74 Tahun Tak Terjawab Teka Teki "3n+1" Mungkin Terpecahkan

Teka teki Matematika yang belum ditemukan jawabannya selama 74 tahun sebentar lagi mungkin akan terpecahkan. Adalah matematikawan dari Universitas Hamburg, Gerhard Opfer yang mengklaim telah menemukan solusinya.

Teka teki Matematika itu yang bernama Collatz Conjecture atau "3n+1" itu diajukan oleh Lothar Collatz pada tahun 1937. Teka teki itu melibatkan operasi bilangan bulat yang dilambangkan n. Singkatnya, ada 2 syarat yang berlaku dalam Collatz Conjecture. Jika bilangan bulat (n) ialah bilangan genap maka dibagi dua (n/2) dan jika ganjil maka dikalikan 3 kemudian ditambah 1 (3n+1).

Nah, dalam Collatz Conjecture diungkapkan, jika operasi terus dilakukan berulang kali, maka berapa pun angka yang dipilih untuk memulainya, akan selalu didapatkan angka 1 sebagai hasilnya. Verifikasi telah dilakukan hingga angka 5,76 x 10 (18). Namun, tanpa pembuktian matematis yang tepat, selalu ada kemungkinan bahwa angka yang sangat besar akan melenceng dari "hukum" ini. Pembuktian matematis inilah yang telah dimiliki oleh Opfer. Ia menuliskannya dalam paper yang kini telah masuk ke jurnal Mathematics of Computation untuk ditinjau ulang sebab bisa saja pembuktiannya tak tepat.

Nah, apakah puzzle matematika ini nantinya akan benar-benar terselesaikan? Kita tunggu saja. Sementara menunggu, mungkin Anda bisa mencoba mengoperasikan angka berdasarkan Collatz Conjecture.

Coba ambil angka 6. Karena 6 genap, maka dibagi 2, hasilnya 3. Nah, 3 adalah bilangan ganjil, maka dikali 3 dan ditambah 1, hasilnya 10. Lalu, 10 dibagi 2 karena bilangan genap, hasilnya 5. Kemudian, 5 dikali 3 dan ditambah 1, hasilnya 16. Angka 16 dibagi 2, hasilnya 8. Kemudian 8 dibagi 2 hasilnya 4 dan 4 dibagi 2 lagi hasilnya 2. Angka 2 adalah bilangan genap, maka dibagi 2 lagi dan hasilnya 1. Nah, jika diurutkan, maka deretannya adalah 6, 3, 10, 5, 16, 8, 4, 2, 1. Untuk angka 6, berarti terbukti kebenarannya. Anda bisa mencobanya dengan mengambil angka lain. Mau lebih menantang, ambil angka yang besar.

sumber: http://www.kaskus.us/showthread.php?t=9035822

Graphing Stories: Balloon and Tower

Yeah! I have rarely been so excited to get an email.  "We just pulled your Graphing Story out of the oven!" The only downside is that my name is prominent, when I really just incited other people to make them.  But I'm still geeked. Huge props to Dan Meyer and BuzzMath for doing this.

First, the Balloon.  Filmmaker - Anna Minnebo, Balloonist - Gregg Minnebo.






Second, The Towers of Hanoi. Stacker/Graphist - Eric Thuemmel; Camera - Monica Leneway.  In the original he just got the fourth stack complete on the 15th second, but it just barely gets cut off here. He was quick.





Still to come: four simultaneous tower builders... can understand why the graph for that is a special problem.

Teka-teki "3n + 1" Mungkin Terpecahkan ....

74 Tahun tak terjawab...

KOMPAS.com — Teka teki matematika yang belum ditemukan jawabannya selama 74 tahun sebentar lagi mungkin akan terpecahkan. Adalah matematikawan dari Universitas Hamburg, Gerhard Opfer, yang mengklaim telah menemukan solusinya.

Teka teki matematika itu yang bernama Collatz Conjecture atau "3n+1" itu diajukan oleh Lothar Collatz pada tahun 1937. Teka teki itu melibatkan operasi bilangan bulat yang dilambangkan "n". Singkatnya, ada dua syarat yang berlaku dalam Collatz Conjecture. Jika bilangan bulat (n) adalah bilangan genap, maka dibagi dua (n/2) dan jika ganjil maka dikalikan 3 kemudian ditambah 1 atau (3n+1).

Nah, dalam Collatz Conjecture diungkapkan, jika operasi terus dilakukan berulang kali, berapa pun angka yang dipilih untuk memulainya akan selalu didapatkan angka 1 sebagai hasilnya. Verifikasi telah dilakukan hingga angka 5,76 x 10 (18). Namun, tanpa pembuktian matematis yang tepat, selalu ada kemungkinan bahwa angka yang sangat besar akan melenceng dari "hukum" ini. Pembuktian matematis inilah yang telah dimiliki oleh Opfer. Ia menuliskannya dalam paper yang kini telah masuk ke jurnal Mathematics of Computation untuk ditinjau ulang sebab bisa saja pembuktiannya tak tepat.

Nah, apakah puzzle matematika ini nantinya akan benar-benar terselesaikan? Kita tunggu saja. Sementara menunggu, mungkin Anda bisa mencoba mengoperasikan angka berdasarkan Collatz Conjecture.

Coba ambil angka 6. Karena 6 genap, maka dibagi 2, hasilnya 3. Nah, 3 adalah bilangan ganjil, maka dikali 3 dan ditambah 1, hasilnya 10. Lalu, 10 dibagi 2 karena bilangan genap, hasilnya 5. Kemudian, 5 dikali 3 dan ditambah 1, hasilnya 16. Angka 16 dibagi 2, hasilnya 8. Kemudian 8 dibagi 2 hasilnya 4 dan 4 dibagi 2 lagi hasilnya 2. Angka 2 adalah bilangan genap, maka dibagi 2 lagi dan hasilnya 1. Nah, jika diurutkan, maka deretannya adalah 6, 3, 10, 5, 16, 8, 4, 2, 1. Untuk angka 6, berarti terbukti kebenarannya. Anda bisa mencobanya dengan mengambil angka lain. Mau lebih menantang, ambil angka yang besar.

Grading: SBG and U

Math Monster
by Mister Awesome @ Flickr

Standards Based Grading to me is the idea that the teacher lays out what students are responsible for demonstrating ahead of teaching, and students have a long period during which to demonstrate them, possibly up until grades are finalized.  And students have multiple opportunities to demonstrate.  (This is a Part II to the previous grading post.)

Other people describe it better and more thoroughly.  Especially Sam Shah and Shawn Cornally.  Also please check out the beginner's wiki started Elissa Miller and the SBG gala hosted by Matt Townsley. (Note that you could be interacting with these outstanding professionals on Twitter: @samjshah, @thinkthankthunk, @misscalcul8, @mctownsley) Frank Noschese is thinking about it powerfully, too, in physics, but I haven't had the chance to interact with him about it.

I will say that I've only used it with preservice teachers so far, but they were mostly an appreciative audience for it, and would like to see it in their content classes.  I will be doing it in my content classes, starting with a graduate calculus class in the fall, but we're so pinched for math educators right now that I don't get to teach any straight content courses.

The preservice teachers have been helpful for improving my practice of it with their feedback.  If you're making the change, I'd encourage you to discuss it with your students, give your reasons, and involve them in the process.  I was only going to do it through in class assessments and similar things in office hours, but I added an SBG option to portfolio submissions and added an interview option for office hours.  The biggest remaining thing is how to communicate it better at the outset, with which the resources in the second paragraph will help.

Another Speedbump Classic

The most powerful concept to the shift has been giving the students a clearer purpose on the assessments: to demonstrate what you understand by communicating your thinking.  Much of the emphasis on the right answer is gone, as is the expectation that test questions will be trivial repeats of tasks already done.  Not that my tests were like that lately (have to go back over 20 years for one of those), but it was a bone of contention with students.  Now it makes (more) sense to them that they couldn't show understanding on a question like that.  I've had a few students reject a problem because they knew how to do it already.  (That's not the majority, but some day...)

It's different from K-12 use because in the university we see the students so much less. We give up class time for independent work outside of class, which minimizes time for summative assessment.  I struggled to provide multiple assessment points.  Put lots of former standards on assessments as choice, and polled students as to what previous standards they wanted on.  My standards were much broader than they would be in a content course, as math ed classes wind up covering things like "all of high school mathematics."  So I made my standards pretty broad, but we looked at examples of more focused grade level standards.  In the future as I reuse, I'll try to add some of those specifics as ways to demonstrate the broad standards.  I also let them know that the final grade would take into account which standards we had covered and assessed in class.  Some of the content I don't set until the preassessment is in, so it's hard to know ahead of the semester.

Here's the policy on my middle school math syllabus.

Standards Based Grading: SBG is a relatively new way to assess students that seeks to get a higher correlation between grade and understanding. On each of the objectives below, you will have opportunities to demonstrate your understanding. These objectives are a bit broader than you would expect in a secondary classroom, since we are seeing content from three years of schooling. In a secondary classroom, the teacher identifies the standard demonstrated, but in this preservice teacher preparation course you will also be trying to identify which of your work is evidence of which standard.

Scores do not mean an answer is right/wrong, but are meant to reflect how much understanding was demonstrated. It is possible to demonstrate good understanding of a concept without even finishing a particular problem. The score for each category is the average of the 2 highest scores. If there is only one score it is discounted by 1; a single A becomes a B, etc. You can reassess on specific objectives during office hours or at arranged times.

A+ complete understanding and can extend on your own
A complete understanding, can apply when appropriate
B some small difficulty applying or missing a small point of understanding
C significant difficulty in application or missing a major point of understanding
D mechanical application of ideas without understanding
F little to no understanding or evidence of understanding

Mathematical Content Objectives

A. Number: representations and operational concepts

1. Integers
2. Operations on integers
3. Rational numbers: fractions
4. Operations on fractions
5. Rational numbers: decimals
6. Operations on decimals

B. Algebra: representation, operations and modeling

1. Patterns: recognizing and generalizing
2. Variable: as unknown and changing quantities
3. Linear and exponential relationships

C. Geometry

1. Similar figures and proportional reasoning
2. 2-D figures: characteristics and sorting
3. 3-D figures: characteristics and sorting
4. 3-D representation

After a messy fall semester of trying to run parallel SBG and traditional, and a messy winter semester of struggling with full implementation, I'm very happy I came down this road.  I have four basic goals for my grading:

From Comically Vintage
Don't be a Dodo!

• fair - reassessment helps this.
• measures real understanding - move away from non-problems helps this.
• not fear or anxiety inducing - students said this was a big improvement.
• measures where the student is at the end of the course - clear improvement.

While I didn't get a lot of out of classroom reassessment until the end of the semester, I did get people using the in class assessments to reassess.  Students were more responsible for their own marks than ever before, and rather than tracking grades, they were attending to objectives. Broad over-generalized objectives, but I had to start someplace!

I strongly recommend you consider SBG, whether you be K-12 or 13-19.  If you do, let's talk!

മലയാളവും ഐടിയും പ്രണയത്തില്‍..!

കണ്ണുനീരില്‍ പൊതിഞ്ഞ ചിരി എന്ന പോസ്റ്റിലൂടെ തന്റെ ഘനഗംഭീരമായ ശബ്ദത്തില്‍ ചാര്‍ലി ചാപ്ളിന്റെ ജീവിതത്തെക്കുറിച്ചുള്ള ഒരു ഡോക്യുമെന്ററിയുമായത്തിയ അഹമ്മദ് ഷെരീഫ് ഗുരുക്കളെ ഓര്‍ക്കുന്നില്ലേ..? ഒട്ടേറെ വായനക്കാരുടെ, പ്രത്യേകിച്ച് മലയാള അധ്യാപകരുടെ മുക്തകണ്ഠ പ്രശംസ പിടിച്ചു പറ്റിയ ആ പോസ്റ്റ് കഴിഞ്ഞ നവംബറിലാണ് നാം പ്രസിദ്ധീകരിച്ചത്. കാസര്‍കോട് മോഡല്‍ ഗവണ്‍മെന്റ് ഹൈസ്ക്കൂള്‍ ഫോര്‍ ഗേള്‍സിലെ അധ്യാപകനായ അഹമ്മദ് ഷെരീഫ് കുരിക്കള്‍ തികച്ചും ക്ലാസ് റൂം പ്രവര്‍ത്തനങ്ങള്‍ക്കായി തയ്യാറാക്കിയ വീഡിയോയായിരുന്നൂ "ചിരിയുടെ രാജകുമാരന്‍ ". പത്താം ക്ലാസിലെ മലയാളം ഒന്നാം യൂണിറ്റ് വിനിമയം ചെയ്യാന്‍ സഹായിക്കുന്ന മറ്റൊരു കിടിലന്‍ ദൃശ്യ ശ്രാവ്യാനുഭവവുമായാണ് ഇത്തവണ ഷെരീഫ് സാര്‍ നമ്മിലേക്കെത്തുന്നത്. ഇതൊന്നു കാണുകയും അവശ്യം കുട്ടികളെ കാണിക്കുകയും ചെയ്യുന്നത് മലയാളാധ്യാപനത്തെ വളരെയധികം സഹായിക്കുമെന്നുറപ്പ്. ഈ വീഡിയോ കാണുന്ന വിദ്യാര്‍ത്ഥിക്ക് പാഠഭാഗം ഹൃദിസ്ഥമാകാന്‍ മറ്റൊന്നും ചെയ്യേണ്ടെന്നു സാരം. അഭിപ്രായങ്ങള്‍ കമന്റുകളായൊഴുകിവരുന്നതിനായി കാത്തിരിക്കുന്നു. ഒപ്പം സ്കൂളില്‍ ലഭ്യമായ ഹാന്റിക്യാമും നമുക്ക് സ്വന്തമായുള്ള ഓപണ്‍ഷോട്ടും ഒഡാസിറ്റിയുമൊക്കെ ഉപയോഗപ്പെടുത്തി, ഇതുപോലെ ഉപകാരപ്രദമാ വീഡിയോകളുണ്ടാക്കി പങ്കുവെക്കെന്നേ..! ഐടി പിരീഡുകളും മലയാളം പിരീഡുകളും തമ്മില്‍ വിവാദങ്ങളുണ്ടാക്കാന്‍ ആര്‍ക്കും കഴിയാത്തവണ്ണം ചേര്‍ന്നിരിക്കട്ടെ, അല്ലേ..? വീഡിയോ കണ്ടോളൂ....



വീഡിയോയെയും വിഷയത്തെയും കുറിച്ചുള്ള നിങ്ങളുടെ അഭിപ്രായങ്ങള്‍ കുറിച്ചിടണേ.

Project Euler

If you like programming, mathematics and you want to rank yourself among peers then Project Euler may be an option for you. It's not for me ( yet ), my game is mathematics at the Open University but who knows, I might not be able to resist the problems. Although some look simple, once you have solved a problem you get access to a database of how others solved the problem. That may be very illuminating.

Add all the natural numbers below one thousand that are multiples of 3 or 5.

Problem #1.

What is the first term in the Fibonacci sequence to contain 1000 digits?

Problem #25.

And so on... with an ever increasing complexity. Knowing some Elementary Number Theory might help, I suppose.

Link: Project Euler

Happy Coding!

Watermelon: any questions?

Dan Meyer, purveyor of pedagogical principles and master of mathematical memes, has popularized or coined the idea of #anyqs on twitter.  A math teacher will post a photo (or video) and ask, "Any Questions?"

Here's my photos:

Post questions in the comments or I'll record any twitter questions I see.

What is the dimension of color ?

Think of color, pitch, loudness, heaviness, and hotness. Each is the topic of a branch of physics.

Benoit Mandelbrot

The prototypical fractal

( I'll continue with my 'watch notes' of the GEB series later this week. But I'll stay on topic with this post. )

Imagine a recursive process which goes to a certain depth until it stops. This 'certain depth' is a natural number which can be assigned a color. This is basically how fractals like the one above are built, pixel by pixel.

Programming 3D graphics can get quite realistic as we all know from watching movies or playing computer games. What is this reality? When does a picture look real to you? It is what made Rembrandt famous, I suppose. Control over light and thus color and shading.

What I am trying to say is that one number is not enough to encode a color. This is rather counter-intuitive, I know, but only if you assume the number of colors is finite or countable infinite. How many colors are there? Finitely many? Countable infinite many? Or not countable infinite? I don't know.

Assume a scene with an object with a certain color, say lime green. This can be coded with one string, the RGB color method uses 32 CD 32 ( Hex ) for lime green. But then you have an object that looks exactly the same everywhere. Because there is no light in the scene. By adding light we have to add the exact location of the light source in the scene. Shadows must be calculated. And light will be reflected. Each pixel will have a reflection vector which has effect on the color. Does this add to the dimension of the color? What if there are other objects in the scene? They partially reflect light and thus become a light source as well.

Do you get the idea? Then what is the dimension of ( the vector needed to encode a ) color?

More:
- The Dimensions of Colour

Goedel, Escher, Bach - Lecture 4(1)

My brain ran those neural network algorithms.

Justin Curry

The reading assignment was chapter 6 'The location of meaning'. It is basically about coding and decoding. Of course Hofstadter mentioned the Rosetta Stone in this context, the key to ancient Egypt. It contained a parallel text in three languages and was deciphered in 1821 by Champollion.

A recent example in the context of chapter 6 is space archeology. Archeologists and Egyptologists were able to interpret satellite pictures of Egypt which lead to the sensational discovery of new pyramids.

Let's go to the lecture. ( This is a 1h46m lecture and will be discussed in two posts. )

Curry talked about Goedel numbering again, a method Goedel used to code strings in formal number theory to numbers. What Curry said about coding a string in formal number theory, playing with it and then code it back is only true in theory. Simply because Goedel numbers become -extremely large-. Think of numbers built from pages full of digits. ( Would Curry ever have calculated a Goedel number? This reminds me of a DBA course I attended once. The trainer talked about all the beautiful properties of the then new RMAN from Oracle as if backups could be recovered in an instant. It turned out that he never worked in the trenches of 7 x 24 administration of large databases. )

Dialog "Contracrostipunctus" on page 75 of GEB is discussed. How this dialog has meaning on several levels. The dialog refers to itself that it contains a hidden message. The concept of 'Self' is introduced here.

Starting with what does "Snow is white" mean? he builds an argument that there is an isomorphism between electrical activity in the brain and the interpretation of symbols. ( Thought reading might be possible after all, one day. Isn't it true that man can create everything he is able to envision? )

Adam and Eve

There are at least two phases in the proces of assigning meaning two a string. The first is parsing the string, the second is the interpretation of the parsed words. Interpretation depends on the context of the interpreter.

Message in a bottle

He introduces the concept of information. For example how physicists reduce complex physical behaviour to a small sequence of symbols. Like for example how a pendulum works.

Pendulum ?

ആദ്യ മൂന്ന് അധ്യായങ്ങളുടെ സമഗ്രാസൂത്രണം

പുതുപുത്തന്‍ പ്രതീക്ഷകളുമായി നവോന്മേഷത്തോടെ പുതിയൊരു വര്‍ഷം കൂടി കടന്നു വന്നു. നിറങ്ങളില്‍ ചാലിച്ച പത്താം ക്ലാസിലെ ഗണിതപാഠപുസ്തകം നമ്മുടെ കൈകളിലേക്കെത്തി. ഇനി അധ്യയനത്തിന്റെ നാളുകള്‍. ഈ വര്‍ഷവും അധ്യാപകര്‍ക്കും വിദ്യാര്‍ത്ഥികള്‍ക്കുമൊപ്പം മാത്​സ് ബ്ലോഗുണ്ടാകും. കൃഷ്ണന്‍ സാര്‍ തയ്യാറാക്കിയ സമാന്തരശ്രേണിയിലെ അധിക ചോദ്യങ്ങള്‍ അവധിക്കാലത്ത് പ്രസിദ്ധീകരിച്ചത് കണ്ടിരിക്കുമല്ലോ. നിങ്ങളുടെ സംശയങ്ങള്‍, കണ്ടെത്തലുകള്‍.. എല്ലാം ബ്ലോഗിലൂടെ നമുക്ക് പങ്കുവെക്കാം. ക്ലാസ് മുറികളില്‍ നിങ്ങള്‍ പ്രയോഗിക്കുന്ന പഠനതന്ത്രങ്ങള്‍, എളുപ്പവഴികള്‍ എല്ലാം നമുക്ക് കൈമാറ്റം ചെയ്യാം. അങ്ങനെ നമ്മുടെ വൈജ്ഞാനികലോകം കൂടുതല്‍ വിപുലമാകട്ടെ. കാലം ഇത്രയേറെ പുരോഗമിച്ചിട്ടും കമ്പ്യൂട്ടറിനോടും ഇന്റര്‍നെറ്റിനോടുമെല്ലാം ഒട്ടും തന്നെ താല്പര്യമില്ലാത്ത അനവധി നിരവധി അധ്യാപകര്‍ നമുക്കൊപ്പം തന്നെയുണ്ട്. അവരെല്ലാം നമ്മുടെ ചര്‍ച്ചയിലേക്ക് വന്നെങ്കില്‍..!!!! ഈ വര്‍ഷത്തെ പത്താം ക്ലാസ് പാഠപുസ്തകത്തിലെ ആദ്യ മൂന്നു പാഠങ്ങളുടെ സമഗ്രാസൂത്രണം ജോണ്‍ സാര്‍ തയ്യാറാക്കിയത് ഈ പോസ്റ്റിനൊടുവില്‍ നല്‍കിയിരിക്കുന്നു. സമഗ്രാസൂത്രണത്തെ ലാ-ടെക് എന്ന ടൈപ് സെറ്റിങ് സോഫ്റ്റ്​വെയര്‍ ഉപയോഗിച്ച് ഡൗണ്‍ലോഡ് ചെയ്യാനാകും വിധം പി.ഡി.എഫ് രൂപത്തിലേക്ക് മാറ്റിയത് ആദരണീയനായ നമ്മുടെ കൃഷ്ണന്‍ സാറാണ്. താഴെയുള്ള ലിങ്കില്‍ നിന്നും അവ ഡൗണ്‍ലോഡ് ചെയ്തെടുക്കാം. ഐടി@സ്ക്കൂള്‍ തയ്യാറാക്കിയ ആദ്യ രണ്ട് അധ്യായങ്ങളുടെ ജിയോജിബ്ര പാക്കേജും താഴെയുള്ള ലിങ്കില്‍ നല്‍കിയിട്ടുണ്ട്. ചര്‍ച്ചയ്ക്ക് തുടക്കമിടുമല്ലോ.

Chapter-1 Arithmetic sequence
Chapter-2 Circles
Chapter-3 Second Degree Equations
Geogebra Package for Chapter - 1 & 2 (33.5 MB)

Goedel, Escher, Bach - Lecture 3

A guy named Euclid.

Justin Curry

Curry briefly explaines the concepts:
- consistency
- completeness
- and geometry.

A consistent system leads to conclusions that are not contradictory in any sense. A statement is either true or false, and never both true and false.
A system is complete if everything that is true in the context of that system can be derived from the axioms.
Regarding geometry he mentioned that there are Euclidean non-Euclidean geometries.

Then he attempts to explain Goedel's Incompleteness Theorems.
1. Any system as powerful as number theory which can prove its own consistency is necessarily inconsistent.
2. Any system as powerful as number theory is necessarily incomplete.
He explains that Goedel managed to transform the idea of provability to a property of numbers by introducing his Goedel numbers.
He says that students should now have a notion of the Goedel theorems and promises that this is just a first glance at Goedel's theorem. ( Not sure if he meant he would come back at Goedel in this lecture series. )

Trying to explain Goedel

He then talks about Euclid and his postulates.
(1) Any straight line segment can be drawn joining any two points.
(2) Any straight line segment can be extended indefinitely in a straight line
(3) Given any straight line, a circle can be drawn having the segment as radius and the
(4) All right angles are congruent.
===
(5) If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles

He explains that the 5th postulate is consistent in Euclidean Geometry but not in spherical and hyperbolic geometry.

Hofstadter Dialog - Little Harmonic Labyrinth is removed from the video due to copyright concerns. It is part of what makes GEB such a difficult book. Here is part of it.

The Tortoise and Achilles are spending a day at Coney Island. After buying a couple of cotton candies, they decide to take a ride on the Ferris wheel.
Tortoise: This is my favorite ride. One seems to move so far, and yet in
reality one gets nowhere.
Achilles: I can see why it would appeal to you. Are you all strapped in?
Tortoise: Yes, I think I've got this buckle done. Well, here we go. Whee!
Achilles: You certainly are exuberant today.
Tortoise: I have good reason to be. My aunt, who is a fortune-teller, told me that a stroke of Good Fortune would befall me today. So I am tingling with anticipation.
Achilles: Don't tell me you believe in fortune-telling!
Tortoise: No . . . but they say it works even if you don't believe in it.

About 1% of the Little Harmonic Labyrinth dialog.

He explains the cardinal arithmetic, the arithmetic of infinities.
An interesting definition of infinity is that a set can be mapped to a subset of itself. I.e. the natural numbers can be bijectively mapped to the even numbers. The points on the real line can be bijectively mapped to the points on the line between 0 and 1.

Non-Euclidean geometries

Simoncino - Beat the Street EP

MATHEMATICS 053

A1. Beat the Street
A2. Touch (Rhythm Beat)
B1. Target
B2. Jungle Dreams

San Laurentino - Traces EP

MATHEMATICS 052

A1. Magnetic Steps
A2. Traces feat Madmatt
B1. The Olympians
B2. Lazy Angels

Goedel, Escher, Bach - Lecture 2

It will get a little bit mathy, but that's ok.

Curran Kelleher

Lecture 2 is given by Curran Kelleher. This lecture is all about recursion and ends with a nice explanation of the Mandelbrot set.

He starts out with the traditional examples factorial:

factorial[0]:=1;
factorial[n_]:=factorial[n-1]*n; 

and fibonacci sequence:

fib[1]:=1;
fib[2]:=1;
fib[n_]:=fib[n-1]+fib[n-2];

Kelleher's factorial program

Kelleher's hand-out ( pdf ) contains examples of Java code for drawings of the Koch curve and Sierpinski triangle. Although I fast-forwarded through this part of the lecture, it may be very interesting for non-programmers.

Explaining the Fern algorithm

Complex number implemented as a class in Groovy 

At around 1:00 he starts with the topic of the Mandelbrot set. Starting with f(z) = z^2 + c he manages to give a nice explanation of the Mandelbrot set. The color of a point in the Mandelbrot set is based on the number of iterations it took f to 'escape' a circle. Since this is done on a pixel by pixel basis and one pixel may generate not one but several iterations this explains the long time it takes to generate a Mandelbrot set.

P.S.
GEB does not seem 'outdated' at all although it was written in the late seventies. A time when there were no mobile phones, no PCs, let alone laptops and the internet was still in its toddler phase.

Goedel, Escher, Bach - Lecture 1

'Understanding Goedel' is one of the major goals I set for myself.

This final unit brings together all the ideas introduced in the course. These ideas constitute the technical machinery that enables us to prove some very important theorems which answer what we have called Leibniz's and Hilbert's Questions. These theorems, Goedel's Incompleteness Theorems, are among the most profound intellectual discoveries of the the twentieth century. Thus you should not be surprised if you find this unit hard going in places.

M381 - Unit 8.

In Goedel, Escher, Bach (GEB) Hofstadter asks the question: what happens when 'things' start referencing themselves? ( Like people do who are in essence not more than a set of linked molecules. )

At last I took the time to watch video 1 of the GEB series.

Justin Curry

The teacher is Justin Curry. He started by telling that most undergraduates don't get through GEB in less than 13 weeks and that it took him seven years to get through the book. I am not sure but I think my first attempt in reading GEB was in 2007 or 2008. It took me almost six months to get through it. Which I thought was really bad. When I finished the book and still didn't understand what he was talking about I started to seriously doubt my learning abilities. I have to admit that I still don't get it but I made progress. And I am getting closer, thanks to M381 Mathematical Logic ( read: Nigel Cutland ).

Anyway, to the point: the lecture.

He starts with the concept of isomorphism. In GEB, Hofstadter explains isomorphism as a map between structures that maps parts with similar purpose to similar purpose ( my words ). This is different than the mathematical definition which states that an isomorphic map is both surjective and injective. Hofstadters definition can be understood immediately, whereas the mathematical definition needs understanding of layer upon layer upon layer. Since what is a map in mathematical sense? What does surjective mean? What does injective mean? Analyzing a mathematical sentence always creates a ( large ) tree structure.

Recursion. The concept of recursive definition. A fascinating concept which I use a lot, since I am a programmer by profession. Curry uses the example of the Fibonacci sequence 1,1,2,3,5,8,13,... and translates it to f(n) = f(n-1) + f(n-2) and the Sierpinski triangle ( fractal ).

Drawing the Sierpinski triangle

( To be continued in the next post. ) Edit: Nope. I'll make a last note about lecture 1 here and continue with lecture 2 next time.

Some remarks, tips for if you want to give it a try ( like myself ). I was not in continuous awe while watching this lecture. You know when like you are watching the latest BBC Horizon or similar. It's not like that. I don't have the feeling as if I have wasted my time, not at all. I am going to watch lecture 2 soon.

- You definitely need the 720+ pages ( 20 chapters ) book. ( Details on the course site. )
- You need to be ( somewhat ) familiar with Bach's music, or at least -know- someone who is. ( What are forums for anyway? ) To fully grasp the genius of Hofstadter's work.
- If you are a religuous person than GEB might not be for you.

There is an audio set in the lecture room. Near the end of the lecture a piece of Bach is played. Students familiar with that music could elaborate on it. Since I am ignorant to most classical music I must have missed a lot of what Hofstadter said. It might be an opportunity to start listening to some Bach, who knows what happens.,

So far for lecture 1,