Unicolor2D graphics can't be spectaculair, let alone breathtaking. Unless you are in the know. Let me explain. I am now able to  almost  print tilings automatically from just a string of numbers as I wrote about in a previous post.
Not spectaculair at all, it's just a collection of 8 tilings containing 72 polygons in total. But if I change ( 4,2 ) by (10,5) and add a colorfunction the program creates immediately the following graphic.
Both are, in fact, representations of the same sequence.{6, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4} but the variations from this sequence alone are endless. The program creates a graphic for every sequence input. Not all graphics will be tilings of the plane by definition. The following ( major ) step is to let the program reject sequences that do not lead to tilings of the plane.
Tne next part will include an implentation of a 'Point in Polygon' predicate. Many of those exist and have been implemented for many languages. That could be a topic for a more indepth post next time.
Sunday, December 15, 2013
Sunday, December 8, 2013
Challenging the eternal truth of mathematics
I learned that 0,999... = 1. I believe it was in M381 that I learned to prove it. The proof is quite simple actually.
Let
x = 0,999..., multiply both sides with 10
10x = 9,999..., now subtract x from 10x
9x = 9, and we have our result
x = 1.
What I like about mathematics is that it is timeless, i.e. we can still read math books of hundreds of years old and still learn things from them. That's not an advisable strategy hfor any other science than mathematics. Of course, new branches of mathematics appear, new discoveries are made, but they don't invalidate the truths of the past.
Today however I found someone who actually is challenging some of the established truths in mathematics, like for example that 0,999... = 1. I don't think that he is a crackpot, although I have no doubt that the mathematical establishment, professors who are 'safe' by all means, will call him like that.
The man is arrogant though, he calls the proof above, 'juvenile' for example.
Who is he, what are his ideas and how did he disproof that 1 = 0,999? The links below will help you abshereing these questions.
 The New Calculus  The first rigorous formulation of calculus in history.
 Proof that 0.999 not equal 1.pdf
Let
x = 0,999..., multiply both sides with 10
10x = 9,999..., now subtract x from 10x
9x = 9, and we have our result
x = 1.
What I like about mathematics is that it is timeless, i.e. we can still read math books of hundreds of years old and still learn things from them. That's not an advisable strategy hfor any other science than mathematics. Of course, new branches of mathematics appear, new discoveries are made, but they don't invalidate the truths of the past.
Today however I found someone who actually is challenging some of the established truths in mathematics, like for example that 0,999... = 1. I don't think that he is a crackpot, although I have no doubt that the mathematical establishment, professors who are 'safe' by all means, will call him like that.
The man is arrogant though, he calls the proof above, 'juvenile' for example.
Who is he, what are his ideas and how did he disproof that 1 = 0,999? The links below will help you abshereing these questions.
 The New Calculus  The first rigorous formulation of calculus in history.
 Proof that 0.999 not equal 1.pdf
Some progress...
If you read the previous posts on my tiling printing algorithms you'll understand the reason for this post: I made some progress that unraveled some serious knots in my stomach.
{ 4, 2, 4 } >
{ 6, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4 } >
The second white polygon is made from the following points :
\begin{array}{cc}
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
0 & 1 \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{1}{2} \left(1+\sqrt{3}\right) & \frac{1}{2} \left(1+\sqrt{3}\right) \\
\frac{1}{2} \left(1+\sqrt{3}\right) & \frac{1}{2} \left(3+\sqrt{3}\right) \\
\frac{\sqrt{3}}{2} & \frac{3}{2}+\sqrt{3} \\
0 & 1+\sqrt{3} \\
\frac{\sqrt{3}}{2} & \frac{3}{2}+\sqrt{3} \\
\frac{1}{2} \left(1\sqrt{3}\right) & \frac{1}{2} \left(3+\sqrt{3}\right) \\
\frac{1}{2} \left(1\sqrt{3}\right) & \frac{1}{2} \left(1+\sqrt{3}\right) \\
\end{array}
Ready to enter the next level of the problem. ;)
{ 4, 2, 4 } >
{ 6, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4 } >
The second white polygon is made from the following points :
\begin{array}{cc}
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
0 & 1 \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{\sqrt{3}}{2} & \frac{1}{2} \\
\frac{1}{2} \left(1+\sqrt{3}\right) & \frac{1}{2} \left(1+\sqrt{3}\right) \\
\frac{1}{2} \left(1+\sqrt{3}\right) & \frac{1}{2} \left(3+\sqrt{3}\right) \\
\frac{\sqrt{3}}{2} & \frac{3}{2}+\sqrt{3} \\
0 & 1+\sqrt{3} \\
\frac{\sqrt{3}}{2} & \frac{3}{2}+\sqrt{3} \\
\frac{1}{2} \left(1\sqrt{3}\right) & \frac{1}{2} \left(3+\sqrt{3}\right) \\
\frac{1}{2} \left(1\sqrt{3}\right) & \frac{1}{2} \left(1+\sqrt{3}\right) \\
\end{array}
Ready to enter the next level of the problem. ;)
Saturday, November 30, 2013
Organizing Research Notes
If only Mindjet's MindManager could handle LaTeX it would be the perfect tool for organizing my notes, unfortunately it can't and their advice is to use stuff like Equation Editor. That's not for me.
Perhaps at a subconscious level I find that the notes I produce aren't worth saving, fact is that I still haven't found a way of organizing my study notes that suits me. I regularly search the web for tools and today I landed on this MathOverflow question. From this page I jumped to Dror BarNatan's Academic Pensieve, which I invite you to visit because it's unique, impressive and might give you some ideas in organizing your own set of study ( or raw research ) notes.
I have tried a Zillion mindmappers but none come close to MindManager. At work I use FreePlane because it is free ( ( That accurately describes my position with that company ) , the alternative would be to struggle with the tools provided like the damned Word. FreePlane does support LaTeX however.
My little search this morning landed me on DocEar. It supposedly is the MindMapper for Scientists, while MindManager is more positioned at creative business people. If it is any good, I will report on it in a future post.
UPDATE:
I am giving DocEar a try. It's based on FreePlane and JabRef tools that have proved themselves. More later.
Perhaps at a subconscious level I find that the notes I produce aren't worth saving, fact is that I still haven't found a way of organizing my study notes that suits me. I regularly search the web for tools and today I landed on this MathOverflow question. From this page I jumped to Dror BarNatan's Academic Pensieve, which I invite you to visit because it's unique, impressive and might give you some ideas in organizing your own set of study ( or raw research ) notes.
I have tried a Zillion mindmappers but none come close to MindManager. At work I use FreePlane because it is free ( ( That accurately describes my position with that company ) , the alternative would be to struggle with the tools provided like the damned Word. FreePlane does support LaTeX however.
My little search this morning landed me on DocEar. It supposedly is the MindMapper for Scientists, while MindManager is more positioned at creative business people. If it is any good, I will report on it in a future post.
UPDATE:
I am giving DocEar a try. It's based on FreePlane and JabRef tools that have proved themselves. More later.
Sunday, November 24, 2013
Open University: TMA Cutoff date
As an OU student you have to make several TMAs for each course you do. TMA stands for Tutor Marked Assignment. A TMA consists of assignments covering all the booklets you studied in the period prior to the cutoff date of the TMA. The cutoff date is the latest date the work has to be received by your Tutor. This is a period of cutoff dates. Usually you'll find a lot of blog, forum, twitter or facebook posts about TMAs that have been done. Completing a TMA is just part of the study process but for an OU student it's somewhat of an event. Not like an exam, but sending it out gives a feeling of relief, achievement perhaps. Completing a TMA is not something most people do in a few hours, some TMAs take weeks if not months to complete.
The average result of the TMA ( after some formula has been applied to it ), is the maximum result you can get from the course because the final result of the course is the minimum of the exam result and the average TMA score. Also, a minimum TMA score is required to be eligible for taking the exam. For example.
TMA result: 15/100 not eligible for exam.
TMA result: 65/100. Exam result: 100/100. Course result 65/100.
TMA result: 100/100. Exam result: 15/100. Course result 15/100 and thus a FAIL.
Completing all the TMAs on time, requires regular study, and regular study enhances the chance on a good exam result significantly. I suppose that's the thought behind it all. The second example may seem rather undesirable, but it just is not a realistic scenario. A student with a 100% exam score usually has no problems with the TMAs.
Personally, the TMAs are no longer my 'major math challenge'. Slowly but steady I am working on my own mathematical projects. I still need to study, of course, but I am an OU student mainly to justify the time I spend on mathematics to the other stakeholders in my time.  When you say you study mathematics as a hobby, people accept it ( at best ), but explaining that, in fact, you are involved in your own mathematical research is worse than telling that you apply the tools of Scientology in your life. So... I am an OU Student, if you know what I mean. ;)
The average result of the TMA ( after some formula has been applied to it ), is the maximum result you can get from the course because the final result of the course is the minimum of the exam result and the average TMA score. Also, a minimum TMA score is required to be eligible for taking the exam. For example.
TMA result: 15/100 not eligible for exam.
TMA result: 65/100. Exam result: 100/100. Course result 65/100.
TMA result: 100/100. Exam result: 15/100. Course result 15/100 and thus a FAIL.
Completing all the TMAs on time, requires regular study, and regular study enhances the chance on a good exam result significantly. I suppose that's the thought behind it all. The second example may seem rather undesirable, but it just is not a realistic scenario. A student with a 100% exam score usually has no problems with the TMAs.
Personally, the TMAs are no longer my 'major math challenge'. Slowly but steady I am working on my own mathematical projects. I still need to study, of course, but I am an OU student mainly to justify the time I spend on mathematics to the other stakeholders in my time.  When you say you study mathematics as a hobby, people accept it ( at best ), but explaining that, in fact, you are involved in your own mathematical research is worse than telling that you apply the tools of Scientology in your life. So... I am an OU Student, if you know what I mean. ;)
Closed PolyLine  (2)
I developed an algorithm that takes sequences like $\{ 6, 2, 4, 3, 3, 2, 4, 2, 3, 2, 3, 3, 4, 4, 3, 3, 4 \}$ and turns them into drawings like this ( in Mathematica, of course ).
In order to determine if the tile above tiles the plane I need to remove all internal lines...
... which turned out a tad more dificult than expected. I got as far as a prototype algorithm which I am currently testing and improving.
To be continued.
Sunday, November 17, 2013
Closed PolyLine
The picture ( below) contains two drawings ( created with Mathematica ), the drawing on the left consists of a hexagon, a square, a triangle and again a square. In order to facilitate an algorithm that decides if this drawing tiles the plane I need a closed polyline to determine if a point is within the borders of the drawing. Oddly enough it is more difficult to create the drawing on the right than the original on the left.
The drawing on the right is a closed polyline, it consists of a number (11) of line segments and has no begin and endpoint, it is closed. ( In another project I am working on we call PolyLines, MultiLines but I haven't seen that name in use elsewhere. )
Geometric cell division
Take a square and add a similar square to one of its sides and remove the shared edge. What remains is a rectangle.
Take an octagon and add a similar octagon to one of its sides and remove the shared edge. What remains is the following polygon.
Take an ngon and add a similar ngon to one of its sides and remove the shared edge. At infinity the ngons will divide into two circles. Unfortunately my computer is too slow to effectively record this in a video.
Take an octagon and add a similar octagon to one of its sides and remove the shared edge. What remains is the following polygon.
Take an ngon and add a similar ngon to one of its sides and remove the shared edge. At infinity the ngons will divide into two circles. Unfortunately my computer is too slow to effectively record this in a video.
n=12 
n=32 
n=64 
Saturday, November 9, 2013
Example of a polygonal tiling
The tiling in the image below is edgetoedge, polygonal, but not uniform because it has different vertex types, i.e. (3,3,4,3,4), (3,4,4,6), (3,4,6,4) and (3,6,4,4). It is therefore NOT an Archimedean tiling,
With the following piece we can tile the entire plane...
... as we can see here.
With the following piece we can tile the entire plane...
Regular polygons, exercise.
With a drawing program yesterday's pictures are easy to fake, of course. But these drawing programs don't give you the numbers.
Exercise.
Place a decagon edgetoedge on a square with sides of length 1 ( see figure ). What is the distance between the two marked points?
( Answer: $ \frac{1}{4} \left(3 \sqrt{102 \sqrt{5}}+\sqrt{5010 \sqrt{5}}+4\right) $ )
Exercise.
Place a decagon edgetoedge on a square with sides of length 1 ( see figure ). What is the distance between the two marked points?
( Answer: $ \frac{1}{4} \left(3 \sqrt{102 \sqrt{5}}+\sqrt{5010 \sqrt{5}}+4\right) $ )
Friday, November 8, 2013
Printing polygons edgetoedge
Recipe
Suppose you have some polygon with cornerpoints { p1, p2, ..., pk } and you want to print a regular polygon with n edges along one of its edges (p(j), p(j+1) then you can simply find the first point of the regular ngon by rotating p(j+1) with centre p(j) over 360/n degrees. You can continue this process until you have found all points or you can calculate the centre of the regular polygon, its orientation and edgelength which you need to print a regular ngon. Here are some examples.
Examples
Suppose you have some polygon with cornerpoints { p1, p2, ..., pk } and you want to print a regular polygon with n edges along one of its edges (p(j), p(j+1) then you can simply find the first point of the regular ngon by rotating p(j+1) with centre p(j) over 360/n degrees. You can continue this process until you have found all points or you can calculate the centre of the regular polygon, its orientation and edgelength which you need to print a regular ngon. Here are some examples.
Examples
3on4, 4on3 and and 5,7,9,11on 4 (darkonlight ). 
The (3,12,12) Tiling
I made ( what programmers would call ) a 'recipe' for creating a program that prints an Archimedean Tiling. Needless to say that I am talking about Mathematica code. I basically have a program that can distribute any set of Mathematica Graphics objects over a lattice of points. So the recipe basically means calculating the points of the motif, putting them in a set and handing them over to the printer. The last one I did is (3,12,12) Tiling.
The next step is to abstract and code the recipe itself, i.e. translating lists like (3,12,12) or (3,3,4,3,4) to graphics. In fact, I calculated the points for (3,12,12) with a first version of that program. 
Tiling with vertex type (3,12,12). 
Tuesday, November 5, 2013
Johannes Kepler and (3,3,4,3,4)
Who doesn't know the name of Johannes Kepler? Kepler (1571  1630 ) formulated the laws of planetary motion and his work provided the foundation for Isaac Newton's theory of gravity. My point being that Kepler was a scientific giant in his days and his name will live on forever.
My current mathematical project ( personal challenge if you like ) is focused towards tilings of the plane, creating ( Mathematica ) software to print and generate tilings. And ultimately find new tilings I haven't seen before. There aren't many textbooks on the subject, the classic work is very recent ( in mathematical terms ), it was published in 1986: Tilings and Patterns, by Branko Grunbaum and Geoffrey C. Shephard
Earlier today I was doodling on the tiling (3,3,4,3,4) of which I uploaded a picture.
Then, to my surprise I read in Grunbaum / Shephard that it was Johannes Kepler (!) who started the mathematical research on tilings and patterns. One of the beautiful books Kepler wrote is called the Harmony of the World, originally published in 1619 but recently translated into English by Aiton, Duncan and Field and published the American Philosophical Society.
The Harmony of the World consists of 5 books.
 Book 1: On the construction of regular figures
 Book 2: On the congruence of regular figures
 Book 3: On the origins of the harmonic proportions, and on the nature and differences of those things which are concerned with melody
 Book 4: Preamble and explanation of the order
 Book 5: ( No title ).
This is part of a picture ( drawing ) from book 2 which has a drawing of (3, 3, 4, 3, 4 ) just like my doodle rype marked as O ( top right ).
This proves once more that mathematics, by itself, does not change over time, its timeless. At least this part ( if not all ) of mathematics has to be discovered. The tilings of the plane have always been there, it just takes us to see them so that we can ultimately categorize them.
The fact that Kepler worked on this makes him human ( but still a giant of course ) to me, I can imagine the joy and excitement he must have felt drawing the illustrations especially since they take a lot of ( behind the scenes ) calculations.
My current mathematical project ( personal challenge if you like ) is focused towards tilings of the plane, creating ( Mathematica ) software to print and generate tilings. And ultimately find new tilings I haven't seen before. There aren't many textbooks on the subject, the classic work is very recent ( in mathematical terms ), it was published in 1986: Tilings and Patterns, by Branko Grunbaum and Geoffrey C. Shephard
Earlier today I was doodling on the tiling (3,3,4,3,4) of which I uploaded a picture.
Doodle of (3,3,4,3,4). 
The Harmony of the World consists of 5 books.
 Book 1: On the construction of regular figures
 Book 2: On the congruence of regular figures
 Book 3: On the origins of the harmonic proportions, and on the nature and differences of those things which are concerned with melody
 Book 4: Preamble and explanation of the order
 Book 5: ( No title ).
This is part of a picture ( drawing ) from book 2 which has a drawing of (3, 3, 4, 3, 4 ) just like my doodle rype marked as O ( top right ).
This proves once more that mathematics, by itself, does not change over time, its timeless. At least this part ( if not all ) of mathematics has to be discovered. The tilings of the plane have always been there, it just takes us to see them so that we can ultimately categorize them.
The fact that Kepler worked on this makes him human ( but still a giant of course ) to me, I can imagine the joy and excitement he must have felt drawing the illustrations especially since they take a lot of ( behind the scenes ) calculations.
Sunday, October 27, 2013
Archimedean (4,6,12) tiling  Color patterns
Once you programmed the printing of a tiling pattern it is very easy to add colors to the tiles. Some examples.
OU exams more difficult than ever ( ... ) ?
I read a rumor on facebook that the Open University exams were harder than ever. No numbers were shown to substantiate the claim however. You may have been aware that the OU rates have been increased dramatically to align them with the rates of the "Brick Unis" ( = how normal universities are called in OU jargon ). Now one of the commenters said that they are doing the same thing with the exams. Suggesting that until now OU exams were much easier than Brick Uni exams.  To be honest I think it's the other way around. Often homework assignments ( for maths at least ) are part of the grade Brick Uni exams while at the OU you get the lowest grade of homework and exam.
Saturday, October 26, 2013
Archimedean (3,4,6,4) tiling
This is ( part of ) the Archimedean (3,4,6,4) tiling.
The (3,4,6,4) means that at every vertex you'll find four tiles with 3,4,6 and 4 vertices respectively. The Archimedean tilings are vertexuniform.
The (3,4,6,4) means that at every vertex you'll find four tiles with 3,4,6 and 4 vertices respectively. The Archimedean tilings are vertexuniform.
Video Lectures about Lie Groups
"... A Lie group is a smooth manifold obeying the group properties and that satisfies the additional condition that the group operations are differentiable. ..." ( Wolfram Site )
The (self) study of Lie Group theory is hard. I found two aids that helped me going somewhat in the subject. A book called Naive Lie Theory by John Stillwell and a series of weblectures by Erik van den Ban of the University of Utrecht. On van den Ban's homepage under Lecture Notes you'll find a Lie Group's prerequisites pdf with explanations of manifolds, tangent maps etc. which appear frequently in texts about Lie Theory.
The (self) study of Lie Group theory is hard. I found two aids that helped me going somewhat in the subject. A book called Naive Lie Theory by John Stillwell and a series of weblectures by Erik van den Ban of the University of Utrecht. On van den Ban's homepage under Lecture Notes you'll find a Lie Group's prerequisites pdf with explanations of manifolds, tangent maps etc. which appear frequently in texts about Lie Theory.
E8 structure visualized 
Visual Mathematics
Years ago when I started studying mathematics besides my job in IT I never considered that I would be able to apply mathematics in my daytoday job. But I do now, because I work on the development of a drawing program ( a specialized drawing program on Android ). Does that make me happy? Just a bit. Because applied mathematics can get as dirty as computer programming. Once applied, mathematics has lost most if not all of its beauty ( although nothing of its power ). I love pure mathematics, and even more so visual mathematics. The three pictures below are the result of applying a function to respectively the integers 1, 2 and three. The construction ( Mathematica programming, if you like ) of that function however required understanding of calculus, geometry, linear algebra, group theory and tling theory ( they are all parts of the Archimedean tiling of the whole plane with vertex type 4,8,8 ). My point being: this is the mathematics I like so much.
When I really like a picture I made I add it to 'The Gallery'. I am far from being able to create art with mathematics but there is a point where mathematics becomes art or where art becomes ( laying the groundwork for future ) mathematics. M.C. Escher explored mathematics decades before general theories about the subject were formulated.
If you are interested in Mathematics and Art then I can recommend this book: Connections: The geometric bridge between art and science.
Tuesday, September 24, 2013
Reading Challenge: The Road to Reality
The Road to Reality. A complete guide to the laws of the universe.
Prior to my more serious interests in Mathematics I challenged myself to read Goedel, Escher, Bach. Not an easy task if you aren't familiar with musical theory at all. It took me about six months to complete the book. Of course I was in doubt if I really understood it all ( I didn't, not sure if I do know, even after M381 Mathematical Logic, but math grows on you ). Anyway, I have been told that The Road to Reality by Roger Penrose is a book of similar importance as GEB. With it's 1100+ pages it's a challenge alright.
I intend to read it while commuting. Who says commute time can't be made productive? Clearly this is a selfmotivating post ;), but an announcement of a log as well. Sort of a summary of the book in parts. More later. I completed one chapter sofar or 25 pages which is less than 3%.
Prior to my more serious interests in Mathematics I challenged myself to read Goedel, Escher, Bach. Not an easy task if you aren't familiar with musical theory at all. It took me about six months to complete the book. Of course I was in doubt if I really understood it all ( I didn't, not sure if I do know, even after M381 Mathematical Logic, but math grows on you ). Anyway, I have been told that The Road to Reality by Roger Penrose is a book of similar importance as GEB. With it's 1100+ pages it's a challenge alright.
I intend to read it while commuting. Who says commute time can't be made productive? Clearly this is a selfmotivating post ;), but an announcement of a log as well. Sort of a summary of the book in parts. More later. I completed one chapter sofar or 25 pages which is less than 3%.
The Big LaTeX discussion
Although it is perfectly acceptable to handwrite your TMAs at the beginning of every course someone starts the "Big Latex Discussion" again.
It's usually a nerd that starts off by listing the irrelevant but impressive specs of his ( not often her ) hardware, or bloats how much computing know how he has ( parttime math students often work in IT ). He then announces that he 'is going to make his TMAs in LaTeX.
Wow. Jaws dropping. Not.
Not often he also provides us with a list of software artefacts required, version numbers included. Nerdy but deep in the autistic spectrum.
I haven't decided how I'll produce my TMAs this time. I see three options. Handwritten, LaTeX typeset or (new!) handwritten but digitally stored in vector graphics format.
LaTeX became a problem ( pita ) for me in the past when I had to add drawings and stuff. Making and including mathematical drawings in LaTeX is not trivial. Besides it is a major distraction, while you should be thinking about math you are figuring out how some latex drawing package works. Serious waste of time.
Since you can draw on a tablet, store it in SVG and thus manipulate it any way you want, doing the formulas in LaTeX and the drawings 'by hand' is probably the route I take.
Since I do Android work Eclipse became more or less my IDE of (only) choice so I'll give TeXlipse with PDF4Eclipse a try.
It's usually a nerd that starts off by listing the irrelevant but impressive specs of his ( not often her ) hardware, or bloats how much computing know how he has ( parttime math students often work in IT ). He then announces that he 'is going to make his TMAs in LaTeX.
Wow. Jaws dropping. Not.
Not often he also provides us with a list of software artefacts required, version numbers included. Nerdy but deep in the autistic spectrum.
I haven't decided how I'll produce my TMAs this time. I see three options. Handwritten, LaTeX typeset or (new!) handwritten but digitally stored in vector graphics format.
LaTeX became a problem ( pita ) for me in the past when I had to add drawings and stuff. Making and including mathematical drawings in LaTeX is not trivial. Besides it is a major distraction, while you should be thinking about math you are figuring out how some latex drawing package works. Serious waste of time.
Since you can draw on a tablet, store it in SVG and thus manipulate it any way you want, doing the formulas in LaTeX and the drawings 'by hand' is probably the route I take.
Since I do Android work Eclipse became more or less my IDE of (only) choice so I'll give TeXlipse with PDF4Eclipse a try.
Tuesday, September 17, 2013
Tilings resources
I am on a OU course again, ( more about that in posts to follow ). The course site opened today, that really kicks off the course for me.
For now some resources you might find interesting.
A tiling is a covering of the whole plane with nonoverlapping tiles, each of which is a topological disc. The classic work on tilings is Tilings and Patterns by Grunbaum, Shephard. A list with other books on the subject can be found here.
M.C. Escher used tilings in his graphics work in an ingenious way. This book contains a nice collection of the work of Escher.
It's interesting to note that in Escher's time (19011972) there was hardly any mathematical theory about tilings. The foundational work on tilings was published five years after Escher died. Yet from Escher's work it is clear that he understood tilings, and the related line symmetries ( Frieze Patterns ), lattices and plane symmetries ( Wallpaper Patterns ) as no other.
If you are interested in puzzles at all it's likely that you came across Jaap's Puzzle Page, a vast resource of information regarding puzzles. The website is maintained by Jaap Scherphuis. You'll find his YouTube site here with many puzzle demonstrations.
To my astonishment Scherphuizen also maintains an impressive collection of tilings on a Tilings Page. His tilings demonstrations Java Applet is as impressive which you'll find on the same page.
For now some resources you might find interesting.
A tiling is a covering of the whole plane with nonoverlapping tiles, each of which is a topological disc. The classic work on tilings is Tilings and Patterns by Grunbaum, Shephard. A list with other books on the subject can be found here.
M.C. Escher used tilings in his graphics work in an ingenious way. This book contains a nice collection of the work of Escher.
It's interesting to note that in Escher's time (19011972) there was hardly any mathematical theory about tilings. The foundational work on tilings was published five years after Escher died. Yet from Escher's work it is clear that he understood tilings, and the related line symmetries ( Frieze Patterns ), lattices and plane symmetries ( Wallpaper Patterns ) as no other.
If you are interested in puzzles at all it's likely that you came across Jaap's Puzzle Page, a vast resource of information regarding puzzles. The website is maintained by Jaap Scherphuis. You'll find his YouTube site here with many puzzle demonstrations.
To my astonishment Scherphuizen also maintains an impressive collection of tilings on a Tilings Page. His tilings demonstrations Java Applet is as impressive which you'll find on the same page.
'Pentagon Flower' ( background tiling is the [4,8,8] Laves tiling ) (c) nilo de roock 2012 
Sunday, July 7, 2013
The mathematics of beauty.
A mathematics of beauty and art. There is no such thing one would expect. Yet, Owen Jones (1809  1874), London architect, wrote the 'General Principles in the Arrangement of Form and Colour, in Architecture and the Decorative Arts ... ' in his book The Grammar of Ornament.
Proposition 4.
True beauty results from that repose which the mind feels when the eye, the intellect and the affections are satisfied from the absence of any want.
You'll find the other propositions at the link above in the Digital Copy of the 'The Grammar of Ornament'.
Proposition 4.
True beauty results from that repose which the mind feels when the eye, the intellect and the affections are satisfied from the absence of any want.
from Page PL. I 
You'll find the other propositions at the link above in the Digital Copy of the 'The Grammar of Ornament'.
Tuesday, July 2, 2013
Mathematics: created or evolved?
Some ( if not most ) mathematicians think about mathematics as being 'created'. Paul Erdos said that there must be a book, The Book, containing the most important and most beautiful proofs. Who other than God could have written The Book? Einstein once said that 'God does not play dice' and Kronecker said that "God made the integers; all else is the work of man." There is still some creationism left among mathematicians. If there are several ways to attack a problem there is always one best, and that's the one described in The Book.
Where do our mathematical insights come from? Some say they are a form of Divine Intervention. Or is the entirety of mathematics somewhere hidden in our DNA? Evolution Theory versus Intelligent Design. We, the human species, are that part of the physical universe where it becomes selfaware. We ( the physical universe ) describe the behavior of the universe with the help of mathematics. In that sense mathematics evolved over the centuries. In the concept of a created mathematics, mathematics is there, for us humans, to be discovered. Is mathematics finite, created, timeless, deterministic and waiting for intelligent beings to be discovered, understood and applied? Or is mathematics human, a property of our species, a way to handle the complex reality around us. Unlikely as it seems there may be alternative bodies of mathematics possible.
Clearly, compared to the other sciences, mathematics evolves ( is discovered ) at a slow pace. But that's because mathematical truths are eternal ( they say ). The fundamental theorems are, but is the same true for the tools used by engineers and scientists in their daily work? Determinants have lost some of their appeal, quaternions may face the same destiny. Mathematics does change. But the idea that there is one ( created ) mathematics goes deep. You can't patent a mathematical idea for example.
Mathematicians lack a sense of urgency, except for their own careers or immortality maybe. If mathematics in its entirety could be owned by a company, would that ( have ) speed up the development of mathematics? And thus bring the solution of major planetary problems forward? What if there were competing bodies of mathematics, like operating systems, say Windows versus Linux? I think that that would have been possible in some alternate version of history. There are different ways of doing things in mathematics. Take geometry for example. In computer graphics you can choose between ( standard ) linear algebra, add quaternions if you like, or a portion of projective geometry so that you can compose translations and linear transformations with matrices, or: you can use geometric algebra which until not so long ago was either abondoned or exclusively used in quantum physics. My point being we may polish the mathematics until its ready for a place in The Book.
Tuesday, June 25, 2013
Live collisions from CERN on your phone.
Developing Android apps can be challenging and rewarding, especially if graphics and thus mathematics is involved. That applies for the apps I develop myself but even more so for this one that I just found.
If you want to view live 3D displays of collisions direct from CERN on your ( Android ) phone then download this free app called LHSee.
There is also a YouTube video about the development of the app which was done at the University of Oxford.
Did you know that there is even a Lego model of the Atlas collider?
If you want to view live 3D displays of collisions direct from CERN on your ( Android ) phone then download this free app called LHSee.
There is also a YouTube video about the development of the app which was done at the University of Oxford.
Did you know that there is even a Lego model of the Atlas collider?
Sunday, June 23, 2013
From Intelligent Design to Geometry
Just some thoughts...
When I express my doubts about the success of the Apollo project people smile behind my back but that's about it. Friendships aren't broken and neither are ( employment ) contracts. ( See: AULIS on Apollofor more info ). In the last century science turned into a multi billion dollar industry and that certainly has changed the world of academics. Questioning a general accepted theory can ( and will ) ruin careers. Examples are questioning the cause of Aids, and questioning Darwin's Evolution Theory.
Evolution Theory basically says that we evolved over time by a process called mutation and natural selection. Those who question the Evolution Theory in fact question that very, very complex machines evolved from 'mud'. Darwin didn't answer that question because when Darwin published 'On the Origin of Species on 24 November 1859' he wasn't even remotely aware of the complexity of the cells making up life. DNA wasn't discovered until ten years later in 1869 by Friederich Miescher and it took until 1953 when James Watson and Francis Crick discovered the double helix structure of DNA. Intelligent Design Theory accept Evolution Theory but only up to a certain point. They argue that somewhere in the beginning some information or 'design' had to be injected into the system. Who put it there? I would ask.
To the point.
Anyway, these thoughts entered my mind because I am thinking of building a 'DNAtypeof' geometry building block for a computer program. With the help of a computer these geometries should be able to construct ( divide ) themselves in a scene graph and evolve, multiply and so on. At the moment it's just an idea. I started to look for a way to understand more about DNA by finding popular science books on the subject. I haven't learned much about biology and chemistry and what I have learned seems forgotten. But I am only interested in DNA as a computer, or data structure. Then I found this website 'DNA seen through the eyes of a coder'. Since I am a coder ( computer, Android, programmer ) by profession that was exactly what I was looking for. Take this for example, DNA is not binary, DNA is quaternary. Computer letters normally consist of 8 bits called a byte, so using that system there are 256 possible letters. The equivalent of a byte in DNA is the codon and has three places. So in DNA language there are 64 possible letters. Read more on the site.
More to follow on this geometry project soon, I expect.
When I express my doubts about the success of the Apollo project people smile behind my back but that's about it. Friendships aren't broken and neither are ( employment ) contracts. ( See: AULIS on Apollofor more info ). In the last century science turned into a multi billion dollar industry and that certainly has changed the world of academics. Questioning a general accepted theory can ( and will ) ruin careers. Examples are questioning the cause of Aids, and questioning Darwin's Evolution Theory.
Evolution Theory basically says that we evolved over time by a process called mutation and natural selection. Those who question the Evolution Theory in fact question that very, very complex machines evolved from 'mud'. Darwin didn't answer that question because when Darwin published 'On the Origin of Species on 24 November 1859' he wasn't even remotely aware of the complexity of the cells making up life. DNA wasn't discovered until ten years later in 1869 by Friederich Miescher and it took until 1953 when James Watson and Francis Crick discovered the double helix structure of DNA. Intelligent Design Theory accept Evolution Theory but only up to a certain point. They argue that somewhere in the beginning some information or 'design' had to be injected into the system. Who put it there? I would ask.
To the point.
Anyway, these thoughts entered my mind because I am thinking of building a 'DNAtypeof' geometry building block for a computer program. With the help of a computer these geometries should be able to construct ( divide ) themselves in a scene graph and evolve, multiply and so on. At the moment it's just an idea. I started to look for a way to understand more about DNA by finding popular science books on the subject. I haven't learned much about biology and chemistry and what I have learned seems forgotten. But I am only interested in DNA as a computer, or data structure. Then I found this website 'DNA seen through the eyes of a coder'. Since I am a coder ( computer, Android, programmer ) by profession that was exactly what I was looking for. Take this for example, DNA is not binary, DNA is quaternary. Computer letters normally consist of 8 bits called a byte, so using that system there are 256 possible letters. The equivalent of a byte in DNA is the codon and has three places. So in DNA language there are 64 possible letters. Read more on the site.
More to follow on this geometry project soon, I expect.
Sunday, June 2, 2013
Diophantus of Alexandria
Diophantus (+/ 250 AD) of Alexandria has been called 'the father of algebra' and an entire branch of mathematics has been named after him, the study of Diophantine Equations. The most famous problem in this field 'Diophantine Equations' is Fermat's 'Last Theorem'. Fermat was reading Diophantus' comments on the Pythagoran theorem when he conjectured that for an exponent n > 2, the equation \[ x^n + y^n = z^n \] has no integer solutions. This theorem was considered the hardest open problem in mathematics until solved by Andrew Wiles in 1994. Diophantus work was lost to the Western world for thousand years.
Anyway, I thought about Diophantus when I came across this beautiful equation which has an infinite number of integer solutions \[ x^3 + y^3 + z^3 = x^2 +y^2 + z^2.\]
Anyway, I thought about Diophantus when I came across this beautiful equation which has an infinite number of integer solutions \[ x^3 + y^3 + z^3 = x^2 +y^2 + z^2.\]
Monday, May 20, 2013
What is mathematical research ?
What is mathematical research? And how is it done? Questions that have been on my mind for very long.
We know what researchers do:
 answering questions asked in the literature;
 discovering a new theorem;
 publishing an article in a journal ( i.e. Journal for Number Theory );
 writing a book;
 lecturing about their work and giving talks;
 being part of a research community.
Researchers are either employed by a university or by a corporation. University researchers have a commitment to teach and corporate researchers aren't free to chose their topics. Both are pressed to publish often in the best journals possible.
But HOW to they do it? What makes them successful in their field? Questions, I can't answer. Let's go and search for answers elsewhere. Starting with Manning, perhaps.
and this one:
Enjoy.
We know what researchers do:
 answering questions asked in the literature;
 discovering a new theorem;
 publishing an article in a journal ( i.e. Journal for Number Theory );
 writing a book;
 lecturing about their work and giving talks;
 being part of a research community.
Researchers are either employed by a university or by a corporation. University researchers have a commitment to teach and corporate researchers aren't free to chose their topics. Both are pressed to publish often in the best journals possible.
But HOW to they do it? What makes them successful in their field? Questions, I can't answer. Let's go and search for answers elsewhere. Starting with Manning, perhaps.
and this one:
Enjoy.
Monday, April 1, 2013
Conceptual knowledge maps for mathematics
About working and studying, the main topic of this blog: well, it is no longer an option. It is mandatory, society needs people who are committed to their jobs and invest in their lifetime education. Everything changes at an ever faster pace and it is thus our responsibility to stay of value in all of our dynamics. It is an issue we must align regularly.
Studying in a particular field will gradually align your life to your new interests. For example, I have recently, more or less landed in a job where some of my colleagues are mathematicians. In fact, a lot of the work I do involves mathematics. More about that in posts to come.
Everything changes. So do study skills. And a vast amount of tools for computer aided learning is now available. From MIT lectures via YouTube to software for flash cards to sophisticated knowledge databases for personal use. I would like to mention two items I have found recently.
InfoRapid Knowledge Builder is a graphical tool to document conceptual knowledge maps, free for personal use at http://www.buildyourmap.com/. ( Before this I used Mindjet and TheBrain. ) I am working on a map with topics in Analytical Number Theory.
Another item I found is a book. It is called On Course, Strategies for success in College and Life. ( And if you work and study read the title as Strategies for successful combining work and study.) The book is packed with information. It is not a book you should read once but it is a companion you can use, consult, for years. I have the Study Skills Plus edition.
Tuesday, March 26, 2013
Fixing mathematics education
Conrad Wolfram spoke about math education at the 2010 Wolfram Technology Conference:
Now, a new initiative has been recently announced at computerbasedmath.org to build a completely new math curriculum with computerbased computation at its heart.
Also see Conrad's Wolfram blog here where he announces that Estonia will be the first country that will completely rewrite their mathematics school curriculum.
Now, a new initiative has been recently announced at computerbasedmath.org to build a completely new math curriculum with computerbased computation at its heart.
Also see Conrad's Wolfram blog here where he announces that Estonia will be the first country that will completely rewrite their mathematics school curriculum.
Sunday, March 24, 2013
One day left...
to vote for the most important British innovation of the 20th Century. Alan Turing's Turing Machine is currently on 2nd place with 13% of the votes.
Vote here.
Vote here.
Saturday, January 5, 2013
Filling a bottle...
A professor stood before his philosophy class and had some items in front of him. When the class began, he wordlessly picked up a very large and empty mayonnaise jar and proceeded to fill it with golf balls. He then asked the students if the jar was full. They agreed that it was. The professor then picked up a box of pebbles and poured them into the jar. He shook the jar lightly. The pebbles roll ed into the open areas between the golf balls. He then asked the students again if the jar was full. They agreed it was. The professor next picked up a box of sand and poured it into the jar. Of course, the sand filled up everything else. He asked once more if the jar was full.. The students responded with a unanimous ‘yes.’ The professor then produced two Beers from under the table and poured the entire contents into the jar effectively filling the empty space between the sand.The students laughed.. ‘Now,’ said the professor as the laughter subsided, ‘I want you to recognize that this jar represents your life. The golf balls are the important things—your family, your children, your health, your friends and your favorite passions—and if everything else was lost and only they remained, your life would still be full. The pebbles are the other things that matter like your job, your house and your car.. The sand is everything else—the small stuff. ‘If you put the sand into the jar first,’ he continued, ‘there is no room for the pebbles or the golf balls. The same goes for life. If you spend all your time and energy on the small stuff you will never have room for the things that are important to you. Pay attention to the things that are critical to your happiness. Spend time with your children. Spend time with your parents. Visit with grandparents. Take your spouse out to dinner. Play another 18. There will always be time to clean the house and mow the lawn. Take care of the golf balls first—the things that really matter. Set your priorities. The rest is just sand. One of the students raised her hand and inquired what the Beer represented. The professor smiled and said, ‘I’m glad you asked.’ The Beer just shows you that no matter how full your life may seem, there’s always room for a couple of Beers with a friend. ( Anonymous )
Tuesday, January 1, 2013
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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.)