A similar conjecture to Fortunate numbers based on sum of the divisors of n

The last months of 2015 I have been trying to get more expressions similar to the ones I found in my previous post "About Fortunate numbers and other similar expressions". So this is a follow-up note with some interesting results. This post is also a mirror of the question I did at MSE (click here).

The generalized idea is finding an expression E(n)$$, whose distances between the value of the expression and the next prime closer to the expression, N(E(n))$$, and from the expression to the previous closer prime, P(E(n))$$, are both prime numbers or 1.

 

I tried several combinations, but the following one seems to work properly so far and is not related with factorials or primorials, so I believe it would be easier to test:

 "Record values" are defined as the subset of values sigma_1(m) of the set of sum of divisors of a natural number m who are bigger than any previous values of the set, up to that value m, so it provides a strictly increasing set of values of sigma_1(m). Those are exactly the values of OEIS A034885: (1,3,4,7,12,15,18,28,31,39,42... )

These are the first elements of the set E(n)E(n), associated to their respective values of OEIS A034885 (= "m's whose sigma_1(m) is part of the sequence OEIS A034885"). They are the Highly Abundant Numbers (OEIS A002093):

{1,2,3,4,6,8,10,12,16,18,20,24,30,36,42,48...}

The point is that tested up to 10^27 both the distances of those elements E(n)E(n) to the previous and following closer primes (when there is a previous positive prime) is also a prime number or 1.

Here is the PARI/GP code to test up to 10^9:

testlimit = 10^9;n=1;exitval=0;prevsumdiv=0;while((n<testlimit) && (exitval==0),dj=divisors(n);sum_div=0;for(t=1,length(dj),sum_div=sum_div+dj[t]);while(sum_div<=prevsumdiv,sum_div=0;n=n+1;dj=divisors(n);for(t=1,length(dj),sum_div=sum_div+dj[t]));prevsumdiv = sum_div;np=nextprime(n+1);npmn=np-n;pp=precprime(n-1);nmpp=n-pp;if((nmpp==1 || isprime(nmpp)) && (npmn==1 || isprime(npmn)), print("n= ",n," sigma1(n)= ",sum_div," N(n)= ",np," N(n)-n= ", npmn ," P(n)= ",pp," n-P(n)= ", nmpp);,exitval=1;));

The big question is.. why does it happen? as far as I can see, it does not happen for other values of sigma_1(m), only for the m's behind the record values of sigma_1(m) as explained above. I can not imagine which possible reason is behind that property. I wonder if there are similar expressions based on other non-factorial functions. So far I have tried using the totient function and some sums of divisors of even numbers instead of sigma_1(n), all of them unsuccessfully. I did not find any reference to this possible conjecture yet.

Update 2015/12/24: I have been able to reduce the expression. It is possible to use instead of sigma_1(m) a reduced version of the sum of divisors, I will call it sigma_e(m) defined as the sum of the even composite divisors of m not including m itself in case of being even. In other words, it is the sum of divisors not including the sum of the prime numbers that are divisors, odd divisors, 1 and m itself. E(n) is:

So it is not required to use A034885 anymore. Apart from this expression, I tried also using the sum of the prime divisors of m, and other combinations of partial sums of the divisors of m, including 1 and m itself, but those ones did not work. The even divisors seem to be the key.

This is an example including the interval [1,99].

Update 2016/01/04: I was able to test it for the Highly Abundant Numbers up to 10^27. No counterexamples found.

Update 2016/01/05: Amazingly ∀h∈ {Highly Abundant Numbers}, h>3,tested up to 10^27 no counterexamples found, the closest prime number p<h located to a distance d=(h−p)>1 is also always at a prime distance, so d∈P. If the test is not wrong, these would mean that the even Highly Abundant Numbers greater than 2 have always at least a Goldbach pair of primes. h=p+d,p,d∈P.

Update 2016/01/06: Regarding the Highly Abundant Numbers conjecture I found a similar question from some years ago at MSE here so this idea is not new. The good point of my version is that I was able to avoid the use of the HAN replacing them by the records of the sum of the even non-prime divisors. I have made further tests and it is possible to do the same with the records of the sum of the non-prime multiples of 3 divisors, the distance between the numbers is bigger but the property seems to hold as well.

Hobbymath's wanderings #12

Graphviz (tool to visualize graphs, recommended!)

The Discovery of Quasicrystals - Dan Shechtman (YouTube).

Natural Density (Wikipedia).

Subset Sum problem (Wikipedia).

The Easiest Hard Problem (about partitioning problem).

Ernst Chladni (Wikipedia).

Journal of Integer Sequences.

Fractals from Voronoi Diagrams.

Playing with the Chaos Game (II): visualizing a complex division transformation of a 4-gon fractal lattice

There is a very nice way of representing the points generated by The Chaos Game, by applying complex division. All my previous post about The Chaos Game can be found here.

For instance, let us start with the typical lattice result of The Chaos Game when there are four attractor points (4-gon) located in the corners of a square. To obtain a lattice like the one in the image below, the ratio applied is 0.4 (you can see some other combinations at Wolfram's website). Let us call the four attractor points a1, a2, a3 and a4. As it can be seen in the image, the lattice points tend to be symmetric in the x and y axes. The four attractors are located in the corners of the global square.

 

Now instead of representing those points as they appear in the image above, let us imagine that the points are complex points in the complex plane, z=a+bi, and so are the four attractors, that we will call z1, z2, z3 and z4. It is possible to apply the following transformation: divide each attractor by every lattice point generated by The Chaos Game, this is: d1(z) = z1/z, d2(z) = z2/z, d3(z) = z3/z and d4(z) = z4/z for each z in the lattice points. The resulting numbers d1(z), d2(z), d3(z) and d4(z) are also complex numbers. Let us call the transformed lattice to the set of all the points d1(z), d2(z), d3(z), d4(z) obtained applying the complex division to the whole lattice. The result is as follows:

 

The properties of the complex division and the symmetry of the original lattice points made the result very interesting!

About the similarities between the patterns found on the Complex Division and some Quasicrystal diffraction patterns (II): trigonometric transformations of the lattice points

This is a follow-up of my previous post about the similarities between the patterns found on the complex division and some quasicrystal diffraction patterns.

"If the doors of perception were cleansed, everything would appear to man as it is: infinite". William Blake's "The Marriage of Heaven and Hell"

In the previous post, I showed the results of a complex division test where the original complex plane points are a lattice of points z=[-a,a]+[-a,a]i where a is an integer. Let us call L to the set of complex points z of this lattice. Those lattice points form a mesh of points in the complex plane. When the complex division is performed over the lattice (this means that every z complex number of L is divided by the rest of the points of L and itself) the result is a pattern that initially looks similar to some quasicrystal diffraction patterns (please see my previous post for videos and plottings). Below, left side, the original complex plane lattice points and in the right side the pattern that is generated applying complex division over the lattice points as explained above. These examples are made for a lattice L whose value a is 41, so z=[-41,41]+[-41,41]i. This means that the original lattice L is including 83x83 = 6889 complex numbers.

I wanted to know how the transformation of the original lattice modified the resulting pattern, so I have started to test the trigonometric functions applied to the original lattice points. The results are also quite amazing. Here is a briefing about it:

1. Transformation of the complex numbers z of L as follows: z=sin([-a,a])+sin([-a,a])i. Left side, the result of the transformation of L and in the right side the pattern that is generated.

Here is a video zooming in and out the pattern:

2. Transformation of the complex numbers z of L as follows: z=cos([-a,a])+cos([-a,a])i. Left side, the result of the transformation of L and in the right side the pattern that is generated.

It is a little bit similar to the sine transformation, but it is not identical. It is possible also to make hybrids of sine and cosine functions and the resulting patterns are variations of the patterns shown above.

Here is a video zooming in and out the pattern:

3. Transformation of the complex numbers z of L as follows: z=cosec([-a,a])+cosec([-a,a])i. Left side, the result of the transformation of L and in the right side the pattern that is generated.

Here is a video zooming in and out the pattern:

4. Transformation of the complex numbers z of L as follows: z=cotan([-a,a])+cotan([-a,a])i. Left side, the result of the transformation of L and in the right side the pattern that is generated. 

Here is a video zooming in and out the pattern:

5. Transformation of the complex numbers z of L as follows: z=arctan([-a,a])+arctan([-a,a])i. Left side, the result of the transformation of L and in the right side the pattern that is generated.

Here is a video zooming in and out the pattern:

Some of the trigonometric transformations look similar, but they are not exactly the same patterns. These are some zoomed plottings of them:

1. Sine pattern:

2. Cosine pattern:

 3. Cosecant pattern:

4. Cotangent pattern:

5. Arctangent pattern:

Finally, here are some plottings of the closest points to 0+0i, from left to right, top-down, cosine, sine, cosecant, cotangent, tangent, secant. The cosine and sine, cosecant and secant and cotangent and tangent patterns look similar:

I am still testing some other transformations of the original lattice L, but initially they are not so interesting. I will post them in a former post. Some days ago Terence Tao posted an article named "Sweeping a matrix rotates its graph" explaining the technique named "Sweeping" performed over squared matrices which is also based on division by selecting a pivot point of the main diagonal of the matrix. I will try to use that technique setting up the lattice L as a matrix of complex numbers and see what happens, the results could be a variation of these ones. If somebody reading this knows more about these type of patterns please let me know to share ideas.

Some random plottings

The plottings on this post are only for reference in other places like LinkedIn or MSE.

1. Plotting of the sequence S = {p(n)^p(n+1) mod (p^n+2)} where x=trunc(sqrt(s(i))) and y=s(i)-(x^2): 

How to plot the Gamma function fractal

The Gamma function is an extension of the factorial (n!) function, with its argument shifted down by 1, to real and complex numbers. And there is a fractal associated to it. I just wanted to learn how to make it because I want to learn how to plot the Mandelbrot set, and the Gamma function fractal seems easier to catch the idea of recursion. So here is how to make it!

First of all, I have been following the great explanation at Mathistopheles website, but I needed to study by myself some details to make it work properly.  

Basically, when the gamma function is calculated for a complex number z repeatedly (nested, and shifting down 1 the complex number according to the definition), this is: Gamma(Gamma(...(Gamma(z-1)...-1)-1), the final value result of the nested calculations either tends to 1 or grows without limit. So it is possible to map each z (=a+bi) value of the complex plane (in Cartesian coordinates) with a color code showing if the value tends to 1 (this the same as saying that it tends to the complex number 1+0i) or not in a specific numbers of calculations of the nested Gamma function. 

The condition to tend to 1 is as follows: let us say we want to know if the nested calculation of the Gamma function applied to z = (a+bi) tends to 1. In a first step, calculate Gamma(z-1) = c+di. If the distance of the real part a to 1 is greater or equal to the distance of c to 1 and the absolute value of the imaginary part b is greater than the absolute value of the imaginary part d (abs(b)>=abs(d)) then it is possible to say that z tends to 1+0i when the Gamma function is applied over z (nested as it was explained above). 

It is important to understand that it can happen that the first calculation for a given complex number z1 of Gamma(z1-1) provides a complex number z2 which is not closer to 1 than the original z1. And that does not mean that z1 does not tend to 1 when the nested Gamma function is applied again to z2. It could happen that when Gamma(z2-1) = z3 is calculated, z3 is closer to 1 than z2, and in the moment that this happens then the result is that from that point z1 tends to 1 when the nested calculations are applied. So sometimes we just need one calculation of Gamma for the initial complex number z1 to know if it tends to 1 and sometimes we need two nested Gamma calculations, sometimes 3, 4, 5, ... etc sometimes hundreds or thousands. But the calculation of the computer is very slow when the Gamma function is applied hundreds of times for each complex number, so usually a limit of for instance 7 nested Gamma calculations is applied to decide if a complex number tends to 1 or not. It is not totally accurate, but good enough to start to see the fractal patterns of the Gamma function. The more nested calculations the more accurate the patterns will be, and of course, the more complex numbers are calculated, the more we can zoom into the fractal pattern. 

And here is the result:

It is the typical image of the Gamma fractal function in the complex plane interval [-5,0]+[-1,1]i. I tried to find a good combination of colors. The complex numbers that tend to 1 in just one Gamma calculation are in black color. Then Dark blue two nested calculations, then Pink... finally up to Yellow (7 nested loops). The complex numbers that do not tend to 1 in less than 8 nested calculations are in the Blue color (if the reader can rememeber the good old Fractint, it was the popular color used for that purpose!).

Just to finish here is another version in polar coordinates:

And here is the Python code, modifying the value of the "method" variable it is possible to render Cartesian, Polar or Spherical coordinates. Use it freely:

def gamma_fractal():
    import matplotlib.pyplot as plt
    from scipy.special import gamma
    from numpy import real, imag, around, arange, float, cos, sin, pi

    import matplotlib as mpl

    from mpl_toolkits.mplot3d import Axes3D

   
    #0 Cartesian, 1 Polar, 3 Spherical
    method = 0

    if method==3:
        mpl.rcParams['legend.fontsize'] = 10


        fig = plt.figure()

        ax = fig.gca(projection='3d')
       
    arraydim = 1000000
    xrange = [arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float)]
    yrange = [arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float)]
    zrange = [arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float),arange(arraydim,dtype=float)]
    hexcolor=["#000000","#131667","#F028B3","#42677C","#3EA670","#6DD02D","#F9B90F","#0000FF"]
    pos=[0,0,0,0,0,0,0,0]
   
    ival = 0+1j

    if method!=3:
        firstx = -5
        lastx = 0
        firsty = -1
        lasty = 1
        incx = 0.0005
        incy = 0.0005
    else:
        firstx = -pi
        lastx = pi
        firsty = -pi
        lasty = pi
        incx = 0.002
        incy = 0.002
    posx = firstx
   
   
    while posx < lastx:
        posy = firsty
        while posy < lasty:
            counter = 0           
            mycomplex = gamma((posx+1)+(posy*ival))
            nextcomplex = gamma(mycomplex+1)

            while counter < 7:
                if (abs(mycomplex.imag) > abs(nextcomplex.imag)) and (abs(1-mycomplex.real) > abs(1-nextcomplex.real)):
                    break
                mycomplex = nextcomplex
                nextcomplex = gamma(mycomplex+1)

                counter = counter + 1
           
            if method == 0:
                xrange[counter][pos[counter]] = posx
                yrange[counter][pos[counter]] = posy
            elif method == 1:
                xrange[counter][pos[counter]] = posx*cos(posy)
                yrange[counter][pos[counter]] = posx*sin(posy)
            else:
                xrange[counter][pos[counter]] = cos(posy)*cos(posx)
                yrange[counter][pos[counter]] = sin(posy)*cos(posx)
                zrange[counter][pos[counter]] = sin(posx)

            pos[counter] = pos[counter] + 1

            if pos[counter]==arraydim:
                newxrange = arange(pos[counter],dtype=float)       
                newyrange = arange(pos[counter],dtype=float)
                newzrange = arange(pos[counter],dtype=float)   

                for i in range(0,pos[counter]):
                    newxrange[i]=xrange[counter][i]
                    newyrange[i]=yrange[counter][i]
                    if method == 3:
                        newzrange[i]=zrange[counter][i]

                if method != 3:
                    plt.plot(newxrange,newyrange,",",color=hexcolor[counter])
                else:
                    ax.plot(newxrange, newyrange, newzrange,",",color=hexcolor[counter])


                pos[counter]=0
           
            posy = posy + incy
        posx = posx + incx

    for c in range (0,8):
        newxrange = arange(pos[c],dtype=float)       
        newyrange = arange(pos[c],dtype=float)       
        newzrange = arange(pos[c],dtype=float)   
        for i in range(0,pos[c]):
            newxrange[i]=xrange[c][i]
            newyrange[i]=yrange[c][i]
            if method == 3:
                newzrange[i]=zrange[c][i]

        if method != 3:
            plt.plot(newxrange,newyrange,",",color=hexcolor[c])
        else:
            ax.plot(newxrange, newyrange, newzrange,",",color=hexcolor[c])

           
        pos[c]=0

    if method!=3:
        ax = plt.gca()
        ax.set_axis_bgcolor((0, 0, 0))
        figure = plt.gcf()
        figure.set_size_inches(30, 20)
        if method == 0:
            plt.axis([firstx,lastx,firsty,lasty])
        plt.savefig("gamma_"+str(method)+".png")
        plt.show()
    else:
        #ax.legend()
        #ax.dist=8
        for ii in range(0,361):
            ax.view_init(elev=-178., azim=ii)
            mpl.pyplot.savefig("movie"+str(ii)+".png")
        #ax.view_init(elev=28., azim=ii)
        #plt.show()


gamma_fractal()

Enjoy the fractals!

A very nice pattern based on Euler's identity

Euler's identity is one of the jewels of Mathematics, it blends in one sight some of the most significant constants of the mathematical universe:

I tried to visualize how would look the left part of the identity in the complex plane by using the following function:

It is a tryout of a generic version of the left side of Euler's identity. Applying the change e^{ib}=cos(b)+sin(b)i, the following function is obtained:

For instance, to be able to see the patterns, taking k1,k2,k3 in the interval [−10,10] stepping in by 0.1, and calculating all the possible combinations of triplets (k1,k2,k3) and their values f(k1,k2,k3), this is the pattern that it generates when the complex values a+bi are represented in Cartesian coordinates (a=x,b=y):

A closer approach to 0+0i:

Euler's identity f(1,1,1)=e^{pi*i}+1=0+0i would be located at the center and the rest of points are the closer values of the more generic function f(k1,k2,k3) as it was defined above for the example. As expected is periodic and symmetrical. This is a little video:

Isn't it hypnotic? I have asked at MSE about this pattern, as far as I know it might be the fist time that has been plotted. I hope somebody will be able to propose some insights about it!  

Update 2016/01/15: Professor Brent Yorgey kindly gave a very nice explanation of the pattern at MSE!

About the similarities between the patterns found on the Complex Division and some Quasicrystal diffraction patterns

As the brilliant and wise professor William Thurston said: "Mathematics is an art of human understanding. … Our brains are complicated devices, with many specialized modules working behind the scenes to give us an integrated understanding of the world. Mathematical concepts are abstract, so it ends up that there are many different ways they can sit in our brains. A given mathematical concept might be primarily a symbolic equation, a picture, a rhythmic pattern, a short movie — or best of all, an integrated combination of several different representations."

In the case of the following test, the concepts are pictures and movies. As in my former post regarding the patterns that can be found in the roots of quadratic and cubic equations, I wanted to know if there are patterns hidden in the complex division, and it seems I found some interesting too!  

Indeed, reviewing the results of my former post, and thanks to the kind insights from the administrator of the Number Theory group in LinkedIn, it seems that they look like the patterns of the refraction of some type of quasicrystal structures. Are they related? I do not know, but so far here is what I have found:

1. First this is the description of the test:  the images are showing the set S of complex numbers a+bi represented in the Cartesian plane (a=x, b=y) obtained from the divisions of the complex numbers: z1/z2 , where z1 = (A+Ai) and z2 =  (B+Bi) being A and B natural numbers both of the interval [-41..41]. The set S contains all the possible combinations of divisions between z1 and z2 in those intervals (except when z2=0+0i which is not represented). Then a zoom is applied to show only a zoomed part of the whole pattern.

2. As the intervals used for the test are the same ones, A = B = [-41..41], this is the original lattice or mesh of complex numbers that will be divided between the whole same mesh of complex numbers (every point of this image represents a complex number and the output of the test is the division of each one of those complex numbers by all the complex numbers of the mesh)

3. This is a zoom-out to see the result of the complex division. This is the complete set S seen from the interval x=y=[-50,50]. As it can be seen in the image, due to the complex division formula, different meshes representing different subsets of the complex division are generated rotating on the center 0+0i. Up to the interval x=y=[-1..1], the closer we are to the center, the more different meshes are swapped, and the denser it is the quantity of complex numbers of the set. For smaller intervals inside x=y=[-1..1] (e.g. x=y=[-0.5,0.5]) the closer we get to the center (I will call it nucleus too) the darker it gets. The peak of density is at x=y=[-1..1], so the image is brighter when it is zoomed on that interval. Clicking in the image it is possible to see details on the structures that are generated due to the swapping of the meshes. They are similar to tessellations.

4. This is a zoom-in in the frame x,y in [-1.2,1.2]. There is symmetry in both axis, X and Y.

5. Zooming-in even more, it is possible to see how it looks like the nucleus near 0+0i, this is x,y in [-0.1,0.1]. The more meshes are swapped the more interesting patterns appear. The image below is obtained by using the intervals A=B=[-81..81] instead of A=B=[-41..41] because the closer we get into the center inside x=y=[-1..1] the darker it is, so it is required a bigger set S to have more density and bright (so the pattern closer to the nucleus is easier to see).

Summarizing, the following video shows a zoom-in, zoom-out into the patterns that can be seen in the set of complex numbers obtained from the divisions of the complex numbers: z1/z2 , where z1 = (A+Ai) and z2 =  (B+Bi) where A and B are natural numbers in the interval [-41..41]. The set generates all the possible combinations of divisions between z1 and z2 in those intervals.

So far those are the results of the test. Now, what similarities can be found between those images and the refraction patterns of quasicrystals? Let us compare. 

1. Something basic,  credits to the Wikipedia, this is the X-ray diffraction pattern of the natural Al₆₃Cu₂₄Fe₁₃ quasicrystal.

2. Credits to Steffen Weber, this is a simulation of diffraction pattern, which could represent either electron diffraction patterns or the zeroth layers of precession photographs (X-ray), dodecagonal QC:  

The pattern above looks similar to the zoom-in in the pattern of the complex division for the interval x=y=[-1..1]. I suspect that the complex division produces those patterns due to a n-fold rotational symmetry. See here for more information too.

3. Credits to Damian OHara. More refraction patterns:

4. In this link there is a very good technical description of the refraction patterns. And in this link there are more examples of n-fold symmetries. In this one there are similar star-like patterns as the ones I found at the roots of quadratic equations.

So far this is all I have found. I will try to look for more information, but at least the similarities are quite interesting.

I would like to finish this post with two of the best videos I found about quasicrystal, tesselations and tiling. The first one is from Sir Roger Penrose, "Forbidden crystal symmetry in mathematics and architecture".

And my favorite one! Professor Marjorie Senechal, "Quasicrystals Gifts to Mathematics":

Please if somebody reading this knows more about the subject let me know, I would like to learn more about it.

Hobbymath's wanderings #11

"Dancing with Fractals: The Chaos Game (With Music!)"(pdf).

"Remainders of Square Numbers (Youtube).

Sounding Number: Chaos Game.

Introduction to Algebraic Topology (Youtube).

Heath-Brown’s theorem on prime twins and Siegel zeroes.

Gamma function (Wiki).

Mathistopheles: Fractal Factorials.

Gaussian Primes.

The Complex Tale of Eisenstein Prime Numbers.

Benford's Very Strange Law (YouTube).

Battle of Eigenvalues of al most periodic matrices (YouTube).

Polymath and the gap between primes (YouTube).

Billiards and moduli spaces (YouTube).

Curtis McMullen on "Manifolds, Topology and Dynamics" (YouTube).

Julia Set (Wiki).

A Century of Complex Dynamics.

Fractal patterns and Chaos Game by prof. Robert L. Devaney.

Cantor and Sierpinski, Julia and Fatou: Complex Topology Meets Complex Dynamics, by prof. Robert L. Devaney.

Fractal Factorials (about Gamma function fractal in complex plane).

Landau Lectures| Prof. Thurston 1995/6 (YouTube).

Visualizing the patterns in the sets of complex and real roots of quadratic and cubic equations

This is an extension of a question I did at MSE: "What methods are known to visualize patterns in the set of real roots of quadratic equations?".

I came across a previous awesome question about the visualization of the distribution of polynomial roots and tried to do a simpler version applied first to the set of real roots of quadratic equations ax^2+bx+c=0 and then to cubic equations ax^3+bx^2+cx+d=0 to visualize the complex roots (meaning all the roots of the equation represented as complex numbers) and only the real roots (a subset of the complex roots when the imaginary part is zero).

To visualize the pattern of the relationship between the set of roots of the quadratic equations, I prepared a Python program to calculate the roots x1,x2 only for the specific values of the intervals a in [−ai,ai], b in [−bi,bi], c in [−ci,ci], a,b,c are natural numbers N. If the limits ai,bi,ci are very big, the set of roots is very dense and it is very difficult to visualize the emerging pattern.

Complex roots of quadratic equations in Cartesian coordinates (x = real part of the root, y = imaginary part of the root). ai=bi=ci=75:

Complex roots of quadratic equations represented in polar coordinates (theta = real part of the root, r = imaginary part of the root). ai=bi=ci=65:

Real roots (subset of the complex roots whose imaginary part is zero) of quadratic equations represented in Cartesian coordinates (x,y)=(x1,x2). E.g. ai,bi,ci=75:

This is an special subset: real roots (subset of the complex roots whose imaginary part is zero) of quadratic equations whose coefficients are irrational numbers, when ai,bi,ci=575 and using only the square roots of the prime numbers of those intervals, sqrt(a)/a in [−ai,ai], sqrt(b)/b in [−bi,bi], sqrt(c)/c in [−ci,ci]. This will show the pattern of the roots only for a) a,b,c irrationals (because the square roots of prime numbers are irrational numbers) and still constrained to b) x1,x2 in R:

Real roots (subset of the complex roots whose imaginary part is zero) of quadratic equations represented in polar coordinates (theta,r)=(x1,x2). E.g. ai,bi,ci=75:

Due to the symmetries the opposite patterns (x,y)=(x2,x1) and (theta,r)=(x2,x1) are similar.

Real roots (subset of the complex roots whose imaginary part is zero) of quadratic equations represented in spherical coordinates (theta,phi,r)=(x1,x2,1). E.g. ai,bi,ci=25:

Here is an animation (it will take a little bit to upload, please be patient)

So basically there seems to be a pattern in the relationship between both real roots that can be visualized.

Here is the same exercise for the three roots (x1,x2,x3) of the cubic equations. 

Complex roots of cubic equations represented in Cartesian coordinates (the x value is the real value and the y value is the imaginary value):

Complex roots of cubic equations represented in polar coordinates, (theta, r), each root will be represented by the angle theta = real part of the root and the radius r = the imaginary part of the root.

Representing the real roots requires more imagination because there are three real roots and is possible to play with the symmetry of the roots.

Real roots (subset of the complex roots whose imaginary part is zero) of cubic equations represented in Cartesian coordinates (x,y)=(x1,x2),(x1,x3),(x2,x3),(x2,x1),(x3,x1),(x3,x2). E.g. ai,bi,ci=25:

Real roots (subset of the complex roots whose imaginary part is zero) of cubic equations represented in polar coordinates (theta,r)=(x1,x2),(x1,x3),(x2,x3),(x2,x1),(x3,x1),(x3,x2). E.g. ai,bi,ci=20:

Finally, the real roots (subset of the complex roots whose imaginary part is zero) of cubic equations represented in spherical coordinates (theta,phi,r)=(x1,x2,1),(x2,x3,1),(x1,x3,1),(x2,x1,1),(x3,x2,1),(x3,x1,1). E.g. ai,bi,ci=25:

I would like to learn other methods to visualize the sets of roots and would appreciate very much any feedback about this subject.

Multifractals in the first occurrence of each k = {0,1,2,3,4,5,6,7,8,9} in the values of f(n)=sqrt(n)−trunc(sqrt(n))

Learning how to generate the Mandelbrot set, I came across the definition of the "escape condition" which is the one that decides the color that is applied to each point of the plane where the Mandelbrot set is being calculated. This is a mirror of the question at MSE about the observation I did below.

I tried to use that "escape condition" concept in a test regarding the fractional part of sqrt(n), f(n)=sqrt(n)−trunc(sqrt(n)) truncated to 7 decimals. The algorithm finds for all f(n) the first occurrence of a specific value k={0,1,2,3,4,5,6,7,8,9} in the decimals of f(n). For instance 0.0123456 has its first decimal value k=0 in position 1, in other hand 0.1230320 has its first decimal value k=0 in position 4. If it is not found, the escape value will be the maximum possible value, 8. As in the Mandelbrot set depending of the escape value, a different color will be used.

A) First I have organized the natural numbers in the positions a_xy of a plane as follows:

a_n0=n^2

a_n1=(n^2)+1

a_n2=(n^2)+2

a_n3=(n^2)+3

...

a_ni=(n^2)+i

while i<n

In the first file there are squares, in the second squares + 1, in the third squares +2, etc. while the values are less than the next square: 

B) Then each element a_ni is replaced by f(a_ni)=sqrt(a_ni)−trunc(a_ni),  truncating up to 7 decimals. 

C) Finally, for all f(a_ni) now it is possible to find the first occurrence of for instance k=0 in f(a_ni). It is found starting from the upper decimal value. The position where it is found is the "escape position", and each position will have its own color. 

This is the plotting of colors of each escape position for the algorithm when looking for the first occurrences of k=0 in f(a_ni):

Same exercise for the first occurrences of k=1:

They are very similar, but they are not the same. It seems that there are star-like patterns acting like local "attractors" (please excuse me if I abuse of the terminology).

This is an animated image with the same graph when looking for the first k=0, then k=1... up to the first k=9. Same color in the animation means that the first occurrence of k is in a same specific position. It is reduced to fit the screen, when clicking in the image it is possible to see it correctly:

When looking to the animation, it seems that the first occurrences of each type of k are "rotating" around the "local attractors". E.g. at column n=500 is easy to observe the rotation.

It seems that there is a symmetry in the background between the upper side and the lower side of the bisection of the triangle that has (0,0) as the bisection vertex. The direction of the "rotation" explained above is inverted between both sides of the bisection (the upper part "rotates" to the right, and the lower part to the left), but the mapping of colors in both sides seem to be initially the same one.

I have plotted the images with a gray scale gradient, so now it is possible to see more details of the above rotating picture:

The following animation is the search of the position of the first decimal value k=0 in the fractional digits obtained by applying the functions f(n)=n^t−trunc(n^t), where t belongs to the interval [0.49995..0.50006] and each frame is increasing by 0.00001. The pattern "bends" in XY and the background star-like patterns are relocated depending on the slight changes of t. 

Finally, as +XY is symmetric in −XY this is how it looks like the complete picture, without the restriction regarding the values being less than the next square, so it fills the XY plane completely:

This is what I have found so far, there is a comment at my MSE question that explained that these patterns are multifractals, and there is a page with similar star-like patterns appearing in the representation of modular arithmetic products. That is quite interesting, because initially my graphs are not directly related with modular arithmetic, but if the same patterns arise, then there could be a link between modular arithmetic and the first occurrence of a specific digit in the fractional part of sqrt(n).

UPDATE: it seems that my algorithm is producing Moiré patterns. I did the same test using the fractional part of n/m and a similar pattern appears. Using the color code in the first graphs should avoid that effect, indeed the pattern is there independently of the zoom (so it would not be Moiré if I can see the pattern independently of the zoom) but in the case of the polar coordinates, it seems too similar to this. Maybe the examples of the modular arithmetic website are also Moiré patterns, the circular examples in the site look exactly like this. 

Once I found the relationship with the Moiré patterns, just found at MSE a question about the same topic, so somehow the algorithm I did and the modular arithmetic calculations both seem to generate Moiré patterns.