Category Archives: Matrix representation

An important calculation of the 7D number tau (circular version).

3D complex numbers, 5D complex numbers, Exponential curves, Matrix representation

I really took the time to compose this post; basically it is not extremely difficult to understand. Everybody who once has done matrix diagonalization and is still familiar with the diverse concepts and ideas around that can understand what we are doing here.

It is the fact that it is seven dimensional that makes it hard to write down the calculations in a transparent manner. I think I have succeeded in that detail of transparency because at the end we have to multiply three of those large seven by seven matrices with each other and mostly that is asking for loosing oversight.

Luckily one of those matrices is a diagonal matrix and with a tiny trick we can avoid the bulk of the matrix calculations by calculating the conjugate of the number tau.

Just like in the complex plane where the conjugate of the number i equals -i, for tau goes the same.

Basically the numbers tau are always the logarithm of the first imaginary component. But check if the determinant is one because you can use the tau to craft an exponential curve that will go through all basis vectors with determinant one.

This post is 10 pictures long (size 550 x 775), in the beginning I use an applet for the numerical calculation of the matrix representation of the first imaginary unit in 7D space, here is the link:

Matrix logarithm calculator
http://calculator.vhex.net/calculator/linear-algebra/matrix-logarithm

Two years back in 2015 after I found the five dimensional numbers tau every now and then I typed in a higher dimensional imaginary unit and after that only staring at the screen of the computer: How to find those numbers as the log applet says…

The method as shown here can be applied in all dimensions and you now have a standard way of crafting exponential curves in all spaces you want. This method together with the modified Dirichlet kernels that provide always a parametrization of the exponential curve form a complete description.
Ok ok those modified Dirichlet kernels always have period pi while this way to calculation the log of the first imaginary unit is always related to the dimension (recall that the 7D first imaginary unit l has the property l^7 while for the complex multiplication in 7D space we have l^7 = -1), but it is very easy to fix the Dirichlet kernels to the proper period in the time domain you want.

The most difficult part of this post is in understanding the subtle choice for the eigenvalues of tau = log l, or better; choosing the eigenvalues of the matrix representation involved. That makes or breaks this method, if done wrong you end up with a giant pile of nonsense…

Have fun reading it and if this is your first time you encounter those matrices with all these roots of unity in them, take your time and once more: take your time.
If you have never seen a matrix like that it is very hard to understand this post in only one reading…

I am glad all that staring to those numerical values is over and we have the onset of analytical understanding of how they are in terms of the angle 2 pi over 7.
The result is far from trivial; with the three or five dimensional case you can use other ways but the higher the dimension becomes the harder it gets.

This method that strongly relies on finding the correct diagonal matrix only becomes more difficult because the size of the matrices grows. So only the execution of the calculation becomes more cumbersome, the basic idea stays the same.

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I have no idea what the next post is going to be, may be a bit of magnetism because a few days back I got some good idea in explaining the behaviour of solar plasma included all those giant rings that shoot up and land in another spot of the sun.

And we also have those results from the Juno mission to Jupiter where the electrons also come from Jupiter itself without the guidance of electrical fields. But in the preprint archive I still cannot find only one work about it, that might be logical because often people do not write about stuff they don’t understand…

Ok, that was it. I hope you liked it & see ya around.

Intro to the calculation of the seven dimensional number tau (circular version).

3D complex numbers, 5D complex numbers, Exponential circle, Exponential curves, Matrix representation, Uncategorized

All details will be in the next post but I succeeded into using matrix diagonalization in order to find this seven dimensional number tau.
For people who do not understand what a number tau is, this is always the logarithm of an imaginary unit. Think for example at the complex plane and her imaginary unit i. The number tau for the complex plane is log i = i pi/2.

The problem with finding numbers tau becomes increasingly difficult as the number of dimensions rise. I remember back in the year 2015 just staring at all those matrices popping up using internet applets like the next one:

Matrix logarithm calculator (it uses the de Pade approximation)
http://calculator.vhex.net/calculator/linear-algebra/matrix-logarithm

Yet back in the year 2015 I was riding on my noble iron horse (a cheap bicycle) through the swamps surrounding the village of Haren and suddenly I had a good idea. Coming home I tried the idea of matrix diagonalization out in 3 dimensions and it worked.

I even wrote a post about it on 23 Nov 2015:

Integral calculus done with matrix diagonalization.

Now I think that most readers who visit this website are familiar with the concept of finding a diagonal matrix D containing all eigenvalues of a given matrix M. Once you have the eigenvalues you can calculate the eigenvectors and as such craft your matrix C containing all eigenvectors.
You can write the stuff as next: D = C^-1 M C.

Suppose you don’t know what M is but I give you the matrices D, C and the inverse of C. Can you find the matrix M?

Yes that is a beerwalk, all you have to do is calculate M = C D C^-1 and you are good to go.

But with the logarithm comes a whole lot of subtle things for making the right choice for the eigenvalues that you place inside the diagonal matrix D. It turns out you only get the desired result if you use arguments in the complex plane between minus and plus pi.
This is caused by the fact that you always need to make a cut in the complex plane if you want to work with the complex logarithm; but it is a bit surprising that only the cut where you leave out all real negative numbers (and zero of course) makes the calculation go perfect and in all other cases it ends in utter and total disaster.

In the next three pictures I show you some screen shots with numerical values of matrix representations and the logarithm of those matrix representations.

The goal is to find mathematical expressions for the observed numerical values that are calculated via the above mentioned de Pade approximation. We don’t want only numerical approximations but also catch the stuff in a mathematical formulation.

At the end of the third picture you see the end result.

So it took some time to find this result, I wasted an entire week using the wrong cut in the complex plane. And that was stupid because I had forgotten my own idea when riding my noble iron horse through the Harener swamps…

The result for the seven dimensional number tau (circular version) as calculated in the next post is a blue print for any dimension although I will never write stuff down like in a general dimension setting because that is so boring to read.

Ok, see you around my dear reader.

On a way to find more equations so that the 1D existence of exponential curves in all possible dimensions is assured.

3D complex numbers, 5D complex numbers, Exponential circle, Exponential curves, Matrix representation, Quantum mechanics

In part this post picks up where I left the stuff of the missing equations back in the year 2015. The missing equations are found inside the determinant equation; for this to succeed we must factorize determinant of the matrix representations of higher dimensional numbers. A well known result from linear algebra is that the determinant is also the product of the eigen values; so we need to craft the eigen value functions that for every X in our higher dimensional number space give the eigen values.

These eigenvalue functions are also the discrete Fourier transform of our beloved higher dimensional numbers and these functions come in conjugate pairs. Such a pair form two factors of the determinant and if we multiply them we can get rid of all complex coefficients from the complex plane.

A rather surprising result is the fact that if we subtract a cone equation from a sphere equation we get a cylinder…

This post is also a way of viewing the exponential circles and curves as an intersection of all kinds of geometric objects like the unit sphere, (hyper) cones, (hyper) planes and (hyper cylinders. Usually I represent it all as some analysis but you can take a very geometric approach too.

I have no idea if the shape of the higher dimensional curves is studied as a geometrical object; I suspect this is not the case since the use of complex numbers outside the complex plane is very seldom observed. The professionals just want their tiny fishing bowl (the complex plane) and declare it an Olympic swimming pool…
Well, let it be because these people will never change.

All in all this post is 20 pictures long (size 550 x 775) so it is a relatively long read.

                                     

The pictures of the graphs were all made with an applet named Animated drawing, here is a link and there you can find it under ´Online calculators and function plotters´±

https://wims.sesamath.net/wims.cgi

For example you can cut and paste the next five dimensional equations that represents a hypercone going through all the coordinate axis:

((1/5)*sin(5*x)/sin(x))*((1/5)*sin(5*(x-2*pi/5))/sin(x-2*pi/5)) +
((1/5)*sin(5*(x-2*pi/5))/sin(x-2*pi/5))*((1/5)*sin(5*(x-4*pi/5))/sin(x-4*pi/5)) +
((1/5)*sin(5*(x-4*pi/5))/sin(x-4*pi/5))*((1/5)*sin(5*(x-pi/5))/sin(x-pi/5)) +
((1/5)*sin(5*(x-pi/5))/sin(x-pi/5))*((1/5)*sin(5*(x-3*pi/5))/sin(x-3*pi/5)) +
((1/5)*sin(5*(x-3*pi/5))/sin(x-3*pi/5))*((1/5)*sin(5*x)/sin(x))

The above thing should give identical zero for all x.
An important feature of exponential curves in spaces with an odd number of dimensions is that they all are inside a hyperplane. The hyperplane says the sum of the coordinates is always 1. If you cut and past the next sum of the five coordinate functions you see that you always get one for all x:

((1/5)*sin(5*x)/sin(x)) +
((1/5)*sin(5*(x-pi/5))/sin(x-pi/5)) +
((1/5)*sin(5*(x-2*pi/5))/sin(x-2*pi/5)) +
((1/5)*sin(5*(x-3*pi/5))/sin(x-3*pi/5)) +
((1/5)*sin(5*(x-4*pi/5))/sin(x-4*pi/5))

At last the link to the original update from 2015 where I found the missing equations for the first time. But all I knew they were hidden inside the determinant. A few weeks ago I decided to take a better look and the result is this post.

From 14 July 2015: The missing equations.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff03.htm#14July2015

Ok, that is what I had to say. Till updates.

On an old idea that did not work…

5D complex numbers, Exponential circle, Exponential curves, Matrix representation

Recently I am working on a relatively long post where I try to take a much more geometric approach to finding exponential circles and exponential curves. That post is also going forward from a few years back when I was searching for the so called missing equations.

The problem of the missing equations does not arise until you start working in five dimensions or higher; the equations as generated by the sphere-cone equations are just not enough to end up with a one dimensional curve.
Back in the time I simply took a few weeks until I had found the answer: The missing equations can be found inside the determinant!

For example if you have a 17 by 17 matrix in 17 variables (the so called matrix representations of 17-dimensional number systems), all you have to do is factorize this determinant and from those factors you can craft the extra needed equations.

Weirdly enough you find a hyper plane and a bunch of hyper-cylinders.

So in the next post I try to show you how you can have a very geometric approach to finding the higher dimensional exponential curves as the intersection of a sphere, a hyperplane, a bunch of cones and a bunch of cylinders.

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In my old notes I found a mysterious looking line of squares of cosines with their time lags. That was from before I solved the problem and this ‘solution’ has all kinds of faults in it.

That is this small post; it is about something that does not work.
It is just 3 pictures long (550 x 775 pixels):

It was just over two years back I wrote that long update on the other website about the missing equations, I was glad I took those weeks to solve this problem because it is crucial for the development of general higher dimensional theory on this detail.

Here is a link to that old update±

From 14 July 2015: The missing equations.

I hope next week I am ready with the new long post, after that I will likely pick up magnetism again because I finally found out what the professionals mean when talking about ´inverted V-s’; that means there is also an electric field accelerating the particles in the aurora’s of earth and Jupiter.

Tiny problem for the professionals: At the Jupiter site, regardless of inverted V-s yes or no, the plasma particles get accelerated anyway…
So that looks like one more victory for Reinko Venema and one more silence from the professional professors.

See ya around!

Derivation of the number tau for the circular 3D number system.

3D complex numbers, Exponential circle, Matrix representation

There are lot’s of reasons for this update; one reason is that the actual calculation is mega über ultra cool. Another important reason is that this collection of plain imitation of how the value for the number i in the complex plane was found serves as a proof in itself that this way of crafting 3D complex and circular numbers is the only way it works.

Don’t forget that on the scale of things the Irish guy Hamilton tried for about a decade to find the 3D numbers but he failed. Yet Halmilton was not some lightweight, the present foundation of Quantum Mechanics via the use of the Hamilton operator is done so via the work of Hamilton…
Wether the professional math professors like it or not; that is the scale of things.

During the writing of this post I also got lucky because I found a very cute formula related to the so called Borwein-Borwein function. I have no clue whatsoever if it has any relevance to my own work on this website but because it is so cute I just had to post it too…

Furthermore I used two completely different numerical applets, one for integration and the other for evaluating the log of a matrix, only to show you that these kind of extensions of the complex plane to three dimensional space is the way to go and all other approaches based on X^2 = -1 fail for the full 100%.

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This post is ten pictures long, size 550 x 775 pixels.

At the end I will make a few more remarks and give you enough links for further use in case you want to know more about this subject. Have fun reading it.

22oct2016-calculation-of-the-circular-tau01

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22oct2016-calculation-of-the-circular-tau06

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22oct2016-calculation-of-the-circular-tau08

22oct2016-calculation-of-the-circular-tau09

22oct2016-calculation-of-the-circular-tau10The applet for the logarithm of a matrix can be found in this nice collection of linear algebra applets:

Linear algebra
http://calculator.vhex.net/function-index/linear-algebra

In this update you might think that via the pull back principle you observed some proof for the value of the integrals we derived, but an important detail is missing:
In 3D space the exponential circle should be run at a constant speed.
As a matter of fact this speed is the length of the number tau, you can find more insight on that in the theorem named ‘To shrink or to grow that is the question’ at:

On the length of the product of two 3D numbers.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff04.htm#06May2016

A bit more hardcore is my second proof of the value of the integrals as derived in this post. On 15 Nov 2015 I published the second proof that I found while riding on my bicycle through the swamps near a local village named Haren. It is kinda subtle but you can use matrix diagonalization to get the correct answer.
The reaction from the ‘professional community of math professors’ was the usual: Zero point zero reaction. These people live in a world so far away from me: overpaid and ultra stupid…

Integral calculus done with matrix diagonalization.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff03.htm#21Nov2015

A link to the online encyclopedia of integer sequences is the next link.
Remark that by writing the stuff as on-line instead of online reflects the fact this website must be from the stone age of the internet. That is why it can have this strange knowledge…

The On-Line Encyclopedia of Integer Sequences
https://oeis.org/ (Just fill in 1, 2, 0, 9, 9, 5, 7 in order to land on my lucky day).

The last link is one of those pages that try to explain as why 3D complex numbers cannot exist, the content of this page is 100% math crap written by a person with 0% math in his brain. But it lands very high in the Google ranking if you make a search for ‘3D complex numbers’.
So there must be many people out there thinking this nonsense is actually true…

N-DIMENSIONAL COMPLEX NUMBERS.
http://www.alenspage.net/ComplexNumbers.htm

Ok, this is what I had to say. Let me close this post, hit the button ‘update website’ and pop up a fresh beer… Till updates.

Curl, curl and more curl.

3D complex numbers, CR equations, Exponential circle, Matrix representation

This post contains seven examples of the differential operator named curl. The motivation for this update lies in the fact that after my humble opinion inside the set of equations known as the Maxwell equations there is a tiny fault: Rather likely electrons carry at least a net magnetic charge.
And because they carry net magnetic charge they not only accelerated by electrical fields but also magnetic field.

If you have an old television set with one of those fancy tubes that contain one electron cannon for a black & white television and three electron cannons for a color television. With the help of a stack of these strong neodymium magnets it is easy to give experimental proof that the electrons indeed get accelerated… Why in the course of over one hundred years not one of the professional physics professors has done this is unknown to me; may be it is the separation of ‘theoretical professors’ versus ‘experimental ones’ a reason for this omission. May be it is the use of dogma (unproven things that live inside a belief system, in this case the belief system that magnetic monopoles do not exist).

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Anyway the Maxwell equations contain a lot of the curl operator, that is not needed per sé but it makes the formulae look sleek and short. Originally Maxwell had a set of like 20 equations or so while at present day only four remain. But if you see those four equations for the first time it is very impressive, only over time you get used to it.

This post is 13 pictures long, size 550 x 775 pixels.
I start with examples that are as simple as possible and very slowly bring in a bit more abstraction.
Therefore I hope it is very readable, have fun with it!

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As far as I can remember, in the first year I opened these investigations into higher dimensional number systems again I calculated the curl for the complex multiplication in 3D space. In this post we only looked at the circular version of stuff.

But I can´t find it back so I cannot place a hyperlink to it.

Anyway, here is a nice wiki with the curl expressed as an integral (often much harder to calculate but nice to observe it can also be done that way):

Curl (mathematics)
https://en.wikipedia.org/wiki/Curl_(mathematics)

And because this post was motivated by all that curl in the Maxwell equations, I tried to find back when I originally started writing about electrons having magnetic charge instead of being magnetic dipoles like they are tiny bar magnets. It was 29 April that inside the math pages I found the first update on that. Here it is:

From 29 April 2014 : Do electrons have spin?
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff02.htm#29April2014

Ok, that was it. Till updates my dear reader.

 

How I found the first modified Dirichlet kernel in 5D numbers.

3D complex numbers, 5D complex numbers, Exponential circle, Exponential curves, Matrix representation, Quantum mechanics

Back in Jan 2014 I was able to solve the circular and complex 5D numbers systems using the so called ‘tau-calculus’. Basically tau-calculus is very easy to understand:

  1. Find a basis vector (from an imaginary number) that has determinant +1.
  2. Craft analytical expressions that you cannot solve but use internet applets for numerical answers. Or, equivalent:
  3. Use an applet to calculate the logarithm of your basis vector once you have put it on a matrix representation.
  4. Start thinking long and hard until you have solved the math analysis…

This is easy to understand but I could only do this in three and five dimensional space, seven or eleven dimensional space? It is now 2.5 years later and I still have no clue whatsoever.

Anyway back in the time it was a great victory to find the exponential curves in 5D space. May be some people with insect like minds think that the Euler identity is the greatest formula ever found but let me tell you:
The more of those exponential circles and curves you find, the more boring the complex plane becomes…

In the months after Jan 2014 I wanted to understand the behavior of the coordinate functions that come along those two exponential curves. But the problem kept elusive until I realized I had one more round of internet applets waiting for me.
I had to feed this internet applet for the log of a matrix over 50 matrices and write down the answers it gave me; that was a lot of work because every matrix had 25 entries.

Originally I planned for ‘digesting’ 10 matrices a day so it would be a five day project, but when I finally started somewhere in June 2014 after two days I was ready to sketch my strongly desired coordinate function of the very first coordinate in 5D space.

I still remember sitting at my table and do the drawing and when finished it was so fast that I realized the next: This is a Dirichlet kernel.

And the way I used it, it was so more simple to write down this kernel and if you think how those exponential curves in higher dimensional spaces work with their starting coordinate function and all the time lags that follow… This finding will forever be in the top ten of most perfect math found by your writer Reinko Venema…

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Ok, what is in this new post?
Nine pictures of size 550 x 775 pixels containing:
Basic definitions upon 5D complex and circular number &
Tau calculus for the exponential curve &
Explaining how I found the graph of the first modified Dirichlet kernel &
A small quantum physics example related to probability amplitudes &
Some cute integrals that are easy to crack now.

Hope you can learn a bit from it, do not worry if you do not understand all details because even compared to the 2D complex plane or the 3D complex numbers this website is about:
The 5D realm is a space on it’s own!

Have fun reading & thinking upon it:

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A few links of interest are the next:

You need a good applet for the log of a matrix representation if you want, for example, crack the open problem in tau calculus for the 7D complex numbers:
http://calculator.vhex.net/calculator/linear-algebra/matrix-logarithm

My original update on the other website about the 5D number systems from Jan 2014:
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff02.htm#30Jan2014

If you are more interested in those kind of weird looking integrals suddenly easy to solve if you use a proper combination of geometry and analysis, the update from July 2015 upon the missing equations is also worth a visit:
The missing equations.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff03.htm#14July2015

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This is what I more or less had to say, have a nice life or try to get one.
See you in the next post.

The new 3D coordinate system; [a, r, t] coordinates.

3D complex numbers, Exponential circle, Matrix representation, Quantum mechanics

About a full week late all is now finished. Also there was that problem of the hacked webpage; the third update on the Schrödinger equation was loaded and loaded with all kinds of weird comments. But if you did read those comments you could see the overwhelming majority was not written by humans but were posted by bots.

Anyway in the end even advertisements for online sales of viagra and stuff surfaced so I took an evening to see if this could be repaired. For the time being it is not possible to post comments although before it was also impossible but that was due to an unknown technical fault…

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In this update we dive into the [a, r, t] coordinate system and this is very similar to using polar coordinates in the complex plane only in 3D we are moving a half-cone along a line and as such sweep through the entire 3D space. In short it goes like this:

With a we control how far the cone is moved from the origin,
with r we denote the distance from a particular point X to the top of the cone and finally,
with t we figure out how much we need to rotate stuff using an applet for the log of a matrix.

I also give once more the coordinate functions of the exponential circle so that you are able to find coordinates on the main cone (that has the origin as it’s top) for yourself.

Without prove I also give a new de Moivre formula, the original de Moivre formula is from about 50 years before Euler so after 3 centuries there is a bit of moving forward on that detail.

I also give a rather strange looking sum of squares of cosine functions that add up to 1.5; the three cosine functions all differ by 120 degrees or 2pi/3 if you want.

It is important to remark I did my best to keep it as simple as possible so I also concentrate on a few worked examples so it is not just theory but also how to find these new coordinates. In relation to other posts like the Schrödinger equations posts, I think this new coordinate system is definitely of interest to people from quantum physics and chemistry that try to calculate those atomic and molecular orbitals.

This update is relatively long: 14 pictures of size 550 x 775 pixels and for the first time I use unshrunk jpg pictures because year in year out the bandwidth of internet connections is still rising fast so why remove fine detail from jpg pictures any longer?

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I hope you learn something new, here are the pics:

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0024_23May2016_the_art_coordinates14Ok, that was it. For myself speaking I more or less expect everything to be the same:
Professional math professors from around the globe will keep on talking about their own research and just how important it is to use complex numbers from the complex plane.

A few useful links:

De Moivre’s formula
https://en.wikipedia.org/wiki/De_Moivre’s_formula

And a good applet for the logarithm of matrix representations is also handy:

http://calculator.vhex.net/function-index/linear-algebra
http://calculator.vhex.net/function-index/linear-algebra

And last but not least is a proof for the latest formula above: the circular multiplication on the [a, r, t] coordinates can be found on the other website:

From 06 May 2016 : On the length of the product of two 3D numbers.
http://kinkytshirts.nl/rootdirectory/just_some_math/3d_complex_stuff04.htm#06May2016

Let’s leave it with that. Till updates.

An important correction on a very very stupid typo…

5D complex numbers, Corrections, Exponential circle, Matrix representation

Let’s not do difficult: there are typo’s and there are typo’s.

The one kind is for every body directly to understand ‘this is a typo’ and the brain pops up what should be the correct thing standing there.

And there is that kind of typo that is a disaster in people trying to understand what this stuff actually means.

I am sorry I messed things up, but it is like picture six on the previous update on the length of complex numbers that must have been a root cause of much disunderstanding.

In the next picture I show you the correction: in that matrix it now reads log r instead of only r.

Why I was that stupid a few weeks ago is non of your business, but by now stuff is repaired and we no longer have this horrible mistake around any longer…

0023=correction_on_the_log_detail

Ok, let’s leave it with that. Till updates.

Third post on the Schrödinger wave equation using 3D complex numbers for atomic & molecular orbitals.

3D complex numbers, Exponential circle, Exponential curves, Matrix representation, Quantum mechanics

This update is 10 pictures long, the pictures are sized 550 by 775 pixels.
This update covers more or less everything, but I still have to explain how you find the six coordinate functions the poeple will need in order to see if these kind of complex numbers give the same result as ordinary complex numbers from the complex plane.

For those that cannot wait: In the post from 03 April I posted a teaser picture with the coordinate functions in 3D, if you multiply this against the e to the power i pi alpha thing in this update you have the six coordinate functions…

Ok ok you neatly have to write them out, but basically it is all there.

At first I was thinking it would be hard to get different results using these higher dimensional complex numbers, but when talking about atomic and molecular orbitals it might be more subtle than it looks. At the end I will post a video where some physics guy shows all kinds of orbitals related to hydrogen but his stuff is different from the pictures we observe in chemistry.
He explains this by saying that the people from chemistry always take a super-position of two wave-blobs and as such it gets oriented along the y-axis say.
If you would take super-positions of my 3D complex numbers you will get very similar results. look at the drawing in the one before last picture:
Take a super-position of an exponential circle and it’s conjugate and observe it must have the same behavior as 2D numbers from the complex plane.

(In that drawing your eys is supposed to be along the line through zero and alpha, so zero is right behind the center of the shown circle…)

Enough of the bla bla bla, here are the 10 pictures:

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Click on the picture to get a larger version of the drawing:

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Now finding these atomic & molecular orbitals is very hard, for simple atoms like hydrogen it is doable but what about uranium or some nice protein with only 3693 atoms in it?

All that kind of stuff falls under what we name n-body problems and for n above 3 it seems impossible to find exact analytical solutions.

There is a nice video out there explaining a bit more on the topic of finding the shapes of atomic & molecular orbitals. It is from Brant Carlson and has the title Hydrogen atom wavefunctions:

0021=13April2016=hydrogen_orbitals

Ok, that was it for today. Till updates.