Showing posts with label Art. Show all posts
Showing posts with label Art. Show all posts

Tuesday, August 08, 2023

How to Handle Super-Hard Questions

For high scorers, only the hardest problems near the end really matter. That’s where top math students get lost, waste time, feel stress, and lose points.

How to avoid freaking out over crazy-hard questions?

Three concrete suggestions follow.  

1. Handle intimidation

It’s normal to feel intimidated by a long, wordy, or complex problem. Most of the time, however, you’ll be able to answer the question correctly in a reasonable amount of time, if you give yourself a chance. Always approach such a problem with the mentality that you can solve it. Boil down the question. Reread and make sure you understand each sentence, part by part. You probably can, in fact, do this. If you slow down a bit, read carefully, take notes, and focus intently, you’ll be successful most of the time.

2. Dodge nightmare questions

Instead of getting hung up on an unsolvable problem, it’s better to surrender, as quickly as possible, and live to fight another day. Prevent the wasted time, loss of energy, frustration, and stress caused by battling impossibly difficult “nightmare questions.” The fight isn’t worth it. Avoid the bottomless pit. Earn points somewhere else. Once you realize a question is over your head, just guess and move on. Do not come back later, even if you have time.

3. Employ Skip-Guessing

Naturally, top students want top scores, and often assume they’ll have to work hard to crack every single question. But this is not true. Students can generally afford to skip-guess lots questions without worry (to confirm this, study the scoring tool for any official practice test). When faced with an uncomfortably difficult question, don't get stuck, whatever you do. Quickly make your best guess, skip the question for now, flag it for later review, and move merrily along. 

What matters is making the utmost of what you’re able to do on test day. To optimize your score, spend your limited time where it can be maximally effective, not on crazy-difficult problems you’ll just get wrong, anyway.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Thursday, September 01, 2022

Gödel, Escher, Bach

Douglas Hofstadter's mind-bending tome about self-reference and meta-thinking published in 1979 is a classic in recreational mathematics. Getting all the way through it is a tantalizing, formidable intellectual challenge.

Self-reference applies to ideas that loops inward (outward?) on themselves. 

Some examples:

This sentence is false. Seeing one's own eyeballs (without using a mirror). Brushing the bristles of a brush with that same brush.

Hofstadter compares the works of three geniuses: Kurt Godel in the domain of pure mathematics, M. C. Escher in the world of fine art, and Johann Sebastian Bach in the realm of western classical music. The similarities are, indeed, surprising and impressive.

All three masters dealt with the concept of circular self-reference, but in different ways. Godel proved the illogical nature of mathematics (which is, itself, based on logic); Escher was famous for stairs that climbed upward to the bottom of the stairs and identical tessellating foreground and background images that seem, somehow, to "cause" each other; Bach would take a short series of notes, and then invert the same motif, play it backwards, string these versions together, etc.

The book is highly intriguing, almost addictive. But it isn't for the faint-of-heart or faint-of-mind. I tried to finish it. Wasn't able to. Made me dizzy. Read it at your own risk.

Buy GEB here.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Tuesday, February 01, 2022

Freerice

At
freerice.com students of all ages can learn everything from basic arithmetic to art history, world geography, anatomy, and English grammar and vocabulary, while earning grains of rice to donate to the United Nations' World Food Program (WPF).

"Rice" is used as a metaphor for donations generated by users of the site that sponsors then fulfill. Each correct answer stores 10 grains of virtual rice for donation to the WFP. Sponsors then give the monetary equivalent of all rice collected to the WPF to fund its charitable work around the globe.

From the site:

"The United Nations World Food Programme (WFP) is the 2020 Nobel Peace Prize Laureate. We are the world’s largest humanitarian organization, delivering life-saving food assistance in emergencies and working with vulnerable communities to improve nutrition and build resilience."
Freerice is a great way to learn while helping others in need throughout the world. The site generates billions of "rice grains" each year, a total of 224 billion to date! Caring students get smarter while making the world a better place in a concrete way."

For anyone who loves learning, it's hard to imagine a better way to have fun.

FAQ here.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Monday, March 01, 2021

Patti Smith at Wesleyan 2016

My daughter graduated with degrees in mathematics and computer science from Wesleyan University in 2016. She's now a senior software engineer at Meta, well married, with a healthy and happy first baby, our first grandchild.

Can you tell I'm proud?

I'll never forget her graduation ceremony. Sitting in the audience, I was nervously flipping through the program, not paying much attention to anything in particular. Time passed slowly, as more and more parents and loved ones filled the chairs facing the stage in front of the beautiful Wesleyan library. Eventually, things got under way.

And then ... that voice!

I recognized it instantly as that of a musical hero of mine, the venerable Patti Smith. Not more than a few words into her talk, I knew it was her. What more fitting a way to mark my daughter's college graduation than with the spoken words of this particular poet extraordinaire:

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I was dreaming in my dreaming 

Of an aspect bright and fair

And my sleeping, it was broken

But my dream it lingered near.

In the form of shining valleys

Where the pure air rarified

And my senses newly opened

And I awakened to the cry.

That the people have the power,

The people have the power.

And where there were deserts,

I saw fountains and like cream the waters rise

And we strolled there together

With none to laugh or criticize.

And the leopard and the lamb

Lay together truly bound

I was hoping in my hoping

To recall what I had found.

I was dreaming in my dreaming

God knows a purer view

But as I surrender to my sleeping

I commit my dream to you.

That the people have the power

To redeem the work of fools

Upon the meek the graces shower

It’s decreed the people rule.

And I believe that everything we dream

Can come to pass

Through our union we can turn the world around

We can turn the earth’s revolution.

For the people have the power,

The people of the power.

Graduates, you are the future, and the future is now.

Congratulations.

Thank you.

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Read the article in The Wesleyan Connection and watch the video here.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Monday, June 01, 2020

Math and the Master

The phrase "Renaissance Man" is epitomoized by Leonardo da Vinci, the master of masters, founder of the High Renaissance. Geometry infused Leonardo's work, and was a particular obsession of his (e.g. The Golden Ratio, perspective, knots, fractals).

An article published by The Mona Lisa Foundation goes into some detail about the geometric underpinnings of Leonardo's design thinking.

It begins:

The important relationship of mathematics to art cannot be [overstated] when discussing Leonardo’s later work, and in numerous documents, letters and notes, the relevance of this is well documented. At times, he seems obsessed with these issues: while working on Mona Lisa for example, Leonardo is reported by Fra’ da Novellara to be concentrating intensely on geometry.

“Non mi legga chi non e matematico.”

“Let no one read me who is not a mathematician.”

-- Leonardo da Vinci

[Continue reading here.]

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Tuesday, January 01, 2019

Math-as-Art Blog



I had planned to store the overflow from the "Math as Art" project in a single post on this blog, but it quickly became apparent that the project would require a site of its own.

Here it is:

Math-as-Art.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Thursday, November 01, 2018

Mandlemap



Everything you wanted to know about the Mandelbrot Fractal. A great gift for the math nerd in your life.

Posters are 24 x 36 inches or 54 x 36 inches.

Buy one here.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Monday, October 01, 2018

The Engine of the World

I remember when tenor Andrea Bocelli sprang out of nowhere as the heir apparent to the great Luciano Pavarotti.

For audiences, it always seems like that; suddenly this great supernova explodes into view, changing the sky forever. 

Of course it's not like that at all.

Fame may happen overnight, but the artistic growth and development of a phenomenon like Bocelli is often excruciating slow and painful. The degree of hyper-focus and self-centeredness required of great artists makes intimate relationships all the harder to create and maintain. No wonder addiction and madness are familiar attendants to those having regular intercourse with the Muses.

Yet somehow the fuse finally gets lit, the coming eruption only a matter of counting down. Those that survive ignition share with us glimpses of other worlds, uncommon truths and ephemeral realities, souvenirs of Heaven.

Dear Veronica, my dear children,

Every life is a wonderful story worthy of being told. Every life is a work of art, and if it does not seem so, perhaps it is only necessary to illuminate the room that contains it.

The secret is never to lose faith, to have confidence in God's plan for us, revealed in the signs with which He shows us the way.

If you learn to listen, you will find that each life speaks to us of love. Because love is the key to everything, the engine of the world. Love is the secret energy behind every note I sing.

And never forget that there's no such thing as happenstance. That's an illusion lawless and arrogant men invented so that they could sacrifice the truth of our world to the laws of reason.

Andrea Bocelli (2017 biopic: The Music of Silence)

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Friday, June 01, 2018

Fractal Coloring Book

Adult coloring books have become popular in recent years. A number have been published already, and more are finding there was to amazon all the time. Below are my favorites:

Coloring Mandalas for Meditation was a hit with my daughter and her friends at Wesleyan who needed to periodically take a break from the stress of studying.

Adult Coloring Book: Fractals by Ben Trube not only offers beautiful fractal designs to color, but also provides a visceral experience with fractals and the math behind them.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Saturday, February 01, 2014

Finite Simple Group (Of Order Two)



Whoever said math isn't musical has never heard The Klein Four.

If you've got a couple of minutes, take a break, give yourself a treat, and check out this hilarious, cleverly written, well performed a cappella composition:

Finite Simple Group of Order Two

Although knowledge of group theory isn't at all required to enjoy the performance, the brilliant plays on words that comprise the lyrics will make real sense only to those with some knowledge of higher mathematics.

Here they are:

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Finite Simple Group of Order Two

The path of love is never smooth
But mine's continuous for you
You're the upper bound in the chains of my heart
You're my Axiom of Choice, you know it's true

But lately our relation's not so well-defined
And I just can't function without you
I'll prove my proposition and I'm sure you'll find
We're a finite simple group of order two

I'm losing my identity
I'm getting tensor every day
And without loss of generality
I will assume that you feel the same way

Since every time I see you, you just quotient out
The faithful image that I map into
But when we're one-to-one you'll see what I'm about
'Cause we're a finite simple group of order two

Our equivalence was stable,
A principal love bundle sitting deep inside
But then you drove a wedge between our two-forms
Now everything is so complexified

When we first met, we simply connected
My heart was open but too dense
Our system was already directed
To have a finite limit, in some sense

I'm living in the kernel of a rank-one map
From my domain, its image looks so blue,
'Cause all I see are zeroes, it's a cruel trap
But we're a finite simple group of order two

I'm not the smoothest operator in my class,
But we're a mirror pair, me and you,
So let's apply forgetful functors to the past
And be a finite simple group, a finite simple group,
Let's be a finite simple group of order two
(Oughter: "Why not three?")

I've proved my proposition now, as you can see,
So let's both be associative and free
And by corollary, this shows you and I to be
Purely inseparable. Q. E. D.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Monday, October 01, 2012

Math as Art (Part 2)

In an earlier post on this subject (click here), I wrote about the pioneering work of Michael Trott, who as "the worlds most advanced Mathematica user" has brought together like few others the realms of visual art and mathematics. Today's post contains a few selected links intended as starting points to ignite and inspire your curiosity and act as further impetus for your own look into the alluring world of mathematical art.

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The tantalizing Wolfram Research Graphics Gallery is the best place I know of to begin enjoying and appreciating the wonders of numbers in visual form:

http://gallery.wolfram.com/.

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Another interesting point of embarkation is the following page highlighting works of mathematical art exhibited at recent major conferences:

http://www.bridgesmathart.org/amsmaa.html.

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The following "Pythagorean Tree" generators are simple ways to create mathematical art of your own:

http://myweb.cwpost.liu.edu/aburns/flash/pyth4.html;

http://www.nsf.gov/news/overviews/mathematics/interactive.jsp
.

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Fractals are fascinating, relatively recent products of mathematical research famous for the alluring beauty and entrancing effect of their images. The internet abounds with sites devoted to the study of fractals (a simple google search will yield upward of 8 million hits!).

Here are a few to get you started:

http://en.wikipedia.org/wiki/Fractal;

http://math.rice.edu/~lanius/frac/;

http://local.wasp.uwa.edu.au/~pbourke/fractals/;

http://members.cox.net/fractalenc/encyclopedia.html.

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Long Island University mathematics professor Anne Burns has published a "Gallery of Mathscapes" showing some of her works as a mathematical artist:

http://myweb.cwpost.liu.edu/aburns/gallery/gallery.htm.

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The National Science Foundation has a superb, highly accessible general mathematics site suited for lay people with more advanced interest in the subject (highlighting recent mathematical news, research, discoveries, etc.)

http://www.nsf.gov/news/overviews/mathematics/index.jsp.

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Last, but absolutely not least, is the Wolfram Functions Site. Containing "the worlds largest collection of formulas and graphics about mathematical functions," a visit here is not for the faint of heart, but for math fans who just have to "have it all:"

http://functions.wolfram.com/.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Saturday, September 01, 2012

Wordle Word Graphs



Once in a while a simple yet brilliant and beautiful idea comes to life online and reminds us that computer technology wasn't invented just to drain our free time, create dependence and then frustration when these vital tools fail to work, and give us another excuse to avoid personal contact with other human beings.

Wordle is one of those ideas. Take a famous speech, a letter to a friend, a junk email message, or any piece of textual information and Wordle can turn it into a thing of beauty and usefulness, marrying form and function to delight the senses and divulge hidden meanings in color, word, and shape.

Think of Wordle as a cross between painting, language, and math that enables anyone armed with a piece of text to engage creatively in all of these activities at the same time.

Wordle takes your text, analyzes which words are used and how often they appear, and creates a verbal montage that shows, based on the size of the words appearing in the wordle, the relative frequency of the particular words comprising that text. By showing graphically the popularity of words contained in a given piece of writing, Wordle reveals with striking clarity those ideas and themes given the most emphasis by the work's author. One gets the distinct feeling that wordles even somehow manages to expose the author's true intentions, giving viewers the opportunity to spy on the writer's unconscious mental processes and discover his or her latent motivations.

Above is a wordle I created out of Barack Obama's victory speech on election night, 11/4/2009.

Users can customize their creations by changing fonts, color schemes, layout, orientation, etc. to form endless variations of the same wordle. There's even a "Randomize" button to generate random versions!

Finally, you can publish your best wordles for the world to see in the Wordle Gallery.

And Wordle costs ... absolutely nothing!

Not only that, but users are completely free (with proper attribution) to use wordles they create in any way and for any personal or commercial purpose they like, under the terms of the Creative Commons Attribution 3.0 United States License that governs images created by Wordle.

Try it out yourself! Click here to go to the Wordle site, and start making your own wordles for fun, learning, or profit.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Thursday, December 01, 2011

Math as Art (Part 1)

Mathematics has gotten a bad reputation for being dry, stodgy, and decidedly uncreative – a very left-brained activity. This reputation is undeserved, however. In fact, math is art!

A recent article on the Apple Web site highlighted the work of Michael Trott, a pioneer in the effort to popularize the creative/artistic side of pure mathematics and the world’s most advanced Mathematica user.

The article begins:

“Fifteen years ago, Michael Trott, a theoretical physicist at a small university in Germany, carried his department’s only available computer, a Macintosh IIfx, from his office to the lecture hall every week to share knowledge about Stephen Wolfram’s groundbreaking mathematics application, Mathematica.”

Read the full text of the article here:

http://www.apple.com/science/profiles/trott/.

You can view examples of some of Trott’s beautiful mathematical art, and works of other Mathematica users, here:

http://www.borlandeducational.com/page10/page10.html.

A Google search for “mathematical art” will lead to innumerable other sites and resources through which to explore this under-appreciated aspect of mathematics, the language of science.

(Click here to go to part 2.)

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Wednesday, April 01, 2009

The Stanford Laptop Orchestra

The personal computer has transformed the world of music.

Just as iTunes transformed the commercial music industry, new hardware and software tools have fundamentally changed the way in which university music departments teach the art of musical composition and performance.

The Stanford Laptop Orchestra (SLOrK ) is a bold experiment by Stanford researcher/programmer Ge Wang that combines the flexibility of Apple MacBook laptops, multi-channel speaker arrays made out of Ikea salad bowls, creative computer programming, and visionary imagination into an intriguing mix that makes for some pretty interesting sounds.

Paul Craft writes in the Stanford Daily:

The set-up is relatively simple. Members of the SLOrk operate black Apple Macbooks, which are connected via a series of cables to a spherical speaker system and control box for volume, among other things. Depending on the piece being played, the set up can include a joystick and other accessories.

The laptops themselves are, as the group’s title suggests, the heart of Wang’s vision. Ensemble members — the “musicians” — use a variety of different programs and configurations to create a variety of different sounds, ranging from a human-like voice to percussion to the ambient noises of a casino. SLOrK relies on a coding language that Professor Wang developed while at Princeton known simply as “CHUcK.” The language is made specifically for music and sound use, prototyping an instrument in a matter of minutes. “The computer itself is not an instrument. We have to craft it into one,” Wang said.

Read the rest of Craft's article here.

An interesting posting in the Pro section of the Apple site further details SLOrk's innovative joining of laptop and audio technology.

For video articles on SLOrk, and to hear the orchestra live in concert, click here and here.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.

Sunday, March 02, 2008

The Elvis of Classical Music?

This young conductor is poised to change the history of music.

Slated to take over as director of the L.A. Philharmonic in 2009, Gustavo Dudamel knows how to communicate the essence and beauty of western classical music in a way no one seems to have seen in decades. He may, in fact, be the one who rescues classical music from increasing cultural irrelevance and puts it back at center stage on radios, ipods, and stages everywhere, where it belongs.

Dudamel was saved by music and the Venezuelan System (a government-sponsored initiative to train poor children to become accomplished musicians as an alternative to impoverishment and criminal street life). Who knows how many other game-changing individuals are waiting for a chance to fulfill basic survival needs so they can share rare and powerful talents with humanity?

Say what you want about Venezuelan socialism; Duhamel's magic is one of its fruits.

If you have a few minutes, give yourself a gift and watch the video piece on Dudamel produced by 60 Minutes. See another excellent Duhamel interview here.

Gustavo reveals, through his genius and passion, the creative artist within each of us. He reminds us, shows us, through music, that this is it! This is what it is to be human. This is what the struggle is for. This is why it all matters.

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Copyright © 2006-present: Christopher R. Borland. All rights reserved.