal spiral, and equiangular spirals using similar metallic ratio from n=1 to 12 (Part 4, Lemniscate of Bernoulli, Hyperbola)

Fig. 66 Concept of powers, sinusoidal spiral, and equiangular spirals using similar metallic ratio from n=1 to 12 (Part 1, Parabola, Cardioid, Tschirnhausen cubic, Cayley's sextic)

Fig. 67 Concept of powers, sinusoidal spiral, and equiangular spirals using similar metallic ratio from n=1 to 12 (Part 1, Parabola, Cardioid, Tschirnhausen cubic, Cayley's sextic)

Fig. 68 Concept of Lemniscate and rectangular hyperbola using sinusoidal spiral and equiangular spirals using similar metallic ratio from n=1 to 12

Fig. 69 Concept of sinusoidal spiral and equiangular spirals using similar metallic ratio from n=1 to 12

Fig. 70 Cardioid, parabola, lemniscates with torus, and hyperbolas with cones about equiangular spirals using similar metallic ratio from n=1 to 12

Fig. 71 Illustrative map of sinusoidal spiral using the power of equiangular spirals about similar metallic ratios(Part 1)

Fig. 72 Illustrative map of sinusoidal spiral using the power of equiangular spirals about similar metallic ratios(Part 2)

Fig. 73 Illustrative map of sinusoidal spiral using the power of equiangular spirals about similar metallic ratios(Part 3)

Fig. 74 Harmony between quadratic curves, (i.e. conic curves such as ellipse, parabola, hyperbola,) and equiangular spirals using right triangles based on similar metallic ratios (Part 1)

Fig. 75 Harmony between quadratic curves, (i.e. conic curves such as ellipse, parabola, hyperbola,) and equiangular spirals using right triangles based on similar metallic ratios (Part 2)

Fig. 76 Geometric visualizations about equiangular spirals using similar metallic ratios (2022)

Fig. 77 Lemniscate, hyperbola, and equiangular spirals on the torus and cones (2022)

Fig. 78 Geometric visualizations about limacon of Pascal, Cayley's sextic, cardioid, and circle using the right triangles about similar metallic ratios (2022)

Fig. 79 Geometric visualizations about limacon of Pascal, cardioid, Cayley's sextic, and circle using the right triangles about similar metallic ratios (Part 2, 2022)

Fig. 80 Geometric visualizations about limacon of Pascal, cardioid, Cayley's sextic, and circle using the right triangles about similar metallic ratios (Part 3, 2022)

Fig. 81 Geometric visualizations about limacon of Pascal, cardioid, Cayley's sextic, and circle using the right triangles about similar metallic ratios (Part 4, 2022)

Fig. 82 Geometric visualizations about limacon of Pascal, cardioid, and circle using the right triangles about similar metallic ratios (2022)

Fig. 83 Geometric visualizations about Cayley's sextic, cardioid, and circle using the equiangular spirals and these right triangles about similar metallic ratios (2022)

Fig. 84 Geometric visualizations about Cayley's sextic, limacon of Pascal, cardioid, and circle using the equiangular spirals and these right triangles about similar metallic ratios (2022)

Fig. 85 Geometric charactrizations about limacon of Pascal, circle, and these right triangles (2023)

Fig. 86 Geometric visualizations of crdioids about Kepler triangles or isosceles right triangles (n=1 or 2, Part 1)(2023)

Fig. 87 Geometric visualizations of crdioids about Kepler triangles or isosceles right triangles (n=1 or 2, Part 2)(2023)

Fig. 88 Geometric characterizations of drawing crdioids using isosceles right triangles (2023)

Fig. 89 Geometric visualizations about crdioids, Kepler triangles, and equiangular spirals (2023)

Fig. 90 Kepler triangles with crdioids or limacons of Pascal (2023)

Fig. 91 Geometric visualizations about stretching or reversing of drawing limacons of Pascal (2023)

Fig. 92 Comparisons about drawing methods of limacons of Pascal (2023)

Fig. 93 Visualizations about similar metallic ratios based on right triangles and cardioids (Part 1) (2023)

Fig. 94 Visualizations about similar metallic ratios based on right triangles and cardioids (Part 2) (2023)

Fig. 95 Geometric visualizations about original metallic ratios using semicircles, squares, and parabolas (Part 1) (2023)

Fig. 96 Geometric visualizations about original metallic ratios using semicircles, squares, and parabolas (Part 2)(2023)

Fig. 97 Geometric visualizations about similar metallic ratios using semicircles, squares, and parabolas (Part 1) (2023)

Fig. 98 Geometric visualizations about similar metallic ratios using semicircles, squares, and parabolas (Part 2) (2023)

Fig. 99 Illusrations of the design about original metallic ratios using semicircles, and squares (2023)

Fig. 100 Illusrations of the design about original metallic ratios using semicircles, and squares (2023)

Fig. 101 Illusrations of nested radicals and extended continued fractions about original metallic ratios times these order using squares (2023)

Fig. 102 Illusrations of nested radicals and extended continued fractions about similar metallic ratios using inverses (2023)

Fig. 103 Illusrations of nested radicals and extended continued fractions about similar metallic ratios using inverses of square roots (2023)

Fig. 104 Illusrations of the design about original metallic ratios using semicircles, and squares (2023)

Fig. 105 Illusrations of the design about similar metallic ratios using semicircles, and squares (2023)

Fig. 106 Illusrations of the design about normalized similar metallic ratios using semicircles, and squares and the related original metallic ratios (2023)

Fig. 107 Nested radicals and extended continued fractions of the design about similar metallic ratios using semicircles, and squares (2023)

Fig. 108 Nested radicals and extended continued fractions of the design about original metallic ratios using semicircles, and squares (2023)

Fig. 109 Nested radicals and extended continued fractions of the design about normalized similar metallic ratios using semicircles, and squares and the related original metallic ratios (2023)

Fig. 110 Visualizezations about original metallic ratios (sush as golden ratio, siliver ratio, and bronze ratio) with semicircles, and squares using related sequences(2023)

Fig. 111 Visualizezations about Japanese platinum ratio with semicircles, and equilateral triangles (2023)

Fig. 112 Harmonies and symmetries about Newton's binomial theorem, applied Pascal's triangle, and generalized Fibonacci sequences with the line using zero (2023)

Fig. 113 Harmonies and symmetries about Newton's binomial theorem, applied Pascal's triangle, and Lucas sequences with the line using powers of two (2023)

Fig. 114 Beauty and truth of visualizations between generalized Fibonacci sequence and Lucas sequence (2023)

Fig. 115 Beauty and truth of visualizations between generalized Fibonacci sequence and Lucas sequence (2023)

Fig. 116 Visualizations about equiangular spirals with Fibonacci sequence and Lucas sequence using Kepler triangles,and Pythagorean theorem (2023)

Fig. 117 Visualizations about equiangular spirals with Jacobstahl sequence and Jacobstahl Lucas sequence using isosceles right triangles, and Pythagorean theorem (2023)

Fig. 118 Visualizations about equiangular spirals with generalized Fibonacci sequence and Lucas sequence using related right triangles, similar metallic ratios and Pythagorean theorem (2023)

Fig. 119 Visualizations of golden ratio using nested radical, extended continued fraction, Fibonacci sequence, Lucas sequence, squares, and circles (2023)

Fig. 120 Visualizations of silver ratio using nested radical, extended continued fraction, Pell sequence, Pell-Lucas sequence, squares, and circles (2023)

Fig. 121 Visualizations of first and second ratio using Fibonaccisequence, Lucas sequence, Jacobstahl sequence, Jacobstahl-Lucas sequence, and circles (2023)

Fig. 122 Extended version of Fibonacci sequences triangleand Lucas sequences triangle using the concept of Hosoya's traiangle (2023)

Fig. 123 Visualization about Padovan sequence using Pascal's traiangle (2023)

Fig. 124 Visualization about Perrin sequence using modified Pascal's traiangle (2023)

Fig. 125 Visualization about Padovan and Perrin sequence using modified Pascal's traiangles with plastic ratio(2023)

Fig. 126 Visualization about Padovan sequence and its spiral using modified Pascal's traiangles with plastic ratio(2023)

Fig. 127 Visualization about golden ratio or plastic ratio and these related sequences using modified Pascal's traiangles or equiangular spirals (2023)

Fig. 128 Visualization about Padovan sequence and Perrin sequence with plastic ratio(2023)

Fig. 129 Visualizations about equiangular spirals with Fibonacci sequence and Lucas sequence using Kepler triangles,and Pythagorean theorem (Part2, 2023)

Fig. 130 Squares and circles about golden ratio and silver ratio with Fibonacci sequence, Lucas sequence, Pell sequence, Pell Lucas sequence, and extened version of continued fraction and nested radical of metallic raios (2023)

Fig. 131 Squares and circles about golden ratio and silver ratio with Fibonacci sequence, Lucas sequence, Pell sequence, Pell Lucas sequence, and extened version of continued fraction and nested radical of metallic raios (Part2, 2023)

Fig. 132 Type k-2 skkiped Fibonacci sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 133 Type k-2 skkiped Lucas sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 134 Concept of the one skkiped generalizecd Fibonacci sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 135 Concept of the two skkiped generalizecd Fibonacci sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 136 Concept of the one skkiped generalizecd Lucas sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 137 Concept of the two skkiped generalizecd Lucas sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 138 Concept of the two skkiped generalizecd Fibonacci sequence and Lucas sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 139 Concept of the one or two skkiped Pell sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 140 Concept of the one or two skkiped Pell Lucas sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 141 Concept of the one skkiped generaluzed Fibonacci sequence with modified Pascal's triangles and similar metallic ratios (2023)

Fig. 142 Concept of the one or two skkiped Jacobsthal sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 143 Concept of the one or two skkiped Jacobsthal Lucas sequence with modified Pascal's triangles and extended Knight's movings (2023)

Fig. 144 Extended nested radical about first and second ratios of similar metallic ratios (2023)

Fig. 145 Extended nested radical about golden and silver ratios of metallic ratios (2023)

Fig. 146 Equilateral triangles using Fibonacci and Lucas, or Pell and Pell Lucas sequenceses (2023)

Fig. 147 Equilateral triangles using Fibonacci and Lucas sequenceses (2023)

Fig. 148 Equilateral triangles using Fibonacci and Lucas, or Pell and Pell Lucas sequenceses (Part2, 2023)

Fig. 149 The golden ratio with Pythagorean theorem, Fibonacci sequence, Kepler triangle, equiangular spiral, and nested radicals (2023)

Fig. 150 Visualization of the ratio such as golden ratio and plastic ratio with the related sequences (2023)

Fig. 151 Spirals using equilateral triangles such as plastic ratio with the related sequences (2024)

Fig. 152 Spirals using equilateral triangles such as plastic ratio with the related sequences (Part 2, 2024)

Fig. 153 Spirals using equilateral triangles such as plastic ratio with the related sequences (Part 3, 2024)

Fig. 154 Spirals using equilateral triangles such as golden ratio with the Fibonacci sequence, Lucas sequence, and Mulatu sequence (2024)

Fig. 155 Spirals using equilateral triangles such as golden ratio, silver ratio, bronze ratio with the Fibonacci sequence, Lucas sequence, Pell sequence, Pell-Lucas sequence, k Pell sequence, and k Pell-Lucas sequence (2024)

Fig. 156 Dynamically modified pascal's triangles for Fibonacci sequences and Lucas sequences (2023)

Fig. 157 Dynamically modified pascal's triangles for 0, 1, and 2 skipped Fibonacci sequences (2023)

Fig. 158 Dynamically modified pascal's triangles for 0, 1, and 2 skipped Lucas sequences (2023)

Fig. 159 Dynamically modified pascal's triangles for 2 skipped Gibonacci sequences (2023)

Fig. 160 Dynamically modified pascal's triangles for weighted Fibonacci sequences and Lucas sequences (2023)

Fig. 161 Concepts of plastic ratios and the related sequences with equilateral triangles (2024)

Fig. 162 Dynamically modified pascal's triangles for Padovan sequences and Perrin sequences (2023)

Fig. 163 Dynamically modified pascal's triangles for Padovan sequences and Perrin sequences (Part 2, 2023)

Fig. 164 Visualization of Matrices for Padovan sequences and these computing (2024)

Fig. 165 Visualization of Matrices and these computing for skipped Fibonacci sequences (2024)

Fig. 166 Visualization modified Pascal's triangles for 1 or 2 skipped Fibonacci sequences and Lucas sequences (2023)

Fig. 167 Visualization modified Pascal's triangles for 1 or 2 skipped Pell sequences and Pell-Lucas sequences (2023)

Fig. 168 Visualization modified Pascal's triangles for 1 or 2 skipped Jacobsthal sequences and Jacobsthal-Lucas sequences (2023)

Fig. 169 Concepts of modified Pascal's matrices and the related metallic ratios (2024)

Fig. 170 Numerical examples of modified Pascal's matrices and the related metallic ratios (2024)

Fig. 171 Numerical examples of modified Pascal's matrices for 1 skipped Fibonacci, Pell, Jacobstahl, Lucas, Pell-Lucas, and Jacobstahl-Lucas sequences (2024)

Fig. 172 Numerical examples of modified Pascal's matrices for 2 skipped Fibonacci, Pell, Jacobstahl, Lucas, Pell-Lucas, and Jacobstahl-Lucas sequences (2024)

Fig. 173 Numerical examples of modified Pascal's matrices for negativeorders of skipped Fibonacci sequence (2024)

Fig. 174 Numerical examples of modified Pascal's matrices for negativeorders of skipped Lucas sequence and Jacobstahl-Lucas sequence (2024)

Fig. 175 GIF Animation of modified Pascal's triangles for several skipped Fibonacci, Pell, Jacostahl sequences and these related Lucas sequencea (2023)

Fig. 176 Double spirals using equilateral triangles such as golden ratio, silver ratio, bronze ratio with the Fibonacci sequence, Lucas sequence, Pell sequence, Pell-Lucas sequence, k Pell sequence, and k Pell-Lucas sequence (2024)

Fig. 177 Triple spirals using equilateral triangles for x powers of 2 which is the sequence basad on Jacobsthal sequence with changing initial values (2024)

Fig. 178 Single, double, and triple spirals using equilateral triangles using plastic ratio, golden ratiom and 2 or these related sequences spirals such as Padovan sequence, Fibonacci sequence, and Jacobsthal sequence with changing initial values (2024)

Fig. 179 Equilateral triangle Spirals for Fibonacci sequence, Lucas sequence,and Mulatu sequence about golden ratio (2024)