It’s called ‘*The Great Bridge Building Contest’* & it’s about the West Virginian carpenter Lemuel Chenowith.

But first, breakfast:

__Lemon & Parmesan Kale with Mini Roasted Bell Peppers__

Oven temp: 450

2-3 stalks of kale

4 mini sweet multi-colored bell peppers, cut into strips

McCormicks Lemon & Pepper seasoning

Parmesan Cheese (I used the kind you shake from a jar)

Garlic Salt

Olive Oil

Pre-heat the oven to 400 degrees & cut up your mini bell peppers.

Place them on a baking sheet with foil & drizzle olive oil with a pinch or two of garlic salt. Make sure the peppers are evenly coated.

Bake for about 25 minutes…They should look like the picture above & be a little browned up.

Put the peppers aside & use the baking sheet with the same foil again for the kale.

Wash & dry the kale. Tear off the leaves from the stalk into bite size pieces. Spread as evenly as possible over the baking sheet. (I learned from Vivian Howard on *A Chef’s Life* not to let veggies overlap too much in the oven!)

Drizzle olive oil over the kale then shake on some lemon-pepper seasoning & parmesan.

Bake the kale for about 8 minutes…Any longer & it wil turn into kale chips. You don’t want that, however there will probably be a few crunchy pieces & that’s not a bad thing at all.

The only thing missing from this kale & bell pepper breakfast bowl is a thick, toasted slice of that amazing spelt bread I made not to long ago!

So just put the kale in a bowl & put the peppers on top. Simple. Healthy. *REALLY* good.

I ate my breakfast outside & finally got a chance to look through the books we found…So funny how these books tie in with my life but that’s a whole different conversation. Maybe over some dinner..*.*?

It’s about a bridge building contest & although the architects bridge designs were grand…Lemuel, who wasn’t so educated, built a model he could prove would stand the test of time.

“Suddenly, using the ring of he chair, he stepped onto the top of the model, and walked across it- from one end to the other. A gasp went through the audience. No way could it hold! Each one of them knew their mathematics. Had this been the actual bridge it would have been as if a six-hundred-foot man stood on it.

But the model held.”

This book is really cool & the illustrations are cute. I love how the picture above shows the men building the bridge on wooden planks.

If you believe you might have a budding little engineer or architect on your hands, I recommend this book for him or *her…*

The last three pages show illustrations of all Lemuel’s covered bridges & one of them is in Missouri (where my grandmother was from)!

Hope as an adult you were able to enjoy this children’s book with me & if you have kids, I beg you to share it with them too.

Enjoy the recipe…I *know* you will!

-HH

]]>It was rumored that the man was one of the greatest mathematicians of the 20th century, but that his best days were behind him because of a mental illness. Many in the outside world were using his work published in the 1950s and applying it to everything from trade negotiations to evolutionary biology, but they did not know whether he was dead or alive. Only a f

The earliest hint of Nash’s genius was spotted by his mum, when, ironically, he received a “B minus” in his fourth-grade arithmetic class. The teacher said Nash, whose immaturity and social awkwardness had defined his time in school, couldn’t do the work well enough. But his mum knew that Nash had found his own, perhaps better, way to solve problems.

That search for original thinking only intensified as he grew up. He cared for achievements but set his own standards for them. He never made it to his high school’s honor roll, but managed to secure a full-scholarship to attend Carnegie Institute of Technology (now Carnegie Mellon University). As an undergraduate student, mechanical engineering classes dulled him, and so did chemistry experiments. But eventually he found his intellectual home in mathematics, where in an accelerated program he graduated with a master’s.

He was accepted to Princeton’s doctoral program with a one-sentence recommendation letter from professor Richard Duffin: “This man is a genius.” And he knew that. He went about his daily life whistling Bach’s Little Fugue, filling up blackboards with equations, and acquiring adjectives such as “haughty,” “detached,” “spooky,” “isolated,” and “queer.”

He feared no one and never minced his words. Nothing mattered to him more than the pursuit of the biggest problem he could lay his hands on. When he had something to show he didn’t hide it either. He accosted Albert Einstein and John von Neumann, both intellectual giants that walked Princeton’s campus then, with ideas he thought were original, but wasn’t let down by their reactions.

He graduated at the age of 22 with a doctorate after writing a 27-page dissertation, that built on von Neumann’s work, despite his supervisor not being sure if it “was of any interest to economists”. In fact, when he formally published the result in a journal, he shortened it further to 317 words, which probably is the shortest paper to have ever won its author a Nobel Prize.

He had made a great contribution to the field of game theory, creating what is known today as the Nash Equilibrium, which provides a compelling way to analyze situations where two or more players are competing. An equilibrium occurs when, according to Nash, no player would change their own strategy knowing fully well what others’ strategy is.

But Nash didn’t know the value of his work then. Instead, he soon began working on other problems that were considered more important by the mathematical community at the time. Even today, it is joked that Nash won the Nobel Prize in economics for what he thought was his most trivial work.

Things seemed to have been going very well for Nash. He married Alicia and fathered a child. He found a job at the Massachusetts Institute of Technology. His work brought him some fame, including being featured on the cover of Fortune magazine. He seemed to be on the path to a great career. But underneath it all, as Sylvia Nasar, author of *A Beautiful Mind, *which later became an Oscar-winning movie and made Nash a celebrity, puts it:

…was chaos and contradiction: his involvements with other men; a secret mistress and a neglected illegitimate son; a deep ambivalence toward the wife who adored him, the university that nurtured him, even his country; and, increasingly, a haunting fear of failure. And the chaos eventually welled up, spilled over, and swept away the fragile edifice of his carefully constructed life…

He would not have known this but, by the age of 30, Nash had finished his life’s most important works. While not unusual for mathematicians to run out of grand ideas at a young age—the mathematicians Srinivasa Ramanujan and Bernhard Riemann were in their 20s when they did their greatest work—Nash’s next few decades would be torturous.

Rationality is a scientist’s tool to unravel the mysteries of the world we live in. But a complete conviction that the universe is a rational place—that everything has meaning—caused Nash to lose 25 years of his life. In 1959 he was diagnosed with paranoid schizophrenia. For many days before then, he had begun codebreaking what he said the voices in his head had told him were secret messages in the pages of Newsweek.

Harvard professor George Mackey would ask of Nash, “How could you, a mathematician, a man devoted to reason and logical proof… believe that extraterrestrials are sending you messages? How could you believe that you are being recruited by aliens from outer space to save the world?”

The world loves a mad scientist, and, in John Nash, it got one. But most of us would not have known his story if it wasn’t for, what Nasar calls, Nash’s third act: brilliance, madness, and reawakening.

Even today, schizophrenia remains poorly understood. And recovery from this “cancer of the mind” is still rare. The support of Alicia and his old friends at Princeton were key to keep Nash going. But it was the enchantment of solving bigger mathematical problems that eventually helped him fight the imagined people in his head. “I emerged from irrational thinking,” he said in 1996, “ultimately, without medicine other than the natural hormonal changes of aging.”

Without this reawakening, Nash would likely not have won the Nobel Prize, which is handed out by a committee that zealously guards the image of the prize. But what probably mattered more to Nash was that his fight against schizophrenia gave him an opportunity to get back to his work, and hope that his son John may successfully fight the disease from which he, too, has been suffering from a young age.

*John Forbes Nash Jr., born on June 13, 1928, was killed along with his wife, Alicia, on May 23, 2015 in a car accident in New Jersey.*

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]]>(2009) *Terengganu – TOV – *Paper 1 & Ans, Paper 2 & Ans

(2011) *SBP – *Paper 1 & Ans, Paper 2 & Ans

(2012) *Terengganu – TOV – * Paper 1 & Ans, Paper 2 & Ans

]]>

Today he’s decided that maths and science should be compulsory for all grade 11 and 12 students.

It seems like a good idea in theory, doesn’t it? Improving the numerical literacy of the country.

But let’s come crashing back to reality for a moment…

Schools are already having to dumb down their subjects (or if we’re being politically correct, it’s *curriculum creep*) in order to teach to the lowest common denominator. Can’t have students actually *failing *anything of course, because that would be bad for their self esteem! So the curriculum gets simplified so that pretty much everyone can pass.

*This is already happening now.*

Australia is progressively getting worse in terms of their educational standards on a global scale.

So imagine that all those students who voluntarily opted out of grade 11 and 12 maths and science suddenly get forced into subjects that they’re either ill prepared for, or they adamantly don’t want to be in.

Imagine for a moment just how much worse that *curriculum creep* is going to get in order to account for students who shouldn’t be in those classes in the first place.

This policy idea is totally ill advised.

It’s a nice idea in theory, but for an education minister, Christopher Pyne seems to have little idea about how education actually works in reality.

]]>**Price**: AUD $35.95

Like masterpieces of art, music, and literature, great mathematical theorems are creative milestones, works of genius destined to last forever. Now William Dunham gives them the attention they deserve.

]]>She mentions love is like a flow from tap.. each drop can add to the volume of love.

Integration is all about area calculations…

Her friend Susan is a teacher who teaches history and geography..

She looks at Zora who is addressing her class with students listening to Zora with good attention…

Susan steps in…

Zora ‘hey class is going on’

Susan ‘Just to add to your bit’

Susan ‘Do you know when the boundaries were getting defined in this world, no one knew the influence of human, animals, plants and environment which would impact the geographical location and hence areas have larger influence of human from same society, caste, color, creed got demarcated. Differentiation came in’

Zora ‘and now our whole attempt is to integrated those slices’

Student ‘It means integration is reverse of differentiation?’

Zora and Susan ‘Absolutely..’

Student ‘Why differentiation needed to be taught?’

Susan ‘because demarcations happen slowly and surely.. the hate doses in human is in slow form. No one hates anyone right from day one.. So as love gets collected from tap.. small doses of hate comes in because of society, caste, color, creed and volume of love is soon differentiated, boundaries are defined’

Student ‘It is too much philosophy behind this Maths’

Zora ‘hence mathematics is important for human’

Student ‘Divide we fall, United we win’

after the class, Zora’s husband Rafi who is an entrepreneur comes to school

Both Zora and Susan discussing Maths

Rafi ‘how does calculus impact an entrepreneur?’

Zora ‘make your product in iterative manner.. that is integration.. small slices added to make whole piece’

Susan ‘differentiate your products and group homogeneous feature list in one group.. creating such groups are nothing but creating multiple modules.. So you first create a story of product, break them into homogeneous groups and then implement each group either phase wise or together’

Rafi is zapped

Rafi ‘dear both, I feel like a novice. you folks expert’

Susan ‘Mathematics is used everywhere.. hence we love maths’

]]>Back in the fall of 2006, I took transport phenomena at my undergraduate university in Florida. I had little idea of what the course was about; people had asked me before if it was a course on automobile traffic flow or supply chain management. However, upon skimming the course text and listening to the professor’s first lecture, it became apparent that the course had nothing to do with daily commutes and everything to do with the mathematical modeling of fluids and the transfer of heat. While most of my classmates could not wait for that semester to be over, I remember that course fondly and with appreciation. The mathematics encountered in that course were far above what I expected to encounter in chemical engineering, but that which does not kill us makes us stronger.

To boil down our motivation, fluids are commonly encountered in mass industrial production, and often these fluids exhibit peculiar properties that make them difficult to process. This motivates the discovery of a general theory for predicting their behavior under general circumstances. There are other situations where the prediction of fluid flows is of direct practical importance, such as the modeling and prediction of the weather and tides.

Think about fluids you encounter in your daily live *other than water*. How might their properties confound industrial production? Shampoo is a good example, as well as products such as toothpaste and Greek yogurt. Volumes have been written on properties of various oils and their usage in the lubrication of metal-metal contacts. The main fluid we deal with in our everyday lives, water, is a Newtonian fluid of low viscosity. However, many fluids deviate severely from Newtonian character, and can exhibit highly nonlinear responses to applied stress, as well as memory effects. This is especially true of liquids that have dissolved within them macromolecules of high molecular weight, such as polymers or proteins. Likewise for liquids such as blood, which have a high content of small particles within them. Suspensions of biological material in general can exhibit non-Newtonian behavior. It would be excessively pedantic to list all of the possible types of non-Newtonian fluid here; suffice to say, there are many of them, and general modeling of their flows is an open research problem.

Most people have little conception as to how difficult the modeling of fluids in general is. The first theories ignored the property of viscosity and instead focused exclusively on zero-resistance (“inviscid”) flow. While there is some use to these equations, their shortcomings are rapidly brought to bear when modeling flows with any significant viscous effects.

In order to model viscous fluid flow, two equations are needed: the continuity equation, and the Navier-Stokes equation. These equations are, in general, nonlinear partial differential equations for which no general solution is known. However, there have been many advances made in the numerical solution of flow fields. This intersecting branch of physics, engineering, and computer science, called computational fluid dynamics, concerns itself with direct solution of the governing equations for a given fluid flow. The advances have culminated in off-the-shelf software, such as FLUENT, which can even model non-Newtonian fluids.

Check out this collection of fascinating films here. Here is one of the videos, a video on Flow Visualization:

In the days that these videos were made (circa 1960), there was no FLUENT, and computers were low-powered, cumbersome devices that would be totally inadequate for flow calculation. Instead of simulating flows, the scientists in these videos had to observe flows directly in the laboratory using experimental apparatus. The video on flow visualization shows many clever techniques used to observe the flow field without disturbing it. The scientists also were unable to solve the nonlinear partial differential equations governing the flow, and thus had to rely on sharp scientific intuition, as well as dimensionless numbers.

]]>But wait! what does this number sequence and the golden ratio have to do with a 15.9 million dollar violin?

Simple, it turns out violins made by the great violin maker Antonio Stradivari and his family are constructed around what are called the golden sections. These sections can be seen throughout the violin.

If we take the section a2 and divide it by a1 we get the golden ratio. b2 divided by b1 will also give us this ratio and so will c1 divided by c2. It turns out that the usage of the golden sections actually improves the tone quality of the violin. However this fact has been known as early as the ancient Greeks.

These splendid violins are so prized that in 2011, the “Lady Blunt”, a violin made by Stradivari for the granddaughter of Lord Byron, sold at auction for £9.8 million or $15.9 million. This topped the previous violin sales by 4 times.

The proceeds went to relief efforts for the 2011 Japanese earthquake.

**Sources Used**

- http://www.goldennumber.net/acoustics/
- http://www.nntdm.net/papers/nntdm-20/NNTDM-20-1-72-77.pdf
- http://en.wikipedia.org/wiki/Lady_Blunt_Stradivarius#mediaviewer/File:Lady_Blunt_top.jpg
- http://www.bloomberg.com/news/2011-06-20/stradivarius-sets-15-9-million-auction-price-to-help-japan-quake-relief.html
- http://commons.wikimedia.org/wiki/Violin#mediaviewer/File:PalacioReal_Stradivarius1.jpg

“All the variety, all the charm, all the beauty of life is made up of light and shadow.” – Tolstoy, *Anna Karenina*

We have already said that in order for something to be considered a work of art that it must communicate. But what else must it do or what quality must it possess? It must “make special” those ordinary things that we see every day. Just as Plato looked at a horse and thought “there must be perfect horse-ness in existence” so we look at something in the world and say “there must be perfect beauty in that thing.” It is a sort of world of forms we live in as artists. What is special to you, you must create in order to make special for another. We have held Michelangelo’s *David* in awe for centuries, because he saw the human form and knew the David story and said to himself “I must share that beauty with the world.” (okay, okay, he was also commissioned to make it, but that’s beside the point). We know from the timeless beauty of *David* that it is a masterpiece and that it is, very simply, a work of art.

Artists look into the world and think, as Mary Oliver so wonderfully put it, “What does it mean that the earth is so beautiful and what shall I do about it?” I believe it is our vocation as artists to delicately and reverently share these moments of beauty with those who haven’t the eyes to see it. I had this very experience when I was in the middle of sculpting my first half life-size figure. I looked at the intersections of the muscles, how you couldn’t see any of them because of the skin, but still instantly understand that *that* is her leg and *that *is her arm. I cried while I was sculpting her because for the first time I had a moment to realize the perfect beauty of God’s creation; the perfect beauty of each muscle; the perfect beauty with which the whole body works together to let you walk, talk, see, think, feel, and love. I knew that each dip and curve of *Selah*‘s figure would showcase the intricate beauties of the human form. Have you ever taken a moment to watch what your patella does when you bend and straighten your knee? Have you ever noticed that there’s a groove that your patella slides up and down so as to make that transition smooth and protect your knee from harm? Have you ever noticed that some people’s patellas are the shape of a teardrop while others the shape of a heart? Think about it. Make it special, if only in your thoughts.

Some may say that mathematics may also complete this task of making special by illustrating with numbers, letters, lines and other symbols frozen moments of perfect time. While mathematics may not be quite a form of art, it certainly runs alongside art in history. Non-euclidean geometry, just like Michelangelo’s *David*, transcends time and is forever beautiful.

The beautiful, however, must also be useful. (I’m just going to throw that statement in here without any explanation, because that can be done later.) And art must communicate, so when the following questions arise, we know how to answer them. “What about Art that makes you shudder in horror like Carl Orff’s *Carmina Burana* or Picasso’s *Les Demoiselles d’Avignon*?” Remember that art is making things special. It does this by making the ordinary extraordinary, no matter the content. During WWII, artists needed to convey how confused, hurt, and angry the world felt that such terrible things were happening. Their art displays this magnificently. Sure, it’s not beautiful in the way we normally think of that word, but it does communicate and make special that agony of loss and destruction.

Art must simultaneously make special and communicate, otherwise it fails in its being and its purpose. It fails to be art. Just as a problem in mathematics without a solution fails to prove anything or illustrate the beauty of the known world.

]]>You know what to do :)

Parts 1 (Hypatia’s Life) and 2(Hypatia’s Work) for anybody interested.

I decided to have these parts reduced to a single post because let’s face it, nothing written here is even remotely close to being as important as the story told so far. I decided not to add to the long list of conspicuous and inconspicuous people who used Hypatia’s name and life in order to tell a different story. I won’t do that to her. All this started as a search for knowledge and a tribute to the person behind the legend and I want it to stay that way.

I know these posts are nothing compared to the tremendous amount of information and legitimate research existing in the web about Hypatia, but if my small drop into the ocean of a study, can give some semblance of justice to Hypatia’s name and some knowledge to that small group of people reading these words, I’m more than satisfied with it.

**3,4) Myths, Literature and people’s opinions**

Let’s start with the history of history. Main historical source:

►*Socrates Scholasticus* (his work The Ecclesiastical History [meaning the history of the church] is important, he was one of her contemporaries, though his reference to Hypatia is short)

(fun fact: Scholasticus [as in *Socrates Scholasticus*] is* σχολαστικός *in Greek and it means fastidious)

►*Philostorgius* (another one of her contemporaries, he also gave us a small description of Hypatia in his own history of the church, titled Historia Ecclesiastica)

►*John of Nikiu* (he wrote an account of Hypatia’s activities during her life and of the circumstances surrounding her death in his Chronicle)

►*Hesychius of Milesius* (his work Onomatologus, a lexicon of Greek writers, included Hypatia. It is not preserved in its original form, but it has been remade using excerpts, which were given as references to his work by other writers)

►*Damascius* (Life of Isidore, is the most important account we have concerning the paganistic life of 5th century Alexandria and it includes an extensive description of Hypatia)

►*Sudae Lexicon* (very important Byzantine lexicon, it was the main source of information about Hypatia for the 19th and 20th century authors and researchers. It is mainly based on Hesychius and Damascius’ reports but was also influenced by Socrates and Philostorgius’.)

And of course,

►*Synesius of Cyrene* (the letters he sent to Hypatia herself and to her other students).

Let me add here, that even *those* sources are not what one would call dependable in their entirety. Specifically: Philostorgius was somewhat contradictory in his accounts. John of Nikiu is the only source we have with a somewhat negative perspective on Hypatia, he reported her as a pagan philosopher who was involved with astronomy, witchcraft, fortune-telling and used to walk around the city talking about philosophy to anyone willing to listen. We know his main source was Socrates, who makes no mention of witchcraft or that kind of public speaking, therefore that sort of information is baseless and generally disregarded. Hesychius of Milesius gives us a list of Hypatia’s work and also informs us that *Hypatia was married to the philosopher Isidore of Alexandria *(make of that what you will). Lastly, Damascius gives us the longest description of Hypatia, along with commentary on her talents and virtues, but his work as a whole is also contradictory at some points and therefore should be treated with caution.

Here I must comment that when examining each and every one of the existing sources, one can easily spot a difference of opinions concerning various areas of Hypatia’s life, personality and work. It’s not necessarily that those early historians were deliberately not being honest or objective (although sometimes they might have been). It’s that their perspectives varied. In an era when people were divided in very different groups with different (and sometimes opposing) goals and agendas, one’s actions (in this case Hypatia’s) could be worded, or understood, in very different terms (e.g. Hypatia using the astrolabe, or having a closed group of students could be interpreted as ritualistic by a person unfamiliar with the scientific and philosophical nature of her activities). Also, let’s not forget the fact that most of these texts were written *centuries* after the facts, which means that alterations could easily occur, due to ignorance, deliberate obscuration of the facts for whatever reason, falsified information etc.

Which brings us to the realisation that in order to uncover what really happened, one would have to study each and every one of these texts, try to view them with an objectively critical eye and uncover what was a blatant error (e.g. Hypatia’s marriage to Isidore) and what was actual fact. It is also what all medieval and most modern historians *didn’t* do (also for a grand variety of reasons).

That conclusion, is one of the reasons why Hypatia’s life has been retold in a hundred different ways, by a hundred different people who had a hundred different agendas. If one doesn’t do a full revision and analysis of all texts available, then depending on which source one chooses to follow different conclusions can be drawn. For example, it is true that up until Hypatia’s time, women scholars were not that uncommon. But the rise of Christianity, while it gave certain liberties to women, also wanted them to be submissive. It’s easy (although not truthful or right) to draw certain conclusions when a brilliant, liberated, beautiful, single, (who may or may not have been a pagan) woman, who freely and easily conversed with the governors of the city and had great influence over the society of the time got brutally murdered by a put off christian mob. And that’s what all those people in and after the 1800s did. They asked themselves the great big “Why?” question we humans adore, they searched for source material and when things turned out to be vague and mixed they drew their own conclusions, added some ideas of their own for spice and to support their causes and spread their stories.

Let’s get introduced with those 18th, 19th and 20th century authors and historians:

►*John Toland* (one of the early ones; he wrote two essays about Hypatia in 1720)

►*Thomas Lewis* (who wrote his own essay titled “The History of Hypatia” in 1721, as an answer to Toland’s work)

►*Voltaire* (makes a short reference to Hypatia in his “Philosophical Dictionary”)

►*Edward Gibbon* (wrote about her in his book “The Decline and Fall of the Roman Empire” in a very similar manner to Toland and Voltaire)

►*Leconte de Lisle* (here we have a famous reference to Hypatia through his homonymous poems, written in 1847 and 1874 and also his play “Hypatia and Cyril”)

►*Charles Kingsley* (one of the main authors behind the legend, he wrote a fictionalised version of Hypatia’s life, entitled “Hypatia” or “*New Foes* with an *Old Face”)*

►*John William Draper* (1869, “History of the intellectual Development of Europe”)

►*Bertrand Russell* (1946, “History of Western Philosophy and Its Connection with Political and Social Circumstances from the Earliest Times to the Present Day”)

►*Carlo Pascal* ( 1908, “Figure e Caratteri”)

►*Mario Luzi *(1978, “Libro di Ipazia”)

►*Ursula Molinaro* ( 1989, “Christian Martyr in Reverse: Hypatia, 370-415 AD, Hypatia: A Journal of Feminist Philosophy”)

Succinctly, both Toland and Lewis used Hypatia’s story to support their respective causes, Toland against the ecclesiastical circles and Lewis in behalf of them. Both Voltaire and Gibbon used her as well, in order to write their own digs against the patrons of the Christian church. Leconte de Lisle was one of the first to clearly fictionalize her story and was followed by Charles Kingsley who is almost single-handedly responsible for the great spread of baseless facts about Hypatia. John William Draper and Bertrand Russell both focused on the scientific nature of Hypatia’s life and they as well used it, this time to support their cause concerning the damage religion can do to science. The last wave of Carlo Pascal, Mario Luzi and Ursula Molinaro focused on the fact that Hypatia was a woman living in a man’s world and how that played a part in her life and in its end.

We can clearly see some motifs in the way all these people wrote about Hypatia. During the 18th century, the main cause behind their many references was their rebellion against the authoritarian character of the church and its rigid principles that worked against their need for intellectual and scientific advancement. We see that when we move on to the late 19th and early 20th century that cause reforms itself and is now a battle between scientific minds and old-fashioned religious ideals. And lastly, when we reach the late 20th century, Hypatia becomes a symbol for feminism. If you want to take it a step further, the 2009 movie Agora shows yet another view where Hypatia becomes a symbol against fundamentalism.

What becomes painfully obvious through those transitions is how Hypatia lost her identity and was reduced to having the stereotypical characteristics appropriate for the model woman of each time frame. A virtuous virgin, a brilliant scientist, a sensual femme fatale, a modern logical thinker and humanitarian. The characteristics that made her unique got lost in all her supposed causes against all things imaginable that have plagued humanity in the last centuries.

It makes one wonder, why all the fuss? Why make such a big story out of her life? Well firstly, her strong and unique personality and her important work is appealing and has all the right ingredients for an intriguing hero. The circumstances of her death give a hell of a reason to fight for. We empathize with her and we all see what we want to see when we read about her. Plus the few information we have provide a good environment for endless speculations. Those elements make her the perfect protagonist for any story. After a couple of them were written it didn’t take long for her to turn from an interesting character to a symbol. If you’ve ever studied semantics you know from Charles Sanders Peirce that a symbol doesn’t have only one meaning. Symbols don’t remain stationary. They evolve dynamically, according to their environment and the definition given to them by their interpreter. Which means that today, Hypatia is whatever one wants her to be.

The goal of these posts was to dig past all the stories and the symbolism and find who Hypatia was before humanity took over. She was a truly singular and awesome lady and she deserves to be remembered as she was. No stereotypes or dramatizations needed. Personally, I still have a dozen questions and doubts in regards to the picture I myself have created of her inside my head. But as I said before, if through all this even one of you out there got the idea of who she might have really been, I’m more than satisfied with the result.

I should mention that since we proudly entered the 21st century, more and more people have started seeing this story with a more critical eye, which means that there’s still hope for us; and if we are extremely lucky and some heroic researcher out there comes up with new material, who knows how this story will end.

If you’ve read so far, I thank you from the bottoms of my heart. I will bid you farewell with all the sweet, sweet tributes humanity has offered Hypatia over the years.

►The first one I stumbled upon, is that there is a crater on the Moon named Hypatia. I found this piece of information through an internet article titled “Hypatia on the Moon”. If that’s not romantic and at the same time the most utterly sappy sentence there ever was, I don’t know what is. I love it.

►There is also 238 Hypatia, a C-type asteroid belt

► And last but not least, The Hypatia Catalogue, which is the largest catalog ever produced for stellar compositions. Awesome.

► I almost didn’t mention it because yuck, but there is a type of moth named Hypatia. Don’t ask me why, I didn’t want to find out, this information got here for the sake of accuracy.

Sources:

John Toland: Hypatia I and II

Thomas Lewis

Voltaire

Edward Gibbon

Maria Dzielska: Hypatia of Alexandria

He died along with his wife in a taxi crash on the New Jersey Turnpike. The Nashes were traveling in a taxi on the New Jersey Turnpike when their driver attempted to pass another car, lost control of the taxi, slammed into the guardrail and then hit another car, according to New Jersey State Police.

In 1994, Mr. Nash and two others shared the Nobel Memorial Prize in Economic Science for their work in game theory, the study of decision-making in competitive environments.

There is ongoing investigation on whether the couples were wearing a seat belt and to know where they were coming from and going to.

Source: wall street journal

]]>I remember thinking at school when will I ever in my “real life” use maths. Well much to my utter surprise I’ve come to the stark realisation that there is a link between mathematics and crochet. Yes its true… In fact, crochet patterns have an underlying mathematical structure — the pattern created by the regular presence or omission of stitches is the very essence of this art form. The similarities to Base2 math, with its series of 0s and 1s, are obvious. That is to say, a present stitch is like a “1”, and a missing stitch is like a “0”. Crochet has been used to illustrate shapes in hyperbolic space that are difficult to reproduce using other media or are difficult to understand when viewed two-dimensionally.

It is believed that the partnership between math and craft dates back to the invention of geometry where the repetitive patterns seen in ancient baskets and weavings first hinted at a mathematical subtext to the world at large.

**Alan Turing**, the theorist and computer scientist, was often seen knitting Möbius strips (a surface with one continuous side formed by joining the ends of a rectangle after twisting one end through 180°.) and other geometric shapes during his lunch break. The Mobius strip in crochet terms is nothing other than the infinity scarf. Below is a beautiful example of an infinity scarf designed by our talented friend Anneke Wiese. The pattern is available **here.**

**Mathematics and hyperbolic crochet**

Hyperbolic Crochet is the name given to applying a mathematical principle to crochet patterns.

A hyperbolic plane expands exponentially from any point on its surface, always curving away from itself. You can easily crochet a hyperbolic surface by increasing at a constant rate throughout the piece. From a crochet perspective this is clearly depicted in the pattern called **Barb’s Koigu Ruffle scarf.**

For hundreds of years mathematicians tried to show that anything like hyperbolic space was impossible, until finally, in the nineteenth century, they accepted the “existence” of this form of geometry. Still many believed it wasn’t possible to model the structure materially. They were thus surprised to learn in 1997 that **Dr. Daina Taimina**** had done just that using the traditional art of crochet.**

Lilian Boloney is a textile artist who uses crocheting to explore the geometry of hyperbolic figures. There is an elegant simplicity to the off-white cotton thread she used to crochet the sculpture “Boy’s surface”. This allows the viewer to explore the complex topology of the figure without the distraction of patterns or color. Lilian not only has a clear understanding of her Mathematical subject, but she transposes their beauty into graceful objects. *“Instead of models of Hyperbolic figures I see them crocheted portraits” *says Lilian*.*

Hyperbolic growth in nature gives rise to the ruffled shapes of coral, kelp, and sea anemone. * The Institute for Figuring* created the concept of the Coral Reef with hyperbolic crochet and have been developing this concept since 2005.

**Chaotic Crochet**

**Dr. Hinke Osinga and Professor Bernd Krauskopf **(Engineering Mathematics, University of Bristol) have turned the famous Lorenz equations into a beautiful real-life object, by crocheting computer-generated instructions of the Lorenz manifold: all crochet stitches together define the surface of initial conditions that under influence of the vector field generated by the Lorenz equations end up at the origin; all other initial conditions go to the butterfly attractor that has chaotic dynamics. The overall shape of the surface is created by little local changes: adding or removing points at each step.

Dr Osinga has been able to crochet since she was seven years old , so she noticed that this is exactly the same way that crochet instructions work – by specifying a “surface” (more usually a poncho or baby’s blanket!) in rows, with the number of stitches increasing or decreasing. From there it was a simple step to turn the algorithm into a crochet pattern, and to start to create a real-life Lorenz manifold.

**Fractal Crochet**

Dictionary.com defines a fractal as: *A geometric pattern that is repeated (iterated) at ever smaller (or larger) scales to produce (self similar) irregular shapes and surfaces that cannot be represented by classical (Euclidian) geometry. Fractals are used especially in computer modeling of irregular patterns and structures found in nature.*

Scientists and mathematicians have discovered how to use fractals to numerically describe coastlines, the shape of trees, our vascular system and much more. Fractal crochet is absolutely beautiful in its form and you can buy a fractal crochet pattern on **Ravelry**

I hope you found this post as insightful and fascinating as I did. I like to think that there is intelligence behind crochet in more ways than one. What I am sure of though is that Crochet is most certainly my second language. I cannot say the same for mathematics!

**May your creativity multiply twofold this week!**

**Yarn blessings**

Stellation-series “wrap around,” so if this is stellated one more time, the result is the (unstellated) great rhombicosidodecahedron. In other words, the series starts over.

The dual of the great rhombicosidodecahedron is called the disdyakis triacontahedron. The reciprocal function of stellation is faceting, so the dual of the figure above is a faceted disdyakis triacontahedron. Here is this dual:

To complicate matters further, there is more than one set of rules for stellation. For an explanation of this, I refer you to this Wikipedia page. In this post, and the one before, I am using what are known as the “fully supported” rules.

Both these images were made using *Stella 4d*, software you can buy, or try for free, right here. When stellating polyhedra using this program, it can be set to use different rules for stellation. I usually leave it set for the fully supported stellation criteria, but other polyhedron enthusiasts have other preferences.

Dr. John Nash, aged 86, was killed along with his wife Alicia, in a taxi crash on Saturday in New Jersey. The pair were returning from a trip to Norway where he had received the Abel prize, a high-ranking honor in mathematics.

Nash is celebrated for his Nobel-prize winning discovery known as the Nash equilibium, and also for being the inspiration behind the major motion picture “A Beautiful Mind” (in which he was portrayed by Russell Crowe). The film won the Oscar for Best Picture in 2001, and raised awareness for the stigma associated with mental illness.

Nash’s award-winning theory has been used in a broad range of fields, including economics, social sciences, evolutionary biologies. He has also been attributed to influencing computing and artificial intelligence. In 2011, the NSA declassified letters he wrote in the 1950s where he anticipated many concepts of modern cyptography. He was a man ahead of his time, who overcame mental illness to pursue excellence within his field. He was a true inspiration and leader. RIP.

]]>Normally I don’t like to blog about something until I’m pretty confident that I have a reasonably good understanding of what’s happening, but I desperately need to sort out my thoughts, so here I go\dots

**1. Primes **

One day, an alien explorer lands on Earth in a 3rd grade classroom. He hears the teacher talk about these things called primes. So he goes up to the teacher and asks “how many primes are there less than ?”.

Answer: “uh. . .”.

Maybe that’s too hard, so the alien instead asks “*about* how many primes are there less than ?”.

This is again greeted with silence. Confused, the alien asks a bunch of the teachers, who all respond similarly, but then someone mentions that in the last couple hundred years, someone was able to show with a lot of effort that the answer was pretty close to

The alien, now more satisfied, then says “okay, great! How good is this estimate?”

More silence.

**2. The von Mangoldt function **

The prime counting function isn’t very nice, but there is a related function that’s a lot more well-behaved. We define the **von Mangoldt** function by

It’s worth remarking that in terms of (Dirichlet) convolution, we have

(here is the constant function that gives ). (Do you see why?) Then, we define the **second Chebyshev function** as

In words, adds up logs of prime powers; in still other words, it is the *partial sums of *.

It turns out that knowing well gives us information about the number of primes less than , and vice versa. (This is actually not hard to show; try it yourself if you like.) But we like the function because it is more well-behaved. In particular, it turns out the answer to the alien’s question “there are about primes less than ” is equivalent to “”.

So to satisfy the alien, we have to establish and tell him how good this estimate is.

Actually, what we *believe* to be true is:

Conjecture 1 (Riemann Hypothesis)We conjecture that

for any .

Unfortunately, what we actually know is far from this:

Theorem 2 (Prime Number Theorem, Classical Form)We have prove that

for some constant (actually we have done slightly better, but not much).

You will notice that this error term is greater than , and this is true even of the more modern estimates. In other words, we have a *long* way to go.

**3. Dirichlet Series and Perron’s Formula **

Note: I’m ignoring issues of convergence in this section, and will continue to do so for this post.

First, some vocabulary. An **arithmetic function** is just a function .

Example 1Functions , , or are arithmetic functions.

The **partial sums** of an arithmetic function are sums like , or better yet .

Example 2The Chebyshev function gives the partial sums of , by definition.

Example 3The floor function gives the partial sums of :

Back to the main point. We are scared of the word “prime”, so in estimating we want to avoid doing so by any means possible. In light of this we introduce the **Dirichlet series** for an arithmetic function , which is defined as

for complex numbers . This is like a generating function, except rather than ‘s we have ‘s.

Why Dirichlet series over generating functions? There are two reasons why this turns out to be a really good move. The first is that in number theory, we often have convolutions, which play well with Dirichlet series:

Theorem 3 (Convolution of Dirichlet Series)Let be arithmetic functions and let , , be the corresponding Dirichlet series. Then

(Here is the Dirichlet convolution.)

This is actually immediate if you just multiply it out!

We want to use this to get a handle on the Dirichlet series for . As remarked earlier, we have

The Dirichlet series of has a name; it is the infamous **Riemann zeta function**, given by

What about ? Answer: it’s just ! This follows by term-wise differentiation of the sum , since the derivative of is .

Thus we have deduced

Theorem 4 (Dirichlet Series of von Mangoldt)We have

That was fun. Why do we care, though?

I promised a second reason, and here it is: Surprisingly, complex analysis gives us a way to link the Dirichlet series of a function with its partial sums (in this case, ). It is the so-called **Perron’s Formula**, which links partial sums to Dirichlet series:

Theorem 5 (Perron’s Formula)Let be a function, its Dirichlet series. Then for any not an integer,

for any large enough (large enough to avoid convergence issues).

Applied here this tells us that if is not an integer we have

for any .

This is fantastic, because we’ve managed to get rid of the sigma sign and the word “prime” from the entire problem: all we have to do is study the integral on the right-hand side. Right?

Ha, if it were that easy. That function is a strange beast.

**4. The Riemann Zeta Function **

Here’s the initial definition:

is the Dirichlet series of the constant function . Unfortunately, this sum only converges when the real part of is greater than . (For , it is the standard harmonic series, which explodes.)

However, we can use something called **Abel summation**, which transforms a Dirichlet series into an integral of its partial sums.

Theorem 6 (Abel Summation for Dirichlet Series)If is an arithmetic function and is its Dirichlet series then

It’s the opposite of Perron’s Formula earlier, which we used to transform partial sums into integrals in terms of the Dirichlet series. Unlike , whose partial sums became the very beast we were trying to tame, the partial sums of are very easy to understand:

It’s about as nice as can be!

Applying this to the Riemann zeta function and doing some calculation, we find that

where is the fractional part. It turns out that other than the explosion at , this function will converge for any whose real part is . So this extends the Riemann zeta function to a function on half of the complex plane, minus a point (i.e. is a meromorphic function with a single pole at ).

**5. Zeros of the Zeta Function **

Right now I’ve only told you how to define for . In the next post I’ll outline how to push this even further to get the rest of the zeta function.

You might already be aware that the behavior of for has a large prize attached to it. For now, I’ll mention that

Theorem 7If , then .

*Proof:* Let be the real/imaginary parts (these letters are due to tradition). For , we use the fact that we have an infinite product

Using the fact that , , converges to some finite value, say . By standard facts on infinite products (for example, Appendix A.2 here) that means is .

The situation for is trickier. We use the following trick:

Now,

for all . By looking term-by-term at the real parts and using the 3-4-1 inequality we obtain

Thus

Now suppose was a zero (); let . Then we get a simple pole at , repeated three times. However, we get a zero at , repeated four times. There is no pole at , so the left-hand side is going to drop to zero, impossible. (The key point is the deep inequality .)

Next up: prime number theorem. References: Kedlaya’s 18.785 notes and Hildebrand’s ANT notes.

]]>

বৃদ্ধির পরিমাণ নির্ণয়:

টাইপ -১:বর্ধিত বর্গক্ষেত্র ক্ষেত্রফল নির্ণয়:

বর্গ ক্ষেত্রের প্রতিটি বাহু x% বৃদ্ধি হলে

ক্ষেত্রফল শতকরা কত বৃদ্ধি পাবে?

টেকনিক:

বর্ধিত ক্ষেত্রফল= x^2/100

উদাহরণ:

সমস্যা: একটি বর্গ ক্ষেত্রের প্রতিটি বাহু ১০ %

বৃদ্ধি হলে ক্ষেত্রফল শতকরা কত বৃদ্ধি পাবে?

সমাধান: বর্ধিত ক্ষেত্রফল= ১১০^2/100

=১২১%

সুতরাং ক্ষেত্রফল বৃদ্ধি =(১২১-১০০)=২১%(উত্তর)

টাইপ -২ বর্ধিত আয়তক্ষেত্রের ক্ষেত্রফল

নির্ণয়:

আয়তক্ষেত্রের দৈর্ঘ্য x% বৃদ্ধি এবং y% হ্রাস

পেলে ক্ষেত্রফলের শতকরা কি পরিবর্তন

হবে?

টেকনিক:

বর্ধিত ক্ষেত্রফল= (বর্ধিত দৈর্ঘ্য X হ্রাসকৃত

প্রস্থ)/১০০

উদাহরণ:

সমস্যা: একটি আয়তক্ষেত্রের দৈর্ঘ্য ২০% বৃদ্ধি

এবং ১০% হ্রাস পেলে ক্ষেত্রফলের শতকরা

কি পরিবর্তন হবে?

সমাধান: বর্ধিত ক্ষেত্রফল= (১২০ X ৯০)/১০০

=১০৮ সুতরাং ক্ষেত্রফল বৃদ্ধি=(১০৮-১০০)%

=৮%(উত্তর)

All these virtual models were made using *Stella 4d*, which you can try and/or buy here.