Question: what is the square root of 36 ? – Part Two

I will continue with answers and results equal to the square root of 36 ( originally answered at quora.com ). This time the results are mostly related to physics.

With physics one has to take into account the units and the corresponding dimensions of the equations and of the constants.

6 and the square root of 36 are dimentionless numbers , so the result must be dimentionless .
If the result is a simple fraction with numerator and denominator , then the units usually cancel out.
In other cases when one deals with logarithms one should multiply with the inverse dimensions to get a dimentionless result.
In one or two results where I didn’t look for the inverse units  I multiplied the equation with a quantity I called (U) ,  which represents the inverse of the units by which one should multiply the result to get a dimentionless number.

Here are the results :

 

One possible way to explain what I have done here is the following:
If some people , living on an isolated fictitious island or on an another hypothetical planet , attached a great importance to and had a fixation on  the square root of 36 (or the number 6) for one reason or another , and got accustomed to the use of 6 as a fundamental constant ,  unit or number , then they would have likely  tried to construct a system of measurement  based on the number 6 , and  to express physics and math formulas ,equations , constants and rules in relation to 6.

After all , 6 or is equal to :

• The floor of  :
  ;

• of the circumference of a circle in degrees.
• It is also  one tenth of 60 seconds which make up a minute , one tenth of 60 minutes which make up an hour , one fourth of 24 hours which equal a day on Earth , one half of  12 months which make up a year , etc.
• A peculiar ‘hexacentric’ system , so to speak.

Or this can be seen as a (creative) exploration of or exercise in advanced math and physics in order to express many equations , formulas and constants in relation to the number 6 (or ) .
Or whatever.

Apologies to Isaac Newton , Leonhard Euler , Bernhard Riemann , Einstein , Stokes , Coulomb , Avogadro , Lagrange , and others (wherever they may be) , for playing around with their equations , formulas , constants , and/or functions.

And one more addiction to this answer :

Does the future of humanity depend on answering what is the square root of  36 , or not?
Have philosophers from Antiquity to the present overlooked this fundamental question , which goes beyond the Kantian categories of space and time set out in his Critique of Pure Reason , and beyond Nietzsche’s Beyond Good and Evil , ushering the transmutation of all values and a defining moment for a new era  in the history of Humankind?
It’s just a square root , for common sense’s sake (or is it?).

Anyway , enough philosophizing.

Here are ( ) more answers to , this time with images :

  is equal to :

— The number subjected to a geometric rotation in the following image (done with Mathematica and some Photoshop) :

— The number expressing the power and the coefficients in the equation of the curve in the polar plot below :

— The number expressing the degree of the root  and the power of the variables in the 3D plot below :

The rotated number  and the polar plotted curve in the first two images  above seem to exhibit symmetry.
Symmetry is an very important property in science , math , physics , equations , nature , and wherever it is found.

Online sources and reference works related to what I have written in this answer can be found in my pages about Science books ,  problem solvers and philosophy books in this site/blog.

Some other online sources:
http://mathworld.wolfram.com/

http://en.wikipedia.org/wiki/Category:Mathematics

http://en.wikipedia.org/wiki/Category:Physics

Pi in the sky , and on the computer

Results and things  related to pi are usually published or made known on Pi Day . But you never know when you get inspired by π or find time to explore this ‘venerable’ math constant , and I have already published a post about pi on Pi Day . By the way there is also a Pi approximation Day (July 22) , so there’s more than one date to talk or write  about π .

Let’s kick off with an image of the value of pi (in the clouds ) with 12 decimal digits , made with Photoshop.I tried to make it realistic and show π and its numerical value ( 3. 141592653589) as part of the clouds and the sky.

Click on the image above to see and enlarged version.

Now for some computer-based mathematical explorations related to π.
There is a known relation between e ( the base of the natural logarithm ) and π :

and is approximately equal to 20 or almost 20 , which is known as an almost integer.
If we try to be more accurate and find the first 1000 decimal digits for the expression above we get (with the help of Mathematica) the following number:

19.99909997918947576726644298466904449606893684322510617247010181721652594440424378488893717172543215169380461828780546649733419980514325361299208647148136824787768176096730370916343136911881572947102843075505750157713461345968680161070464780150721176248631484786057786790083331108325695374657291368002032330492961850463283115054452239990730318010838062172626769958035434209665854687644987964315998803435936569779503997342833135008957566815879735578133492779192490846222394896357465468950148911891909347185826596341254678588264050033689529697396648300564585855142666534919457239163444586998081050100236576797224041127139639108211122123659510905094871070706680635934325684092946890616346767578519812785089761055789304041857980123101280905543416254404987679233496308302396952371198509012175432057419088516489412743155057902167919927734272964964116423666794634333328342687902907792168390827162859622042360176355034576875485783678406122447755263475337650755251536818489395213976127148481818560841182505647  ,
which shows that does not approach 20 completely or uniformly.

Here are some more expressions and calculations  involving π (calculated with Mathematica):

I think it remains to be seen if the numbers above are transcendental. The Mathematica (version 10)  command Element[z , Algebraics ] cannot determine whether these numbers belong to the domain of algebraic numbers or not.

Now let’s consider an expression containing π , e and i (the imaginary unit complex number). The following expressions are equivalent:

A 2D complex plot of the last function above (with z between -4 and 4) gives the following graph (made with Maple):

A 3 D complex plot of the same function ( with z between -4-4i and 4+4i)   gives the following graph (with the help of Maple):

A general solution of the function f(z) above for f(z) = 0 is (calculated with Mathematica):

Here is an interesting result:

Using the Mathematica commands Element[z , Algebraics ] and Not[Element[z , Algebraics ]] , it seems that the solutions z of f(z)=0 above ( for different values of the constants ) do not belong to the domain of algebraic numbers , and are therefore  transcendental numbers.

And with this I bid π farewell for now.

A note about the sinc function as a solution to a differential equation

The (infamous , or famous , take your pick) sinc function is known to be one of two solutions of the differential equation:

This is a linear second order ordinary differential equation with dependent variable x and independent variable y.
I have tried to explore and find the solutions to this differential equation using mostly computer math software and programs.

The Texas Instruments 92 Plus scientific calculator and the Maple computer algebra system agree and give the same solution :

Mathematica gives the following solution:

After converting the exponentials to trigonometric functions the expression above and the solution of the differential equation given by Mathematica becomes:

 

This solution is less simple than (1). Note that if we make the assumption in Mathematica that λ > 0 ,we get the solution:

If we try to find the graphical representation of solution (1) for different values of the arbitrary constants c1 and c2 and with λ (or n ) between -3 and 3 ,we get the following graphs:

Then I tried to take the absolute value of λ in the numerator of (2) for different values of the arbitrary constants , and compared their graphs  with the graphs of solution (1) for the same values of the constants. λ in (2) and n in (1) are between 1 and 4.

The graphical results are similar but not exactly the same.

Using the Manipulate built-in function in Mathematica , here is an animation of the graphs for solution (1) obtained by varying λ , c1 and c2:

Finally , I will give the solution of a generalized form of the differential equation above

obtained with Mathematica :

In the solution above , Jμ(x) is the Bessel function of the first kind , and Yμ(x) is the Bessel function of the second kind.

Some more 3D graphics related to the sinc function

I’ll finish exploring the sinc function by showing a few 3D graphs of sinc related curves.
Here is a first set of 3D curves of sinc related functions:

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As an example , here is the Mathematica code for the 3D curve √(x² + y²)sinc(√(x² + y²)) above:

And here is the second set of 3D surfaces related to sinc:

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The Mathematica code for the curve given by sinc(x² -y²) is:

I’m somewhat fed up with sinc , so soon I will move on to other subjects.

Update: I will add a last group of six 3D curves related to sinc . They include curves where sinc as a function of x is multiplied by sinc as a function of y , such as sinc(x)×sinc(y) , sinc(ln(x))×sinc(ln(y)) with ‘ln’ the natural logarithm to the base e  rendered as ‘log’ by Mathematica in the image , sinc(sin(x))×sinc(cos(y)) , and sinc(x²)×sinc(y²)  .

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2D and 3D graphics related to the sinc function

After having explored calculus results related to the sinc function , here are a few 2D and 3D graphic results.

I’ll start with the integral :

The graph of the function above is (made with Mathematica):

We’ll consider more than one way to graph this function in 3D.
First , by taking a specific function of x and y and using the following Mathematica code :

 
We get the 3D result :

We can use the RevolutionPlot3D buit-in function in Mathematica to get the graph of the solid obtained from the function by revolving around the z-axis:

The graph of the solid obtained by revolving about the x-axis is:

The Mathematica code for the graph above is:

And the graph of the solid obtained from the function by revolving about the y-axis is:

The Mathematica code for the graph above is:

Next is a comparison  between the 2D graph and the 3D graph of the function sinc(tan(w)):

Below is a  set of six 2D graphs of sinc as a function of hyperbolic trigonometric functions:

Mathematica code for the graphs above( rd will be used in the 3D graphs later):

Here are the six 3D graphs related to the above six 2D sinc graphs:

And the Mathematica code for the 3D sinc graphs above:

Rosetta’s path and the encounter with comet 67P/Churyumov–Gerasimenko

Here is a video animation I have  made of the Rosetta spacecraft on it way to  comet 67P/Churyumov–Gerasimenko . It shows the path of Rosetta (in yellow) since its launch in 2004 until it approaches and meets the comet (orbit in blue). The animation was made using the Starry Night astronomy program.

The same  video with shorter length , less effects and a slightly better resolution can be viewed at this link.

The coordinates of comet 67P  were not prebuilt in my version of Starry Night , so I had to program the orbital elements of Churyumov–Gerasimenko myself. After researching I used the following  elements :

Eccentricity (e): 0.640980 ;  pericenter or perihelion distance (q): 1.243230 AU ;

Node : 50.1423 °  ;  argument of pericenter ( w ) : 12.7854°  ;

Inclination (i): 7.0402°  ; pericenter time (Tp) : 2457247.5683  ;

Epoch : 2456967.5  ;  Magnitude : 11 ;

 North pole right ascension: 69.00°  ;  North pole declination: 64° ;

Rotation rate : 2 rotations/day  .

Here is also a slideshow of the position and orbit of Rosetta at different dates . In the last three images the position of the Dawn spacecraft is shown as well. Dawn entered the orbit of the Vesta asteroid (or protoplanet)  in July 20 , 2011 , and completed  its fourteen month survey mission of Vesta in late 2012. Dawn is getting nearer to the dwarf planet (or large asteroid) Ceres and will arrive there in early 2015. The images were created and prepared with the Redshift astronomy software.

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Last but not least , here is a an image I’ve prepared ( with Starry Night and Photoshop ) showing the path and orbit of the Rosetta spacecraft from its launch in 2004 to the end of 2014 , including the orbit of comet 67P/Churyumov–Gerasimenko ,  with annotations recounting briefly the important steps of the Rosetta mission.

 

Fractal graphics , curves and art

I’m presenting fractal images made with various fractal programs.

Simply put ,  fractals are  (mathematical) objects displaying self-similarity at different scales.

The images below show the Mandelbrot set , a set of complex numbers   obtained by iterating the equation z² = z + c , where c is a constant number.I used Mathematica and Fraqtive to create these images.

A set related to the Mandelbrot set is the Julia set :

Equations in 3 D and a surface

I have made two 3D drawings ,   one representing the Einstein field equations of General Relativity and the other for the Dirac equation . The drawings were made using MathType , Photoshop , Illustrator and Mathematica . The Dirac and Einstein equations are two of the most important equations of advanced physics of the twentieth century.

I have already inserted a less complex drawing of the Einstein equations in the slideshow for the books page .

Here is the drawing for the Einstein field equations:

I tried to represent at the bottom  the curvature of space-time by a massive object (the sphere or spherical body in the middle of the picture).

The equation at the top is the geodesic equation of motion in General Relativity.

I have studied General Relativity and Tensor Calculus in the past and I’m going to describe some elements of the equations in the two drawings . For more details one can refer to books , courses and works about advanced physics and about  these subjects .

In the geodesic equation of motion , the Christoffel symbols of the second kind are related to the Christoffel symbols of the first kind [ρσ,ν] and are given by

I think the field equations of General Relativity are the most important physics equations elaborated by Einstein . These equations were part of a successful attempt to unify Gravity and electromagnetism . They contain within them as special cases and at certain conditions other known equations such as the mass-energy equivalence equation ( E = m c² ) , Newton’s law of universal gravitation  , and Maxwell’s equations . 

The Einstein equations with the cosmological constant added are: The left-hand side of the field equations without the cosmological constant is the Einstein tensor , and the right-hand side is the constant χ multiplied by the stress-energy tensor.

Here are some more details about the elements of the equation (which represents a set of partial differential  equations):

 And here is the drawing for the Dirac equation:

I’ve placed in the middle of the drawing the condensed simple form of the Dirac equation with the partial differential symbol ∂ in Feynman slash notation . The background of the picture pertains to particle physics , particle accelerators and quantum mechanics , which are domains where the Dirac equation is used and applied.

Here is a brief description of this equation :

At the top one can see the expanded  Dirac equation ( in terms of the gamma matrices ) written around the 3 D ring . Any similarity with the ring from ‘The lord of the rings’ novel and movies is unintentional.

Finally , here is a  nice looking 3D surface I obtained with Mathematica and customized with Photoshop. It took a little more time to render but I think it was worth it. I was inspired by the book  ‘Graphics with Mathematica  Fractals, Julia Sets, Patterns and Natural Forms ‘ , and I made my own modifications and changes to the Mathematica input and output . Click on the picture to see an enlarged version.

 The Mathematica code for the surface is:

About the gamma function , math , and language dictionaries

I have a good knowledge of  English (see my literary books page) , and I have a number of  reliable language dictionaries I consult from time to time. I also know about science and mathematics (see my science books page and the problem solvers page) , and I sometimes find mathematical expressions or physics formulas interspersed between scientific definitions in dictionaries. But occasionally I have noticed that math expressions or formulas published in these reference works contain errors and inaccuracies or seem to have been published hastily and without enough care. Whether they have been published using a math writing software or html related coding , the lack of accuracy or completeness is visible.

One example from physics is that of the Schrodinger equation (HΨ = EΨ)  inserted in a scientific definition in a renowned English language dictionary .The summation sigma symbol ( ∑) in the expanded Schrodinger equation is shown followed by the index of summation  and by the lower and upper bounds of summation as if they were multiplied by it . At other times parentheses are lacking  or a plus (+) sign , a letter or variable is omitted.

In one known language dictionary there was an error in the integral definition of the gamma function. I will not name any dictionaries  here , but in any case the error remained for a few years and  was later corrected in newer editions. The gamma function is related to the factorial by :

and is defined by the improper integral

In the dictionary the definition was :

So the variable of integration changed from dt to dx . In the online version of the dictionary ,the same integral expression is shown with the correct variable of integration , but it is missing a minus (-) sign and a letter variable.

Out of curiosity , I had the idea to find out what the value of the integral would be with a change of variable or if more than one letter or parameter were changed.

For the integral above with dx as integration variable , exp(-t) is constant , and the solution of the integral is :

A plot of the solution gives (done with Mathematica):

Here is a 3D plot of the absolute value of the function   in the complex plane (with the help of Mathematica) :

A generalized solution of the integral with dx and with arbitrary bounds of integration is the following:

Another definition of the gamma function is :

If we change the integration variable we get:

One could go on making variations to the gamma function definitions and finding the corresponding solutions.

That was an excursion caused by a mistake in a math definition in a dictionary . Errors can be sometimes found even in science dictionaries and they are often corrected , but I think  if language dictionaries are able to write mathematical expressions and formulas more carefully and clearly , they would become more accurate and subsequently more appreciated.

Heart to heart with 3D Math

I have studied mathematics and worked with Mathematica for quite some time , as can be seen from my problem solvers page and the science books page .

Some curves and surfaces have nice 3 D heart shapes when plotted the right way .Maybe it’s Valentine’s day by coincidence and maybe not  , anyway here are some examples of ‘math’ hearts with Mathematica ( and sometimes a little Photoshop):

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For those interested in math and coding , I’ll give as an example the Mathematica code for the big red heart with white background  :

 And here are some more (customized ) heart surfaces. Shows  there is beauty in mathematics .

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The ‘plump’ red heart with the orange background is a heart-shaped surface given by the Taubin equation:

And also by the equation :

The Mathematica code for the Taubin heart-shaped surface is: