G’day. This post is formatted in three parts: The protagonist, The chapter, and finally any relevant conclusions on the chapter. I’ll avoid going too heavy on the math from here on, as the chapters become increasingly complex, building on each other as they progress, as they should. I really don’t feel like being a pedant or the Maths Gestapo this evening.

I’ve gotten some good feedback on this series, and it’s been pointed out that the admittedly poorly chosen title convention previously used in this series of reviews has caused some confusion about the actual title of the book, hence the change in the this entry’s header for this installment and for each one hence, using the format of main title/chapter number/chapter name.

Also, what I’ve learned since the start of this series bears correcting technical errors in the earlier installments, so I’ll be updating those over the next fortnight.

Links to some of the concepts mentioned are provided where needed.

The protagonist:

This time we’ll first take a brief look at the life and accomplishments of this installment’s costar, one Lazarus Fuchs, whose relevant work is noted here.

Born in 1833 close to present day Poznan in Poland, Fuchs held prominent membership in the Berlin mathematical circle. Much of his work involved research into differential equation systems applied to the complex plane, to see the changes in solutions made by circuiting singularities in which a given term acquired an infinite value (as with 1/z when z=0).

He soon understood the simplicity of looking into two solutions’ ratio, as circuiting a singular point this ratio alters in the manner of a Möbius map. Rounding distinct paths around differing singular points generates a group from all of these maps together.

Not long after his May, 1880 paper was published, he began correspondence with Henri Poincare, then of the University of Caen, who observed that the ratio-function’s inverse was in fact one applied to a group’s associated surface.

This led to Poincare realizing that Möbius transformations and non-Euclidean geometry were somehow linked — But that’s a matter for the review of the book’s final chapter, “Epilogue.”

It was Poincare who named the resulting groups he found *Fuchsian*, as a tip of the proverbial hat to Lazarus Fuchs.

Here are some details of the chapter:

It begins with a discussion of fractal curves formed using circles in the complex plane, either “kissing” tangential Schottky circles or those disjoint.

This allows the creation of circles that nest into each other, with each nest reducing in scale with the limit set approaching zero.

This allows the circles to retain the same degree of complexity and preserving detail no matter their magnification.

As this is also an art book, illustrations, some rather gorgeous, are provided for the chains of circles produced, by variations in the transformations used for each.

There is of course art in earlier chapters, but beginning here they become truly impressive.

The first diagram, figure 6.1, shows a rather cool looking linked fractal chain — Indra’s necklace indeed.

There are also rules given for setting up the initial parameters for the formulas to generate this and similar shapes.

These rules, or necklace conditions are (paraphrased):

- We must correctly line up the tangent points using our transformations “a” and “b,” and image points on the plane must be accurately matched.
- We must make certain that the fixed point of transformation “abAB” is parabolic.
- We must be sure that transformation abAB’s trace is -2. This lets us get a parabolic commutator and also makes the chain of fractal dust shrink at ever smaller scales, with no loss of complexity.

This generates a series of nested, “kissing” circles with looped limits, resembling links in a chain or gems on a necklace. These looped limits are *quasicircles*, and Möbius transformations with a quasicircular limit set are Q*uasifuchsian*, after Lazarus Fuchs’s work.

The concept of Fuchsian and Quasifuchsian groups is further developed in some detail, the difference between the limit set and its separate regular or ordinary set is made clear, and though the math can be a bit hairy-looking, it’s manageable once you get used to playing with it.

The conjugation of already existing Möbius transformations, done by altering their coordinates on the Riemann sphere, may be done to switch perspective, making new images from old by modifying the generating matrix, though without fundamental changes in behavior.

The rest of the chapter gets more technical, but also more interesting, detailing the mathematics and program parameters for generating with and experimenting with the shapes and transformations discussed. The projects at the end provide things to practice on, and act as steppingstones to chapter 7 and beyond.

Conclusion:

Who knew that such cool things could be done with something as ordinary as circles? I’m already inspired to get to work on the Kleinian group raytrace formulas on my copy of Ultra Fractal, just to see what strangeness results, maybe even write my own algorithms.

As with last time, there’s no need for gratuitous mystery-mongering here, but plenty of fun to be found in the solving of mysteries, and the feeding of very human curiosity.

]]>A team of researchers says the widely repeated advice isn’t feasible in practice, and they’ve provided the math they say proves it. The burden stems from the two foundations of password security that (A1) passwords should be random and strong and (A2) passwords shouldn’t be reused across multiple accounts. Those principles are sound when protecting a handful of accounts, particularly those such as bank accounts, where the value of the assets being protected is considered extremely high. Where things break down is when the dictates are applied across a large body of passwords that protect multiple accounts, some of which store extremely low-value data, such as the ability to post comments on a single website.

Very detailed, very understandable too: Still, I’m not gonna change my own policy of “every login has its own password”. Granted, I cheat as well on some points, but I’d find it very worrisome if I used the same password across many logins. For me, snoopy2 is not gonna make it. Sorry, Snoopy.

]]>**When I was a little kid, somebody, probably my father, drew a picture like that and challenged me to draw the same thing without lifting my pencil from the paper. That seemed like a pretty easy thing to do. It wasn’t.**

**Many years passed before my first and last attempt. Not that I spent days and nights continuously working on this, but there were a lot of classes and then some meetings to sit through. And I guess I was competitive, stubborn and/or obsessive.**

**But one thing that kept me going off and on through the years was the belief that I had successfully met the challenge once, couldn’t remember how I did it and should be able to do it again (too bad I didn’t keep notes).**

**Of course, I eventually concluded that this was a false memory. The thing cannot be done!**

**What brought all this back to me was an article at Three Quarks Daily called “A Square Peg for Every Round Hole”. It’s about mathematical puzzles, the most famous being Fermat’s Last Theorem (“I have discovered a truly marvelous proof of this, which this margin is too narrow to contain”). In particular, it’s about **

another enticing mathematical morsel which is still unsolved: the Square Peg Problem (SPP). The history is a bit murky, but it is generally credited to Otto Toeplitz in 1911. The SPP is the conjecture that if you draw a curve on a sheet of paper without picking up your pencil and which begins and ends at the same place, then you can find four points on the curve which form the corners of a square.

**For example, I drew this wavy curve in black and was then able to overlay a square with its four corners intersecting the curve. **

**(Ok, I cheated and put in the square first and then drew the curve. The other way may be mathematically possible in every case (or not) but it’s not that easy and my little obsession lies elsewhere.)**

**Maybe mathematicians proved long ago that it’s impossible to draw a picture like the one at the top of this post. Maybe there’s even a name for this particular “mathematical morsel”: the Square With Lines Around It and a Cross in the Middle Problem (SWLAIAACITMP). **

**On the other hand, if you know a way to draw the damn thing without lifting your pencil or pen from the paper or your index finger from the mouse – or know why it can’t be done – please let me know.**

*Update:* That didn’t take long. A person going by the name of “X” gave the answer in the comments at *Three Quarks Daily *after I described the SWLAIAACITMP problem:

This is a cool problem called “Euler Paths”. You can prove it’s impossible for this graph because there are too many vertices with an odd number of edges coming out of them. So there will always come a time when you go into a vertex and can’t get out. This page has the rules: Euler’s Graph Theorems.

**Thank you, X, whoever you are.**

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It’s even got a search engine, so that if you have the start of a sequence — say, “1, 4, 5, 16, 17, 20, 21” — it can find whether there’s any noteworthy sequences which begin that way and even give you a formula for finding successive terms, programming code for the terms, places in the literature where it might have appeared, and other neat little bits.

This isn’t foolproof, of course. Deductive logic will tell you that just because you know the first (say) ten terms in a sequence you don’t *actually* know what the eleventh will be. There are literally infinitely many possible successors. However, we’re not looking for deductive inevitability with this sort of search engine. We’re supposing that our sequence starts off describing some pattern that can be described by some rule that looks simple and attractive to human eyes. (So maybe my example doesn’t quite qualify, though their name for it makes it sound pretty nice.) There’s bits of whimsy (see the first link I posted), and chances to discover stuff I never heard of before (eg, the Wilson Primes: the encyclopedia says it’s believed there are infinitely many of them, but only three are known — 5, 13, and 563, with the next term unknown but certainly larger than 20,000,000,000,000), and plenty of stuff about poker and calendars.

Anyway, it’s got that appeal of a good reference tome in that you can just wander around it all afternoon and keep finding stuff that makes you say “huh”. (There’s a thing called Canada Perfect Numbers, but there are only four of them.)

*On the title: some may protest, correctly, that a sequence and a series are very different things. They are correct: mathematically, a sequence is just a string of numbers, while a series is the sum of the terms in a sequence, and so is a single number. It doesn’t matter. Titles obey a logic of their own. *

If not, then, let me fill out this post by bringing up Donna A Lewis’s Reply All, surely one of the most alleged comic strips to be in print today. And it is literally in print, like, running in newspapers. Comic strips have always valued writing over artwork — a comic strip with funny final panels will go farther than a bland strip with great artwork, however odd that might seem for a visual medium like comics — but here, well. I can *see* the humor, but it’s up against such a wall of bad artwork I just do not understand how this is more than just the comic strip that runs in the daily student newspaper because unspeakably terrible things get to run in the daily student newspaper. And Lewis gets two comics, with the panel-strip version Reply All Lite, which I just do not understand at *all.*

I don’t want to be cranky, and I don’t want to sound like I’m calling for Lewis’s head or anything. I have no reason to think she’s anything but a pleasant person with good friends and a useful day job (the only cartoonist who can afford to live just on the comic strip is Charles Schulz) and appealing hobbies and all. I just offer this for you to gape at and not understand.

]]>Freshman year, I was enrolled in the Liberal Studies Introductory Colloquium titled “Weather & Society” taught by Dr. Christopher Godfrey. At the time, this course opened my eyes to the world of emergency management and other opportunities for atmospheric scientists. In my pursuit of a meteorology degree, I became passionate with the impacts of weather and disasters, more than the actual science of weather or natural disasters. I can now say that Dr. Godfrey’s class was the ignition of my passion; that freshman course has profoundly changed my career, future aspirations, and consequently, my life.

This newfound passion directed me towards 4 wonderful internships that I pursued over a 2 year period with the state and local governments. The leadership and guidance I received from the many emergency managers I encountered during that time has contributed to my change in career path. After graduation, I applied for a variety of jobs, including: emergency management coordinator, natural hazards planner, continuity planner, statistician, hydrologic technician, among many others.

I accepted a position at East Carolina University in Greenville, North Carolina, in June 2013. I was hired as the Emergency and Continuity Planner within the Office of Environmental Health and Safety. Here, I have many responsibilities, including: emergency notification, hazardous materials response, business continuity planning, crisis communication, emergency operations planning, as well as disaster planning and preparedness. I still utilize my meteorology degree, almost every day, in making decisions for the safety of our students, employees, patients, and visitors.

Within the world of emergency management, I have learned more about my majors (especially atmospheric sciences) and the connection between the disciplines. Meteorology impacts everyone’s lives; weather may impact how you dress, what vehicle you drive, outdoor events, and even your mood. My position looks at the impacts of weather on people and a society, and determines ways to mitigate the effects, prepare for the disruption, respond to a disaster, and recover from crises.

While I get to utilize my atmospheric sciences degree, emergency management deals with other types of hazards, including man-made and technological disasters. This is where my liberal arts degree comes into the picture. While I sometimes grumbled about going to the humanities and general education courses, these disciples taught me patience, humility, compassion, and creative/critical thinking. Those attributes have proven to given me the peace of mind and confidence I need to make decisions that may affect thousands of lives on campus.

I would have never imagined 6 years ago, when I graduated high school, that I would be where I am today. It is with hard work, dedication, and guidance that I was able to make it this far. At this time I would like to present you with a few things that I learned during this experience:

- Talk with your advisor on a regular basis, not just when you need to register for classes. This person is knowledgeable and has many contacts that can help you find the right graduate school or job. Don’t like your advisor? Choose someone else within the department as your advisor, or consult with someone at the career center.
- Join student organizations and honor societies on campus, especially those related to your major. Not only do these look great on your resume, but they can be a ton of fun and you could meet your lifetime best friend. Make sure you go to the special events and activities. Take advantage of the freebies!
- Start building your resume your freshman year, and edit it after each semester. If you do not have a job, you should be listing extracurricular activities and related course work. Visit the career center; those folks are amazing at fixing resumes and assisting with letters of intent.
- Get a job, internship, or complete an undergraduate research project that interests you. Whether it’s working for Campus Recreation, interning with a local company downtown, or doing research out in the field (or in the mountains rather), getting that hands-on experience will give you invaluable intel to your future wanderings.
- Step out of your comfort zone! Go to a conference within your discipline, and present your research poster or talk with other undergraduate/graduate students who are presenting their posters. I cannot say this enough: network, network, network! Through networking you will meet new friends, find your future employer, or interest you in a new career path.
- Go exploring Asheville and the surrounding area. Walk downtown and enjoy the music, art, and restaurants. Go drive the Blue Ridge Parkway on a beautiful fall afternoon. Take your friends and go camping (Campus Recreation rents out the equipment!). Rent a bike and ride around the city. Asheville is a beautiful place, so make beautiful memories!

Good luck in your future endeavors, bulldogs!

]]>All you have to do is fit the pieces into a square shape. You can turn any of them over if you want to.

Why is it called a stomach ache? Well, there are 17,152 different ways to solve this puzzle and, as you fiddle about realising you cannot even figure out ONE of them…. trust me, it’s sickening!

Last weekend we went to Siracusa, birthplace of Archimedes, and tried our hands at solving this puzzle at the Tecnoparco Archimede. We were guided about the open air museum by a pair of charming and highly knowledgeable old fellows, who let us play with the catapults, set tissues on fire using the burning mirrors, and tell the time using the water clock. I shall be posting more of those fun things later, but let’s get back to this stomach ache, shall we?

Print this square out. Before cutting out the pieces, cut another piece of paper the same size to use as your base… then see how many ways you can fit all these shapes into the square.

If you can figure out any, that is.

The tecnoparco guides began by presenting my son with a wooden stomachion, which he managed to solve in three different ways before we continued our tour. Later in our hotel, Hubby and I fiddled with it for ages before coming up with a single solution.

How many can you manage?

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**The Singapore Philosophy**

*Earlier this term, I attended a course on Singapore Maths. I hope my summary is a fair interpretation of the facts and that my comments are taken as my own – in no way representative of those of the speaker.*

Singapore has not always been a world leader in Maths education. In fact, in 1983 it achieved mid-table mediocrity in a SIS study of 26 countries. They decided it was time to act. Maths teaching needed a philosophy- it was time to go back to basics. The philosophy that was decided upon is known as** CPA** – *Concrete, Pictorial, Abstract*. If this reminds you of Bruner’s *Enactive, Iconic, Symbolic*, then you would be spot on – Bruner’s research provides the foundations for Singapore Maths. Another guiding principle is that problem solving should be at the heart of each lesson. If this sounds like something from the Cockcroft Report, well you would be right, because it is. Teachers in Singapore would laugh at the thought of a classroom of English teachers learning about *Singapore* Maths. The underlying philosophy comes from Hull University (Cockcroft) via the USA (Bruner).

So, here is a philosophy that any Maths PGCE student would be able to drop into an essay. Apart from something called the bar method, I didn’t see anything new during the training day.

Could it be that simple? Implement CPA and watch your results improve?

This seems a bit like telling the England team that the ability to pass the ball is integral to World Cup success. Oh, hang on…

Well, we can only judge a team by its results… [slides courtesy of http://www.mathsnoproblem.co.uk/]

Ok, this particular team seems to be doing quite well. If the philosophy is nothing earth-shattering, which other factors are at play?

**The Singaporean Context**

Singapore Maths does not rely on individual teacher knowledge and expertise as much as the UK context. This is seen as far too risky and liable to produce variable results. There is a greater reliance on centrally vetted textbooks and a uniformity of approach to teaching new concepts. Class sizes are large (40 – 50 not unusual), so ready-made, tried and tested resources are provided for all Maths teachers. Here is the Head of Maths CPD in Singapore, Dr Yeap Ban Har giving an overview:

There are less schools and teachers in Singapore – this means that centralised coaching and co-ordination of approach is possible. The key ingredients seem to be:

- rich problems

- quality textbooks built around CPA

- good CPD backed up with Lesson Study in schools (more time is allocated to teacher development)

**Cultural Differences **

It is widely accepted that cultural differences account for much of China’s progress in school mathematics – it is not unusual for private tuition to rank second only to food on household budgets in Shanghai. As Sean Harford writes here,

*“My diagnosis of the difference in our systems is that mathematics in the Chinese curriculum is not seen as an elite subject. It is viewed as an essential of life, and one in which everyone can be highly competent if you work at it. ‘Maths gets you everywhere’ is a common phrase used in China and far from turning pupils off the subject, this focus and respect for it gives their pupils confidence and purpose.”*

Malcolm Gladwell also writes persuasively on how inherent cultural differences may have an impact on Maths learning here,

In a similar way, students’ attitude to learning seems to be healthier in Singapore. As Liz Truss put it, “Diligence redeems lack of ability”.

There is also the question of *time*.

As a fellow delegate asked, “This is all great, but we would get hammered by OFSTED for giving the same task to a large group of kids and then waiting until they ‘got it’ in their own time. Where is the evidence of progress for each individual in the classroom?”

Giving students *time* to work out problems for themselves and strive for mastery is a common theme in Singapore Maths. The discrepancy in classroom pace is balanced with increased teaching time across the academic year. This is something the recently departed Gove-Truss axis spotted and wrote about here,

“…the amount of time spent teaching maths in England is also low – we are 39th out of 42 countries, with 116 hours a year spent teaching maths at age 14. This compares with 166 hours a year spent teaching maths in Chinese Taipei, 138 hours in Singapore, 138 hours in Hong Kong and 137 hours in South Korea – some of the highest performing education jurisdictions (TIMSS 2011).”

This is before homework and private tuition.

**An off -the-shelf solution?**

All of this seems to suggest that the success of Singapore Maths is down to much more than the philosophy of CPA. Truss et al may want to take it off the shelf and import it to our shores, but there is a fundamental problem with this: *we know the philosophy already* – some would say we invented it. If the difference does not lie in the philosophy, I would argue that it must lie in the Culture and Context – attitudes to learning, time devoted to learning, teacher training, lesson study. If we want to perform on the global stage this is a long term revolution, and we will have to do it *our* way – with *our* students in mind.

My mind wanders…

*As England are dumped out of another major footballing tournament, the inevitable soul-searching begins:*

*Why are we not able to compete with the big boys? **How is the rest of the world beating us at our own game? We taught them how to play it! *

*Why can’t we produce world class talent through our youth system? **What is our philosophy anyway?*

*Let’s look at what the big boys do – the Germans and Spanish built footballing dynasties on their philosophies. They nailed their colours to the mast and allowed time for their systems to produce results…*

*Yes, but if we prioritise international football, we sacrifice the domestic priority – our league tables…*

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Science:-

Complete the Exercise of the Textbook.

Note: Bring chocolate chip cookie, scissors, U pin, paper plate and glue stick for the activity.

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It is important to have **good problem solving skills** as we face problems in our daily lives that we need to solve.

*Click on the image below to complete a puzzle.*

In a **comment** below, **explain** how you could **solve the problem** and * how* you reached your

**Problem #1**

A large box contains 18 small boxes and each small box contains 25 chocolate bars. How many chocolate bars are in the large box?

]]>When people have something to say, they manage to find a way to communicate.

Today’s American teenagers are writing. Our textbook says, “Look at what today’s adolescents are doing on their own! They are doing more *authoring* than any young people in the history of the world.” (Daniels et al., 2007, p.3) And *that* was written *before* Facebook, Twitter, Tumbler, and Reddit; before smartphones. Now, everyone is writing. All the time.

The challenge is in getting the students to stop writing long enough to do their schoolwork.

There always have been and always will be articles and editorials saying that our schools are failing in some capacity and students aren’t accomplishing what somebody thinks they should. But seeing room for improvement is not the same as a crisis.

Standardized tests measure our students’ ability to produce writing on demand, on a topic in which they have no interest. Writing when you don’t have something to say is a hard skill to master. More important is making sure that our students have something worth saying.

I love several of our textbook’s short writing-to-learn strategies. Taking a two or three minute writing break in the middle of math class for students to respond to a short prompt looks interesting.

Using brief writing as an exit ticket from math class is especially appealing to me. I can imagine asking the students to assess their understanding, or to describe what they learned today, with full points for participation. After a math class on how to combine exponents, an exit ticket asking students to write in their own words what is an exponent may make someone stop and think. That would be a hard question to answer, but simply getting the students thinking about the question would be a good place to end a class.

An exit ticket asking students about the next day’s content could also be interesting. For example, asking “What is a negative number?” on an exit slip would allow the teacher to identify misconceptions, but it may also awaken some memories and prime the students anticipation for the next day’s lesson.

Daniels, H., Zemelman, S., Steineke, N. (2007). *Content-Area Writing: Every Teacher’s Guide*. Portsmouth,NH: Heinemann.

By Jason Padgett and Maureen Seaberg

**Synopsis: **The remarkable story of an ordinary man who was transformed when a traumatic injury left him with an extraordinary gift.

No one sees the world as Jason Padgett does. Water pours from the faucet in crystalline patterns, numbers call to mind distinct geometric shapes, and intricate fractal patterns emerge from the movement of tree branches, revealing the intrinsic mathematical designs hidden in the objects around us.

Yet Padgett wasn’t born this way. Twelve years ago, he had never made it past pre-algebra. But a violent mugging forever altered the way his brain works, giving him unique gifts. His ability to understand math and physics skyrocketed, and he developed the astonishing ability to draw the complex geometric shapes he saw everywhere. His stunning, mathematically precise artwork illustrates his intuitive understanding of complex mathematics.

The first documented case of acquired savant syndrome with mathematical synesthesia, Padgett is a medical marvel. *Struck by Genius* recounts how he overcame huge setbacks and embraced his new mind. Along the way he fell in love, found joy in numbers, and spent plenty of time having his head examined. Like *Born on a Blue Day* and *My Stroke of Insight*, his singular story reveals the wondrous potential of the human brain.

**Published: **April 2014 | **ISBN-13:** 978-0544045606

Book’s Homepage: http://www.struckbygenius.com

Book’s Facebook Page: https://www.facebook.com/StruckbyGenius

Jason’s Twitter: https://twitter.com/Jasonquantum1

Maureen’s Twitter: https://twitter.com/SynesthesiaGal

Huffington Post Article by Author

Kirkus Reviews Book Review

Publishers Weekly Book Review

The Star Book Review

NY Journal of Books

[Image Credit: http://www.kurzweilai.net/images/9780755364589.jpg ]

]]>Dr. James McLurkin has a swarm of robots. Individually, theyre not that smart, but a crateful of them behaves in some very complex ways, like the bees that inspired them. Gizmodo got to see the wee machines in action, and while theyre adorable, they represent some serious future bot capabilities.

Dr. McLurkin, a professor of computer science, runs the Multi-Robot Systems Lab at Rice University. He and his team research distributed algorithms for multi-robot systems. In other words, using the combined abilities of several rather simple robots to perform complex tasks. Dr. McLurkin has spent the past three years developing Robot Swarm, an exhibit of his hive-mind bots set to debut at Manhattans Museum of Mathematics in early 2015. This week, Dr. McLurkin gave a sneak preview of the exhibit, and Gizmodo was there.

READ MORE Bee-Inspired Bots Skitter and Swarm at NYCs Museum of Mathematics | Gizmodo

]]>I’m betraying my age, but when I was in high school, pi was 22/7 or 3.1416, and the answers to the geometry problems were to be rounded off to two decimal places. We had our slide rules, but most of the problems had to be done “the long way”. Calculators were not allowed to be used on tests or exams. I can think of two good reasons: most of us students couldn’t afford the expensive calculators, and the teacher wanted to make sure that we knew how to “do the math”.

A lot of new technology has been invented since my high school days, but circles will always be an important part of mathematics and its application to our lives. God must like circles because He sure made a lot of them! Psalm 19:1 in the Bible says: “The heavens are telling of the glory of God …”. I invite you to go to a nearby park or even take a close look at your own backyard (if you have one). I’ll bet you run out of time or give up before you count everything that is circular or spherical in its shape. Look inside your house and you will find man-made circular objects galore. Why? Symmetry, beauty, efficiency, and order, to name a few reasons.

I believe I’m correct when I say that pi is a real, irrational, transcendental, infinite, constant, prime number. It’s the only one of its kind. Correct me if I’m wrong.

National Geographic did an article with pictures of “almost” perfect circles in nature and in the universe. Such examples as: the rings of Saturn, the pupil of the human eye, the arc of a rainbow, tissues in the cross-section of a plant stem, a ripple, “fairy circles” in the desert grasslands. I believe they show eight pictures and descriptions in all. Type “almost perfect circles in nature” in your web browser and it will direct you to the site. You’ll be amazed!

A question that is often asked: Is Pi in the Bible? Yes it is! In the Old Testament, in First Kings chapter 7, and verses 23-26, where God gave to king Solomon the instructions for constructing the sea of cast metal. If we divide the circumference of the cast metal sea (“thirty cubits”) by its diameter (“ten cubits from brim to brim”), pi is 3. It may be pi to the “zeroith” decimal place, but it’s still pi. I’m not a mathematician, but Roy A. Reinhold, in his website, ad2004.com/prophecytruths/articles/mathmysterys.html gives a picture and description of this huge sea of water, and does some mathematical calculations that result in a more accurate approximation of pi. Please check it out. It’s a short and easy-to-read article. If you type “biblical math mystery solution for pi” into your web browser, it will take you directly to that site.

By the way, isn’t pi always an approximation? You can calculate pi to a million decimal places, but you still haven’t come to the exact number. There is more calculation to be done, isn’t there? Frustrating, but amazing!

I hope this short message has given you an opportunity to think about the myriad of applications of pi in nature, how these shapes came into being, and the One who put it all together. I hope it has also caused you to think for a moment about infinity/eternity.

Happy Pi Approximation Day to you this 7/22/14, and may all your approximations be sufficient for their applications!

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For some background, in 1895, Lewis Carroll wrote this paradox in the philosophical journal, *Mind*. Carroll is so wonderfully wacky, and I love the intersection of math and logic that he worked on.

Anyway, the main idea is that there are two statements which seem to have a logical implication. I believe the implication is logical, and you probably would too. Achilles definitely thinks so, but the Tortoise doesn’t seem to agree. Achilles compromises by letting the Tortoise come up with his own hypothetical addition to the statement, which make sense to me.

The main argument is that if

A) Things that are equal to the same are equal to each other, and

B) Two sides of a triangle are equal to the same, then

Z) The two sides of the triangle are equal to each other.

The Tortoise accepts statements A and B, but does not agree that statement Z is implied by A and B. So the Tortoise comes up with another hypothetical proposition, C, which states

C) If A and B are true, then Z must be true.

At this, Achilles demands that if the Tortoise accept A, B, and C to be true, it must accept Z as well! But alas, the Tortoise jumps at turning this into a new hypothetical, D, which states

D) If A, B, and C are true, Z must be true.

The idea is that this could go on forever, or let’s say n times, such that the statements are as follows-

1) Things that are equal to the same are equal to each other, and

2) Two sides of a triangle are equal to the same, then

3) 1 and 2 ⇒ Z.

4) 1, 2, and 3 ⇒ Z.

…

n) 1, 2, …, (n-2), and (n-1) ⇒ Z.

Z) The two sides of the triangle are equal to each other.

Every time, the Tortoise argues that even though he accepts premises 3 through n, there is some further premise (that if all of 1 through n are true, then Z must be true) that it still needs to accept before it is compelled to accept that Z is true.

Does your head hurt yet? Mine does, but in the exact right way! I truly enjoy reading these and thinking them through.

I’m out!

Ingi

We can think of an arrangement or *permutation *of as a way of placing each number into one of distinct slots. Given any permutation we can write for the contents of slot and we can write any permutation as a map .

Since each slot must be filled and no two slots can be filled by the same number the map is in fact a bijection. Now consider another permutation where each is mapped to . We say that if and only if for all . This clears up the problem of what it means for two arrangements to be distinct. This makes (some) sense since there are different bijections from to itself

Since any two permutations are bijections we can compose them in the usual way. For example let and .

A particularly important permutation is the identity permutation . The identity permutation simply sends each number to itself. One can easily check that for any other permutation the identity permutation really does behave like the identity: .

In fact the set, , of all bijections from to itself with the binary operation composition of maps is a group.

There is yet another way to think of permutations. Given some permutation we put down labelled points and draw an arrow from each to to build what is called a directed graph. Since is a bijection each point has precisely one arrow coming in and one arrow leaving it. For example the following graphs correspond to and above.

Each connected component of the directed graph associated with a permutation is called a *cycle *and every permutation is made up of disjoint cycles that partition . Furthermore, there exists another notation for a permutation where we write each cycle in brackets. For example we can write and ; this is called disjoint cycle notation.

Now that we have set up arrangements of more carefully we can ask more difficult questions. For example, let be the number of permutations such that . Question: Calculate .

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(N.B.: No, you cannot invoke AAT to conclude that anyone who disagrees with you is ignorant or unreasonable, even though it is a very tempting prospect to lend an air of irrefutability to your self-aggrandising claims using mathematical rigor…)

We could say that, given how precisely the terms are being defined in AAT, there really is no other way the result could have turned out — if all the exacting epistemic conditions are met, then it becomes a matter of course that the agents involved would have to agree. So we might end up with the impression that, since AAT seldom applies in real life, it is therefore rather useless. But we could also think that AAT is nonetheless important, for it reminds us of just how extraordinarily difficult it is to actually achieve full agreement on non-trivial issues — after all, any slight deviation from the definitions used in proving AAT would render it invalid. (With this consideration in mind, we may better come to appreciate Paul Graham’s sentiment that, if you do not disagree on any important issue with the other members of the society in which you live, you are most probably not doing enough independent research and thinking — for if you did, chances are very high that you would depart from the mainstream on at least one issue.)

I think AAT could have done an even greater service if it had a secondary part, which explicitly points out that agreement may be achieved even in the case of starting out with different prior probabilities and then progressing with different sets of information or assumptions. I have come across quite a number of reasonably intelligent people who think that the lack of dissent with their opinions is a reliable sign of an above-average intellect. They may articulate such a view directly, or insinuate it indirectly — e.g., through preferential treatment towards people who agree, perhaps by paying more attention to them when they speak, or by soliciting their feedback more regularly. Of course, such a mentality is not unique to intelligent people — it would probably not be an exaggeration to say that nearly everyone falls victim to it, though one would hope that intelligent people would be more capable of avoiding it.

To belabour the message that it is possible for people to agree with you for completely misguided reasons, let me offer a contrived example. Suppose you live in a world where most adults are so mathematically challenged that they either do not realise or vehemently disagree that x^{3} > 0 if x > 0. (If you do happen to be already living in such a world, then I offer my deepest sympathy.) Being a keen student of mathematics, for many years you have had to live in abject intellectual loneliness. However, one day, you fortuitously come across a rare person who is informed of the fact that x^{3} > 0 if x > 0 — you are delighted, for you have finally found someone with whom you can discuss mathematics! In tremendous joy, you instantly regard him as an intellectual peer; but, unbeknownst to you, the line of reasoning which leads him to conclude that x^{3} > 0 if x > 0 follows thus:

Premise 1:If x > 0, then x^{2}> 0.

Premise 2:If x^{2}> 0, then x^{3}> 0.

Conclusion:If x > 0, then x^{3}> 0.

Clearly, Premise 2 is utterly wrong. However, when coupled with Premise 1 (which is correct), it nevertheless produces the correct conclusion that if x > 0, then x^{3} > 0. The premises in absentia, listening to this conclusion leads you to mistakenly believe that your new friend is mathematically competent.

I described this toy scenario in order to highlight that, even in discussions of issues free of connotations, interpretations and vagueness – such as elementary mathematics — it is still theoretically possible for someone to arrive at an uncontroversially true conclusion in spite of employing faulty premises. In the case of elementary mathematics, however, once such erroneous premises are revealed, they can immediately be identified and corrected, and you may very quickly be disabused of your flawed impression that the other person is competent or knowledgeable. On the other hand, in many other issues that are far more accommodating of imprecision and subjectivity, even when dubious premises are uncovered, they may still fail to raise red flags in your mind – for it is far easier for someone to rationalise using them.

In summary, it is very unusual to achieve full agreement on a non-trivial and non-scientific issue. It is, however, probably not that unusual for someone to arrive at the same conclusion as you do due to luck. These are the facts of which I constantly remind myself, so that I would remember to actively seek out reasonable people who disagree with me.

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