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Sunday, July 31, 2011

The K programming language


The decimal number, $585 = 10010010012_2$ (binary), is palindromic in both bases. Find the sum of all numbers, less than one million, which are palindromic in base 10 and base 2.

Problem 36  - Project Euler


+/&(&/{x~|x}'2 10_vs\:)'!_1e6

That is, in the K programming language. If Java is verbose then I would say that K is autistic. It has been estimated that there are not more than 1000 professional K-programmers. Most of them highly paid and employed in the high-end financial sector ( London, New York ).

Another example.


The nth term of the sequence of triangle numbers is given by $t(n) = \frac{n (n+1) }{2}$, so the first ten triangle numbers are $1, 3, 6, 10, 15, 21, 28, 36, 45, 55$. By converting each letter in a word to a number corresponding to its alphabetical position and adding these values we form a word value. For example, the word value for SKY is $19 + 11 + 25 = 55 = t(10)$. If the word value is a triangle number then we shall call the word a triangle word. Using words.txt, a 16K text file containing nearly two-thousand common English words, how many are triangle words?

Problem 42  - Project Euler



Again, in the K programming language. ( I used 4 lines in Mathematica, it would take 20+ lines in Java. )

Saturday, July 30, 2011

Pluperfect Digital Invariants

A Pluperfect Digital Invariants or PPDI aka Armstrong Number is a number which is equal to the sum of a power of its digits.

For example:

$153 = 1^3 + 5^3 + 3^3$.
$1634 = 1^4 + 6^4 + 3^4 + 4^4$.
$54748 = 5^5 + 4^5 + 7^5 + 4^5 + 8^5$

The largest ( known? ) PPDI is $$115 132 219 018 763 992 565 095 597 973 971 522 401$$ which is equal to the sum of the 39th power of its digits.

These facts belong to the realm of mathematics, of course. But we get rather close to what I call anti-mathematics: numerology.

As soon as you discard scientific rigor, you're no longer a mathematician, you're a numerologist.

Sol Robeson in Pi

Friday, July 29, 2011

Mathematica is self-documenting.


Mathematica is self-documenting. ( That is if you use a mild form of Literary Programming. ) Anyway let me try to prove my point by looking at some code and compare it with two other languages.

Using names.txt (right click and 'Save Link/Target As...'), a 46K text file containing over five-thousand first names, begin by sorting it into alphabetical order. Then working out the alphabetical value for each name, multiply this value by its alphabetical position in the list to obtain a name score.

For example, when the list is sorted into alphabetical order, COLIN, which is worth 3 + 15 + 12 + 9 + 14 = 53, is the 938th name in the list. So, COLIN would obtain a score of 938 53 = 49714.

What is the total of all the name scores in the file?

Source: Euler 22

My solution was:

file = ToFileName[{"C:", "My Dropbox", "Mathematica", "Project Euler"}, "names.txt"];
dta = Sort[ReadList[file, Word, WordSeparators -> { ",", "\",\""}]];
Timing[Sum[ k*Plus @@ (ToCharacterCode[#] - 64 &) /@ Characters[dta[[k]]], {k, 1, Length[dta]}]]

Do you really need the following comments to get an idea what the program does?
Line 1. Address file.
Line 2. Read and parse data
Line 3. Process ( time and present ) data ( ToCharacterCode[] assigns 65 to A and so forth, the file contains names in all uppercase. ) Timing shows the time it took the computer to process line 3. Alternatives for @@ and /@ are the usage of Map and Apply.

Compare this to a solution in Java:

import java.util.*;
public class Problem22 {
    public static void main(String[] args) throws Exception {
	BufferedReader br = new BufferedReader(new FileReader(new File(\"names.txt\")));
	String[] names = br.readLine().split(\",\");
	long sum = 0;
	long count = 1;
	for (int i = 0; i < names.length; i++) {
	    sum += count++ * sum(names[i]);

	System.out.println(\"The sum is: \" + sum);
    static long sum(String name) {
	char[] letters = name.toLowerCase().toCharArray();
	long sum = 0;
	for (int i = 1; i < letters.length - 1; i++) {
	    sum += letters[i] - 96;
	return sum;
Java is known to be verbose, but therefore very readable as well.

Some people get high from coding Haskell... ( I can't guarantee this code works, Haskell is not my thing. No thanks. )

import Control.Monad
import Data.List
import Char
import Control.Applicative
main = (read :: String -> [String]) `liftM` readFile "" >>= \list -> putStrLn $ show $ sum $ zipWith (*) [1..] $ sum . map (\x->ord x - ord 'A' + 1) <$> sort list

I tried to show that Mathematica is self-documenting. Not that any other language is bad or otherwise inferior.

Thursday, July 28, 2011

Marcus du Sautoy in the footsteps of Max Cohen

The Code, part 1 of the new BBC / Open University mathematics documentary presented by Marcus Du Sautoy is in one word excellent. Du Sautoy is on a journey, looking for the code which governs all the patterns we see in the world around us. - Pi ( The Movie ) revisited! The documentary begins almost exactly as the scene in Pi where Max Cohen leaves his flat and meets Jenna, a young girl living in the same building who likes number games. She asks 'How much is 73 divided by 22? ): Max answers with three point three, one eight, one eight, one eight, one eight, one eight, one eight, one eight, etc. until Jenna can no longer hear him. - Once Max enters the street we hear him think:

1. Mathematics is the language of nature. 2. Everything around us can be represented through numbers. 3. If you graph the numbers of any system patterns emerge. 4. Therefore there are patterns everywhere in nature. Evidence. The cycles in disease epidemics, sun spot cycles, etc.

What Marcus du Sautoy does in Code is collecting evidence, much more evidence for the rules of Max.

Max was searching for a code too, I hope the ending of the series of The Code will be more uptone than Pi. We shall see, two more episodes to view. - Click on the image above to watch it if you qualify, or download a torrent from your favorite site.

Wednesday, July 27, 2011

The ColorFunction Option in Mathematica

Scenegraphica progress: programming ( how I prefer to do it anyway ) involves a lot of throwing away of code. Start again from scratch often but with the know-how ( and portions of code ) of the previous version. Eventually entire classes survive and a system emerges. There must be a name for this 'method'. Probably something with agile in it.

I just started on the third cycle of scenegraphica: x3. Not much of x2 survived. I am concentrating on color, surface properties of objects, light, shadow and viewpoints in this release. - I am redesigning the root scenegraph class. It gets a property of type collection to store all the light sources in the universe. - Instead of using Mathematica's own List I am developing an OO-System List class. It will implement Java's Collection interface. Think of a little brother of ListArray.

To get ideas for scenegraphica I look a lot at professional ( Mathematica ) generated graphics. Here for example.

It will take a while ( if at all ) before I can create graphics like that. Because the author of the article, Jean-Pierre Hebert, was so kind to provide his formulas it inspired me to try something like this:

ContourPlot[{Sin[x + y + Sin[x] + Sin[y]]}, {x, 0, 12 \[Pi]}, {y, 0, 12 \[Pi]}, Frame -> False, ColorFunction -> "Rainbow"]

and this:

ContourPlot[{Sin[x + y + 3 Sin[x] + 4 Sin[y]]}, {x, 0, 4 \[Pi]}, {y, 0, 4 \[Pi]}, Frame -> False, ColorFunction -> "Rainbow"]

Note the usage of ColorFunction which is an option for graphics functions which specifies a function to apply to determine colors of elements. "Rainbow" is just a standardfnction that comes equipped with Mathematica. The fun starts when you start making your own color functions.

It would be nice if I could use these patterns as textures for 3D objects. Not sure yet how to do that though.

Sunday, July 24, 2011

About tori in Mathematica

There is no graphics primitive in Mathematica which enables us to plot a torus with one simple command. So we have to make it ourselves. If you read the post 'About spheres in Mathematica' you may have already tried it for yourself. In any case creating a torus in Mathematica is intuitively simple.

We use a red point, as a reference only and start ( again ) with a circle and view the graphical universe from a Left viewpoint.

point = {Red, AbsolutePointSize[10], Point[{0, 0, 0}]}
to1 = First[ ParametricPlot3D[{(1), (Cos[u]), Sin[u]}, {u, 0, 2 \[Pi]}, {v, 0, 2 \[Pi]}]];
Graphics3D[{point, to1}, ViewPoint -> Left]

We move the circle 3 units in the y-direction.

to2 = First[ ParametricPlot3D[{(1), (3 + Cos[u]), Sin[u]}, {u, 0, 2 \[Pi]}, {v, 0, 2 \[Pi]}]];
Graphics3D[{point, to2}, ViewPoint -> Left]

We can then simply rotate this circle as follows:

to3 = First[ParametricPlot3D[{(Cos[v]), Sin[v] (3 + Cos[u]), Sin[u]}, {u, 0, 2 \[Pi]}, {v, 0, 2 \[Pi]}]];
Graphics3D[{point, to3}, ViewPoint -> Left]

But this is not entirely what we had in mind, but this can be easily corrected as follows.

to = First[ParametricPlot3D[{Cos[v] (3 + Cos[u]), Sin[v] (3 + Cos[u]), Sin[u]}, {u, 0, 2 \[Pi]}, {v, 0, 2 \[Pi]}]];
Graphics3D[{point, to}, ViewPoint -> Top]

Finally, a torus, viewed from the top.

Friday, July 22, 2011

Magic squares - Revisited

Regular readers of this blog know that I am fascinated by magic squares. Finding the 3 by 3 magic square with digits 1,2, ..., 9 can be formulated as an Integer Programming problem. An Integer Programming problem is a Linear Programming problem with the additional constraints that all variables must have integer values.


{2, 7, 6, 9, 5, 1, 4, 3, 8}

Mathematica does find a solution in less than a second. An interesting ( Mathematica ) programming exercise would be to generate the code for solving this integer programming problem for magic squares of size n. With increasing n the number of constraints increase fast. This could become an interesting benchmark. - For a 3 by 3 magic square 34 constraints are used above ( although 32 would have sufficed, probably even less ) but these constraints can be systematically generated.

Wednesday, July 20, 2011

New mathematics documentary

The OU produced another mathematics series.

The Code is a three-part TV series about maths in the natural world, presented by Marcus du Sautoy. Why do bees make hexagonal honeycomb? Where's the best place to stand to get on a train first? How can dozens of wrong answers make a correct one? Join Marcus on an exciting journey to discover how maths shapes the world around us.

Starts on Wednesday 27th July, 9pm on BBC Two. - One day after copies will be all over the internet if you are in a place where you can't receive BBC Two.


Course fees up : change of plan

For UK students the course fee will be set to GBP 5000  per 120 points. Considering they charge twice as much for non-UK students, expect GBP 10000 per 120 points. Or 2,850 Euro for a 30-point course. I had 90 points in mind for 2012. Study is entertainment for me, I won't make a single Euro more if I add a Math degree to my c.v. I could spend time solving Project Euler problems for free for years to come. I never call a tutor, need additional books to understand the booklets ( i.e. the books the authors used to write the booklets in the first place ) , so I would pay 3000 Euro just to be able to say that I passed a 30 point OU exam. To who? No.

Math is entertainment to me, I wrote. That is not entirely true. I don't know how my life would be without mathematics. I think I would go insane. Perhaps I already am. Don't worry I am not on psychiatric meds like half of the population it seems. I haven't really read what's going to happen to the fees for UK students. It seems things stay more or less the same for them bottom line wise.

To be continued!

Tuesday, July 19, 2011

Optimizing profits

Mathematicians like space. Like in vector space, metric space, topological space. In fact, any set with structure can be called a space. In "simple" Euclidean geometry there are all sorts of problems of how to fill a space with as much objects as physically possible. How many spheres of a certain radius can we store in a cube? And so forth. - The Japanese people excel in many areas: architecture, technology. This combined with the special conditions ( a lack of =space= ) in Tokyo lead to extraordinary inventions. The following =must= have been the solution to an optimization problem of some kind. Like maximize the number of hotelrooms ( they must have started with rooms in mind ) given a number of constraint. Or simply maximize profits because I bet this 'cheap' hotel is making more profits than their five star competitors. Thanks to optimization and applied mathematics. No matter how much they love space mathematicians can minimize it for you.

I am Dutch, so somewhat exposed to overpopulation and too many people on limited ( office ) space. I can comfortably stay in any hotel though, no matter how smallish the room.  Not even the German U-boat movie 'Das Boot' gave me the same feeling of claustrophobia.

" Which version? " ... of the future.

Below you'll find a video about the future. "The" future? Definitely a possible future. A future you and I can choose to create. Many people already did  by supporting the Zeitgeist Movement. - Off-topic?! I would not dare, really. If you are privileged enough to study ( or have studied ) mathematics than you can make a difference. But watch it first, then you'll understand.

Waking Up forest scene from Harald Sandø on Vimeo.

Sunday, July 17, 2011

About spheres in Mathematica

Plotting a sphere in Mathematica is simple.


What really happens ? Another method to plot a sphere is the following. But let's do a circle in 3D first.

ParametricPlot3D[{Cos[t], Sin[t], 0}, {t, 0, 2 \[Pi]}]

We can then simply rotate this circle as follows:

ParametricPlot3D[{Cos[t] Cos[u], Cos[u] Sin[t], Sin[u]}, {t, 0, 2 \[Pi]}, {u, 0, 2 \[Pi] }]

ParemetricPlot3D generates data we can use in Graphics3D:

sphere = First[ ParametricPlot3D[{Cos[t] Cos[u], Cos[u] Sin[t], Sin[u]}, {t, 0, 2 \[Pi]}, {u, 0, 2 \[Pi] }]]

A very large output was generated. Here is a sample of it:
Show Less\[ThinSpace]Show More\[ThinSpace]Show Full Output\[ThinSpace]Set Size Limit...


Note the difference between Sphere[] a Mathematica function, and the variable sphere which we assigned the first element in the list generated by the ParametricPlot3D function.

Saturday, July 16, 2011

Collatz sequence - Revisited

This following Mathematica code prints the Collatz sequence for n to 1.

For example

Problem 14 of the Euler Project is about the Collatz sequence. Officially it is a conjecture that every sequence starting with n ends in 1.

I spend so much time on 3x+1.... My first blog post in 2005 was on the Collatz sequence. I am able to generate sequences of any length ( thus including very large lengths ) but they end with 1 by definition. It's sort of a paradox.

Thursday, July 14, 2011

A date to remember

Scenegraphica ( = my pet programming project ) entered the embryonic phase.

I.e. it has ( at least ) one demonstrable purpose : it reads a propriety XML file and converts the contents to Mathematica 3D graphics primitives and displays them. I don't know how much further I'll push Scenegraphica but since every man needs a program ( * ) to toy with anyway I think I settle for Scenegraphica.

Now that I think of it.... Mathematica is supposedly very good in the ( mathematical ) generation of sounds ( which is probably more than noise and less than music ). Something to investigate soon.

Wednesday, July 13, 2011

Random times Random is not Random

Some things in Mathematics are counterintuitive, well at least surprising. The first thing I think of is that there are different 'infinities'. Try to explain to your neighbor that the number of even numbers is equal to the number of numbers divisible by three. Or that there are more points on the ( real number ) line between 0 and than there are integers.

Or this one. Random times is Random not Random? Not uniformly random to be precise. In the experiment below I threw a coin with sides 0 and 1 99,999 times. The first graph shows the number of 0s and 1s thrown. Practically equal at 50,000. Then I did the same with two coins 99,999 times but multiplied the product ( mod 2 ). The second graph shows the result. 75,000 zero's and 25,000 1s.

Click to enlarge
Now imagine the same for any number. A number ends with 0,1,2,..., or 9. The number of 0s,1s, 2s will be equal if you take a large set of random numbers. But if you multiply two random numbers the last digits will not have a uniform distribution.

Interesting is that the distribution depends on the type of operation.

Tuesday, July 12, 2011

Trig challenge exercise

Show that :
$$\frac{1}{\sin^{2}\frac{\pi}{14}} + \frac{1}{\sin^{2}\frac{3\pi}{14}} + \frac{1}{\sin^{2}\frac{5\pi}{14}} = 24$$

Well, in any case it is a beautiful way to represent the number 24.

The Dunning-Kruger effect

We learn and learn, become more knowledgeable every day. Time passes, your skills improve. Then one day you'll have to face ( much ) younger colleages at work. For example one who thinks older people are trash no matter what.

The Dunning–Kruger effect is a cognitive bias in which unskilled people make poor decisions and reach erroneous conclusions, but their incompetence denies them the metacognitive ability to appreciate their mistakes. The unskilled therefore suffer from illusory superiority, rating their ability as above average, much higher than it actually is, while the highly skilled underrate their own abilities, suffering from illusory inferiority. Actual competence may weaken self-confidence, as competent individuals may falsely assume that others have an equivalent understanding. As Kruger and Dunning conclude, "the miscalibration of the incompetent stems from an error about the self, whereas the miscalibration of the highly competent stems from an error about others"

Years ago I thought ( had the illusion ) that I knew quite a lot about computer science, programming, even mathematics ( ... ). Today I seriously have the feeling I know almost nothing. I attributed that process to a sort of more = less principle. It seems to be a psychological law. Part of growing older, maturing, becoming more proficient in your field. - A side-effect is that it creates a division between younger and older workers. - Have you ever worked with a say 10 ( or more ) years younger colleague who thought he was a G_d in his field? ( They tend to call everything that is not coded by their golden genius fingers -trash- ). The tensions that a relation like that can create can be explained by the D-K effect, I suppose.

See this question at /* Programmers */ ( although closed as off-topic ).

Sunday, July 10, 2011

Project Euler - Revisited

In order of difficulty I have now solved the first 12 least difficult problems, of the 300+ total problems. I am still in the zone of the not so difficult stuff because I haven't reached a level yet.

A Pythagorean triplet is a set of three natural numbers, $a, b, c$, for which, $a^2 + b^2 = c2$ For example, 3^2 + 4^2 = 9 + 16 = 25 = 5^2. There exists exactly one Pythagorean triplet for which a + b + c = 1000. Find the product abc.

Project Euler - Problem 9. This problem has been solved by 75913 people, ( problem 1 by 159750 people ).

When you have solved a problem you get access to the answer book. It then becomes clear what a difference it makes if you are somewhat literate in a tool like Mathematica. I have solved most problems so far, if not all. with one line of functional Mathematica code while the model answer is a three or four page pdf full of procedural code. - For a while I had the feeling I was cheating by using Mathematica, but the goal is clear. Solve. no matter what. I am not sure but I think Euler is mainly geared for computer geeks, although the problems so far could have been solved by intelligent programming alone, knowledge of mathematics makes a difference. Take this problem for example.

2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder. What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

Project Euler - Problem 5.

Solved this one without using my computer, thus simple. Because I have done an introductory course in Number Theory of course. Maybe it is not so simple otherwise, I don't think so actually.

If you are ever enjoying leisure time on Sudoku's, Euler is worth considering. More fun!

Watched the movie Source code yesterday. Liked it so much that I will give Proof ( Jake Gyllenhaal has a part in it ) a second chance. Proof is not an action movie, it is more in the line of A Beautiful Mind. An attempt to answer "Where does genius end before it turns into undeniable insanity?"

Saturday, July 9, 2011

Weekend thoughts...

2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder. What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?

- Problem 5 of the Euler series of problems.

I had to bite hard to solve problem 4: ( find the largest numerical palindrome which is the product of two three digit numbers.) Mathematics, like programming or playing an instrument, is something you have to practice =every day=. Doing problems you have solved before doesn't make you any better.

About M381 and the URM. There is a Java program called JURM written by Charlton D. Rose and is available for download, including source. Have fun! Compliments to the author.

More interesting software. Mnemosyne is the open source version of Supermemo. I have written several times about Supermemo and efficient studying. The big plus of Mnemosyne is that you can use LaTeX on the flashcards.

Mathematica programming is not restricted to one paradigm like Java which enforces you to use Object Oriented programming. With Mathematica you can do Functional programming, Procedural programming, Rule-based programming and Object Oriented programming. I am close to entering the realm of rule-based programming. Used by programming experts as the paradigm of choice and unleashes the real power of Mathematica. More on this to follow.

Thursday, July 7, 2011

Mathematics on the internet

An extremely valuable internet resource are the Stack Exchange user forums. No matter what your level ( from beginner to professor ) you'll find your peers at Stack Exchange. Well, at least if you are a programmer or a mathematician. There are more professions active though. I am working on a TMA ( when am I =not= working on a TMA? ) and wanted to do something in LaTeX I hadn't done before. I posted my question here. at the TeX user forum. As you can see, I got a reply real soon. Isn't that amazing? I posted other questions in the Mathematica forum with the same result. People are communicating, exchanging ideas, helping each other in different fields, at all levels all over the planet. In a sense this is how the internet got started but this trend is still accelerating. It is phenomenal.

Gauss, Riemann, Hardy, Ramanujan just to name a few mathematicians, had no internet to benefit from. Hardy once sent a letter to Ramanujan and had to wait weeks ( if not months ) for an answer. Alex Jones, a master in the usage of internet media, recently said at a lecture in New York that the speed in which the internet is developing is still increasing. Stack Exchange is less than three years old. Twitter is about that age ( if I am correct. ). At the website The Sage Notebook you can use free, online a state-of-the-art mathematics workbench. This is possible since, five years or so. With Wolfram Alpha we have instant access to many of the features of Mathematica ( and more ). Wolfram Alpha is less than three years old. - From that perspective ( ... ) I say: we live in exciting times!

Tuesday, July 5, 2011

Mathematical art ( read: scribblings )

I am developing a Mathematica package ( with the original name 'SceneGraphica' ) to create mathematical art. Below you'll find some very early stage test drawings. I'll regularly post pics with a mathematical touch in one of my facebook photo albums.

Sunday, July 3, 2011

Michael Schneider

Twelvealone made a comment about the Egyptian math video. The mathematician / presenter is Michael Schneider. He wrote a book called 'Constructing the Universe'. If you have seen the movie Pi, you'll recognize many similarities between Schneider's and Maximillian Cohen's views about mathematics and the physical reality. Apple Computer employs Evangelists. The mission of a Mac Evangelist is simply to disseminate the beauty of an Apple computer. Unlike teachers who teach from a position of superiority, even if they only want to help their ( engineering ) students to get through Calculus, Michael Schneider does not teach: he disseminates. He invites you to investigate the beauty of mathematics, for yourself. One of my favorite ( alternative ) radio stations is the Swedish Red Ice Radio. It turns out they interviewed him last year about his book, mathematics and so much more.

Parts 2, and following at the Red Ice Radio channel.

Friday, July 1, 2011

Words in mathematics

If you browse through an index of any mathematics book you will see a lot of words that have a meaning outside mathematics. I don't think this is true for the other sciences in the same degree. Take physics for example. If words from physics are used in some non-scientific conversation, 'speed' or 'force' for example then they still ( roughly perhaps ) have the same physical meaning.

When defining new mathematical ideas, concepts, objects or methods mathematicians often borrow words from the natural language and assign a mathematical meaning to it. The mathematical language is hard to grasp for the non-initiated. When a mathematician hears the word 'group' it isn't likely that he associates it with a group of -people-, groups appear everywhere in mathematics. Ring, field, graph, network, category are also examples of words borrowed by mathematics.

People from the early 20th century already knew the word 'computer'. Becoming a 'computer' was a career-option in those days. The insurance industry needed lots of them. To 'compute' premiums for example. The field of computing life-insurance premiums has evolved in a science of its own, the actuarial sciences. A mix of business administration, statistics ( they maintain 'death-tables' ) , mathematics and law. So the word computer got a different meaning over time. I am not sure but I have the impression we use it less and less. We have laptops, phones, tablets and what have you. Nobody calls their phone a computer although it is more powerful than the first 30,000 dollar costing IBM PC.

I am currently studying Linear Programming. Another borrowed word dating back to the 1940s. Although Alan Turing was coding these days we know he was decades ahead of his time. How many times has it been explained that Linear Programming is not at all like computer programming? Any serious Linear Programming however can't be done 'by hand'. Computers are required to do the number crunching. Think of matrices with hundreds if not thousands of rows and columns. Even the simplest problems have five or more unknowns in canonical form.

Besides Linear Programming there are the fields of Integer Programming and Non-linear Programming. The latest books in these fields have the word Programming replaced by Optimization. Which, incidentally, is also the name of the OU course in that field. Otherwise the course would have been called 'Programming'. Try explaining that to your neighbor.

I have subscribed to Project Euler. I couldn't resist. It is the challenge, I suppose. I have solved the first problem. Well, as long as I practice math every day. Don't try quitting mathematics for a few months or so. You 'think' your brain can handle it, but that is merely memory-recall. Just stay mathematically fit.

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Welcome to The Bridge

Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before

(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)