## Saturday, September 2, 2023

### Origins of Life’s Sensoria: Endless Discoveries Most Fascinating: Emergence, Receptors, Perception and Consciousness

Available on Amazon:

This book may change your view of life. We take our senses for granted. Rarely do we seek to understand them; yet we depend upon them our entire lives. This book represents more than thirty years of scientific research to understand how cellular receptors create our sensoria (senses), perceptions, and consciousness. This book may answer many of your questions and may also answer questions that you did not previously consider.

## Saturday, October 16, 2021

### Simple Observations on Ford Circles (Speiser circles or Haros Circles) with the Harmonic Sequence that Suggest Deeper Connections to Number Theory

Simple Observations on Ford Circles (Speiser circles or Haros Circles) with the Harmonic Sequence that Suggest Deeper Connections to Number Theory

Richard G. Lanzara

These are vast areas that need further exploration. I only present my humble and simple observations that may help those with a deeper knowledge of number theory to make future discoveries.

Like pearls on a necklace, there are two parabolas that intersect the center of the Ford circles (Speiser circles or Haros circles) that include the Harmonic sequence (Hn) (1/2,1/3,1/4,1/5,…) (y=(1/2)x^2) and the sequence ((n-1)/n) (1/2,2/3,3/4,4/5,5/,…) (y=(1/2)(x-1)^2). To this author’s knowledge, these parabolas were previously overlooked, but appear to be an essential connection between the Farey sequence, Ford circles and the Harmonic sequence. They may have deeper connections to number theory that are beyond this author’s expertise.

Figure 1: Connections between the Ford circles and the two parabolas (y=(1/2)x^2 and y=(1/2)(x-1)^2).

Looking at the tangent lines between the two adjacent Ford circles (for example at, x=1/2 and 1/3 on the left side and x=1/2 and 2/3 on the right side), we see that for each pair of Ford circles on each side of the middle Ford circle, at x=1/2, the tangent lines always converge on the x=1/2 line (proof not shown).

Figure 2: The tangent lines between the Ford circles meet on the x=1/2 line.

All the y-intercept points where these pairs of tangent lines meet are on the x=1/2 line, and are given by the following progression of intercept points: -1/5, -5/7, -11/9, -19/11, -29/13, -41/15, -55/17, -71/19, -89/21, -101/23… These intercept points have the odd numbers as denominators with the numerators being of the form n+(n+1)^2, which is found in the integer sequence data base (oeis) as A028387 (a(n)=n+(n+1)^2) and A110331 (related to the Pell numbers) and A165900, which is also the Fibonacci polynomial n^2 – n -1 (which is the same as n+(n+1)^2 when n+2 is substituted for n in n^2 – n -1).

***Note that these intersections of the tangent lines progress down the negative side of the y-axis. They all converge at (0,1) or (1,1) for each side of the x-axis (0,1) respectively.

Figure 3: Showing just the tangent lines between successive Ford circles that all intercept the x=1/2 line.

Table I: Equations for the paired tangent lines from the Ford circles surrounding the circle at x= 1/2, (x-1/2)^2 + (y-1/8)^2 = (1/8)^2

 Equations for the tangent lines between the two Ford circles For the Ford circles at x: y-intercept at the line x=1/2 y=1-12x/5 1/2,1/3 -1/5 y=12x/5-7/5 1/2,2/3 -1/5 y=1-24x/7 1/3,1/4 -5/7 y=1/7(24x-17) 2/3,3/4 -5/7 y=1-40x/9 1/4,1/5 -11/9 y=1/9(40x-31) 3/4,4/5 -11/9 y=1-60x/11 1/5,1/6 -19/11 y=1/11(60x-49) 4/5,5/6 -19/11 y=1-84x/13 1/6,1/7 -29/13 y=1/13(84x-71) 5/6,6/7 -29/13 y=1-112x/15 1/7,1/8 -41/15 y=1/13(112x-97) 6/7,7/8 -41/15

Checking oeis.org for the general forms for the numerators and denominators of the tangent line intercepts with the x=1/2 line yields the following general equation for these intercepts, which has the following limit for tangent line intercepts equation:

The Appendix lists more details about the tangent line intercepts, Pythagorean triangles, modified Farey sequences and modified Ford circles (see Appendix below).

Mapping the Ford Circles to the Complex Plane

These figures show the before and after mapping of the Ford circles to the complex plane. The two Ford circles at x=0 and x=1 with both upper and lower bound parabolas are included to better visualize this mapping. Figure 4: The Ford circles with two bounding parabolas, y=(x^2)/2 and y=1-(x^2)/2.

If we allow for a mapping of the Ford circles to the complex plane by changing x to z, then we have the following: Figure 5: The mapping of the Ford circles with two bounding parabolas, y=(x^2)/2 and y=1-(x^2)/2 to the complex plane. Upper and lower plots Demonstrate the Mirror Symmetries Through the Intercepts on the Line y = 1/2.

Figure 6: By translating the y-axis to z = 0.5, the symmetries around the tangent line intercepts from the Ford circles become evident.

The tangent lines of intersection intersect on the y=1/2 line in the complex plane. Note their mirror symmetry and that these lines would extend from the Ford circles to symmetrically intersect the points on the line y = 1/2.

If we consider a picture of an Apollonian gasket, or a Ford configuration of Apollonian type, or Ferry gasket (not shown), then the Ford circles at the top of the gasket (the line y=1 in our case), touching the upper line, will have tangent lines that intersect in a mirror image to these we have seen for the tangent lines for the bottom Ford circles.

Figure 7: When mapped onto the complex plane, these tangent line intersections proceed down both the negative and positive sides of the z-axis at y=1/2.

Figure 8: The symmetries suggesting an Apollonian packing.

Concluding remarks:

Perhaps Figures 1 and 7 are two dimensional representations of a three-dimensional Apollonian packing (also called Apollonian sphere packing, which have many beautiful examples on the internet). Since all proper fractions are underneath the umbrellas of the two parabolas (1/2x^2 and 1/2(x-1)^2) and the unique properties of the Farey sequence insures the orderly inclusion of all proper fractions, the circle at x=1/2 represents a boundary condition. One might visualize this as picking up the x=1/2 line (y=1/2 line in the complex case, Figure 7) with the parabolas forming the sides of this unique three-dimensional shape.

Appendix

Tangent line intercepts:

Table I: Table of the paired tangent line intercepts at x=1/2 with the equations for the tangent lines from the Ford circles surrounding the center circle at x= 1/2, (x-1/2)^2 + (y-1/8)^2 = (1/8)^2.

 y-equations for the tangent lines For the Ford circles at the x values y-intercepts* at the line x=1/2 (x,y) values for the centers of the two circles Perpendicular equations to the tangent line equations x-intercepts** at y=0 y=1-12x/5 1/2,1/3 -1/5 (1/3,1/18) (1/2,1/8) y=5/12x-1/12 5/12 y=12x/5-7/5 1/2,2/3 -1/5 (1/2,1/8) (2/3,1/18) y=-5/12x+1/3 7/12 y=1-24x/7 1/3,1/4 -5/7 (1/3,1/18) (1/4,1/32) y=7/24x-1/24 7/24 y=1/7(24x-17) 2/3,3/4 -5/7 (2/3,1/18) (3/4,1/32) y=-5/12x+1/3 17/24 y=1-40x/9 1/4,1/5 -11/9 (1/5,1/50) (1/4,1/32) y=9/40x-1/40 9/40 y=1/9(40x-31) 3/4,4/5 -11/9 (4/5,1/50) (3/4,1/32) y=-5/12x+1/3 31/40 y=1-60x/11 1/5,1/6 -19/11 (1/6,1/72) (1/5,1/50) y=11/60x-1/60 11/60 y=1/11(60x-49) 4/5,5/6 -19/11 (5/6,1/72) (4/5,1/50) y=-5/12x+1/3 49/60 y=1-84x/13 1/6,1/7 -29/13 (1/7,1/98) (1/6,1/72) y=13/84x-1/84 13/84 y=1/13(84x-71) 5/6,6/7 -29/13 (6/7,1/98) (5/6,1/72) y=-5/12x+1/3 71/84 y=1-112x/15 1/7,1/8 -41/15 (1/8,1/128) (1/7,1/98) y=15/112x-1/112 15/112 y=1/13(112x-97) 6/7,7/8 -41/15 (7/8,1/128) (6/7,1/98) y=-5/12x+1/3 97/112

*All the y-intercept points where these pairs of tangent lines meet are on the x=1/2 line, and are given by the following progression of intercept points: -1/5, -5/7, -11/9, -19/11, -29/13, -41/15, -55/17, -71/19, -89/21, -101/23… These intercept points have the odd numbers as denominators with the numerators being of the form n+(n+1)^2, which is found in the integer sequence data base as A028387 (a(n)=n+(n+1)^2) and A110331 (related to the Pell numbers) and A165900, which is also the Fibonacci polynomial n^2 – n -1 (which is the same as n+(n+1)^2 when n+2 is substituted for n in n^2 – n -1).

**Roots: Subtracting the complimentary equations for y (eg. y=(1-12x/5)-(12x/5-7/5)) always gives x=1/2 as a root, whereas dividing them gives the value for the first x-intercept (eg. y=(1-12x/5)/(12x/5-7/5) gives x= 5/12 as the root).

The general equation for the tangent line intercepts with the x=1/2 line:

y = (-(((n-1) + n^2)/(2(n+1) + 1)))

which has roots:

n = 1/f and -f,

which are the reciprocal and negative values of the golden ratio (1/2(1 + sqrt(5)). This appears to connect the Fibonacci sequence (the ratios of successive terms equal the golden ratio) with the tangent lines between successive Ford circles.

Pythagorean triangles:

Another fascinating property of these tangent lines is that they form groups of Pythagorean triangles with perfect square rational numbers. If we look at where the extensions of the two adjacent circles’ radii meet with the tangent lines, we see that the base of the triangle is formed by the distance between the two circles centers and the height is two times the distance where the y-axis hits the lines extended from the two circles’ radii.

The tangent line intercepts and representative Pythagorean triangles:

It is interesting to note that the Ford circles have a symmetry around, x=1/2, that appears to connect their parabolic, circular and tangent relationships with rational Pythagorean triangles.

Table II: Pythagorean triangles: Equations for the tangent lines from the Ford circles surrounding the circle at x= 1/2, (x-1/2)^2 + (y-1/8)^2 = (1/8)^2, and containing the Harmonic sequence.

 y-equations for the tangent lines x-intercept* at y=0 y-intercept at x=1/2 Rational Pythagorean triples** y=1-12x/5 5/12 -1/5 (6/5,1/2,13/10) y=12x/5-7/5 7/12 -1/5 y=1-24x/7 7/24 -5/7 (12/7,1/2,25/14) y=1/7(24x-17) 17/24 -5/7 y=1-40x/9 9/40 -11/9 (20/9,1/2,41/18) y=1/9(40x-31) 31/40 -11/9 y=1-60x/11 11/60 -19/11 (30/11,1/2,61/22) y=1/11(60x-49) 49/60 -19/11 y=1-84x/13 13/84 -29/13 (42/13,1/2, 85/26) y=1/13(84x-71) 71/84 -29/13 y=1-112x/15 15/112 -41/15 (56/15,1/2, 113/30) y=1/13(112x-97) 97/112 -41/15

**In general, the equation for the Pythagorean triangles for the tangent lines of the Ford circles for the Harmonic sequence is:

For circle, c1, where x1 = 1/a gives the position along the x-axis for c1, and c2, where x2 = 1/b

(2/(a+b))^2 + (1/(ab))^2 = c^2

For example: for the Ford circles at 1/5 and 1/6, this gives:

(2/11)^2 + (1/30)^2 = (61/330)^2

In lowest terms this is:

60^2 + 11^2 = 61^2

Each Pythagorean triangle can be divided into two equivalent triangles where the tangent line crosses the x-axis. In fact, there is a larger Pythagorean triangle related to these other three. This larger triangle is formed where the tangent line crosses the y-axis and where the lowest point of the other triangles meet (where the tangent line crosses the perpendicular line through the center of the larger circle -not shown).

Modified Farey sequence and modified Ford circles:

There is much to discover here, but in the future, I attempt to write more about modified Farey sequences with their accompanying modified Ford circles and their corresponding rational Pythagorean triangles in part II.

For the modified Ford circles:

Modified Farey Sequence and Modified Ford Circles:

Beginning the Farey sequence at 0/2 instead of 0/1, creates a modified Farey sequence that starts as: 0/2, 1/3, 1/1. The next level is 0/2,1/5,1/3,1/2(2/4),1/1. This continues with the right-hand side from 1/2 being the regular Farey sequence, while the left-hand side between 0/2 and 1/3 generates the reciprocals of all the odd numbers, while the region between 1/3 and 1/2 reverts into the original Farey sequence (This also works for the even numbers (not shown)).

In the region of the reciprocal odd numbers, modified Ford circles can be generated using the squares as denominators for the y-values of the circles (y=1/x^2) instead of twice the squares for the regular Ford circles (y=1/(2(x^2)).

The reciprocals of all the odd numbers can now be related to this modified Ferry sequence and their modified Ford circles.

Solving for these tangent lines to the successive, modified Ford circles (only for the odd fractions 1/3,1/5,1/7, etc.), gives the following equations for y:

Table III: Equations for the tangent lines from the modified Ford circles surrounding the circle at x= 1/3, (x-1/3)^2 + (y-1/9)^2 = (1/9)^2

 y-equations for the tangent lines** For the Modified Ford circles at the x values y-intercept at x=1/2* y=1/8(4-15x) 1/3,1/5 -7/16 y=1/12(6-35x) 1/5,1/7 -23/24 y=1/16(8-63x) 1/7,1/9 -47/32 y=1/20(10-99x) 1/9,1/11 -79/40 y=1/24(12-143x) 1/11,1/13 -119/48 y=1/28(14-195x) 1/13,1/15 -167/56 y=1/32(16-255x) 1/15,1/17 -223/64 y=1/36(18-323x) 1/17,1/19 -287/72 y=1/40(20-399x) 1/19,1/21 -359/80 y=1/44(22-483x) 1/21,1/23 -439/88 y=1/48(24-575x) 1/23,1/25 -527/96 y=1/52(26-675x) 1/25,1/27 -623/104 y=1/56(28-783x) 1/27,1/29 -727/112 y=1/60(30-899x) 1/29,1/31 -839/120 y=1/64(32-1023x) 1/31,1/33 -959/128

*For y=1/2 all x=0. The numerators are all of the form 4n+3 and prime, except for 119 (7x17) and 287 (7x41) and 527 (17x31) and 623 (7x89) and 959 (7x137) and 1519 (7^2x31). The general equation: y= (1/(4+4(n-1)))(2n-(4n^2-1)x).

**For the y-equations, the numerators for the slope of x in the tangent equations are of the form 4n^2-1 (A00466, oeis). The denominators are of the form 8+4n, which can be seen to also be the simple sum of the denominators of the x values for the two circles (eg. for c1,c2 with x values, 1/3,1/5; 3+5 =8, the denominator in the equations for the tangent lines.

For the Pythagorean triangles of the modified odd Ford circles:

For circle, c1, where x1 = 1/a gives the position along the x-axis for c1, and c2, where x2 = 1/b

(2/(a+b))^2 + (2/(ab))^2 = c^2

Where the difference between these Pythagorean triangles and the other Pythagorean triangles for the normal Ford circles is the numerator 2, in the (2/(ab))^2 term.

For example: for the Ford circles at 1/3 and 1/5, this gives:

(2/8)^2 + (2/15)^2 = c^2

Which gives c=17/60.