7

Caustics

The envelope of light rays reflected or refracted by a curved surface is called a Caustic.

All the light rays incident on the caustic surface are tangent to it.

The next time you take a sip from your coffee mug, be sure to take a look at its soul that it benevolently exhibits.

In the case of a circle, the caustic shape/curve formed is the cardioid.

Even water produces some really interesting caustic patterns

and so does wine in a glass.

Some of the images that have portrayed in the post have been made using computer graphic rendering systems. They trace the possible paths of light beam, accounting for refraction and reflection.

Have a great day!

Sources:   caustics  harvard  mr-cad

We know that there are infinitely many prime numbers, because if you multiply the first n prime numbers and add 1, the result cannot be a product of any of the first n prime numbers, and, since it isn’t 1, must be the product of different prime numbers.

What if you add 2? This result must be composite for n > 0, because the product itself is even, so the sum of it and 2 will also be even.

For n > 1, this works for adding 3 as well, because the product will be divisible by 3. Since the product is still even, adding 4 also yields a composite number.

For any n, you can add any integer between 1 and the (n+1)th prime (exclusive), and the result will be composite, since the integer you add must be divisible by one of the first n primes. This gives you a sequence of at least p(n+1) - 2 consecutive composite numbers.

As mentioned above, there are infinitely many prime numbers, and since prime numbers are integers, this means they become arbitrarily large. This is one way to show that there are arbitrarily long sequences of consecutive composite numbers.

(Perhaps there is some way to connect this to the Twin Prime Conjecture, but it is not immediately obvious, since this doesn’t imply either an injection or surjection between twin prime pairs and sequences of composite numbers bounded by two prime numbers that have prime cardinality. Sometimes adding 1 or p(n+1) to the product of primes yields a composite number, and there are such sequences, such as (14, 15, 16), that do not follow a product of the first n primes.)

Hypatia was a real person (here portrayed in a painting by Charles William Mitchell in 1885).  She was a world-famous mathematician and teacher of astronomy and the last librarian of the Library of Alexandria. So why did the painter depict her naked? Because she was stripped by a mob before they murdered her during violent Christian-Jewish clashes in Alexandria in 415 CE

5

Cats + Mathematics = the Furbonacci sequence

The Fibonacci sequence is found in many places in nature, including the branching of trees, leaves on a stem, the flowering of an artichoke, or an uncurling fern. But what if you applied this famous approximation of the golden spiral to our feline friends? Suddenly cats become even more awesome than we already thought they were.

Bored Panda is building a large collection of cute kitties demonstrating their mathematical prowess, nearly 40 as of the writing of this post. Click here to view them all.

Photos by thecheat42, Akimasa Harada, songbardbird, jacquiep, naux, and Beverley Goodwin respectively.

[via Bored Panda]

MARCH MASTERPOST MADNESS PT I as part of a follower milestone and celebrating spring break, i present to you march masterpost madness: a series of masterposts/guides on studying by yours truly!! first up we have math & how to study for it!!

i love math so much it looks like i spend almost all my free time doing it. anyway, i got a few requests asking me how to study best for math and about resources, so i thought it’d be best to start off with this one!!

SO, HOW DO I STUDY FOR MATH?

  • practice, practice and practice!! this is so important, omg. math is heavy on problem-solving, and most problems require analytical thinking and application of logic — the best way to hone these skills is through plain practice! besides, once you get used to math questions, you find them less intimidating and you get faster and more accurate at solving them! i suggest using practice books (your textbook should be a good starting point) or finding problems online (Khan Academy is great!). A pro tip is to look around your school community (ask teachers, students, librarians) for practice books — most of them will gladly lend you a book they’ve gone through! it’s ideal to practice/do at least one-two questions everyday, or even more! if you’re stuck on a practice problem and have no one to ask to, feel free to ask the mathblrs, wolfram alpha or the internet! (google your question: it has saved me numerous times!!)
  • pay attention in class!! this is definitely recommended, but it’s really not a must. if you don’t have a strong aptitude for math, i really urge you to do this! listen actively, you don’t have to take note of every word the teacher/lecturer says, but do listen to the most important parts! when a teacher emphasizes something, pay extra attention to that (but don’t leave out any parts of the chapter!).

RESOURCES

CALCULATORS

NOTES/CHEATSHEETS

  • algebra I: x, x, x, x
  • algebra II: x, x, x, x
  • geometry: x, x
  • trigonometry: x, x, x, x, x, x, xx
  • calculus: x, x, x, x, x, x, x

A FINAL NOTE

  • remember all your formulas!! know the syllabus!! know how to apply the concepts in problems!! those three things alone will get you to pass — i’m sure!!
  • try not to cram — study a week (at least) in advance for an exam!! (bigger exams need more prep time!!) an all-nighter won’t do you good!!
  • always remember, if you fail, you are not a failure!! there’s always a next time that you can get a better score in!!

Since it’s the first day of Black History Month, it seems appropriate that I share this portrait of Marjorie Lee Browne, who was one of the first African-American women to receive a doctorate in mathematics. Quoted from Wikipedia: 

Browne’s work on classical groups demonstrated simple proofs of important topological properties of and relations between classical groups. Her work in general focused on linear and matrix algebra. Browne saw the importance of computer science early on, writing a $60,000 grant to IBM to bring a computer to NCCU in 1960 – one of the first computers in academic computing, and probably the first at a historically black school. Throughout her career, Browne worked to help gifted mathematics students, educating them and offering them financial support to pursue higher education. Notable students included Joseph Battle, William Fletcher, Asamoah Nkwanta, and Nathan Simms. She established summer institutes to provide continuing education in mathematics for high school teachers. In 1974 she was awarded the first W. W. Rankin Memorial Award from the North Carolina Council of Teachers of Mathematics for her work with mathematics education.

I completed this illustration a while back for Women’s Work, a really awesome zine on women - past and present - working in STEM. The project was put together and curated by the wonderfully talented Shannon May and Celine Loup. You can learn more about the zine and see more art here

*EDIT* A math teacher approached me about purchasing a print of this illustration for her classroom. I’ve made 7x10 prints available in my shop, here

sciencealert.com
This ancient Babylonian map of Jupiter just changed history as we know it
You're looking at the foundations of modern calculus.
By Bec Crew

Analysis of an ancient codebreaking tablet has revealed that Babylonian astronomers had calculated the movements of Jupiter using an early form of geometric calculus some 1,400 years before we thought the technique was invented by the Europeans.

This means that these ancient Mesopotamian astronomers had not only figured out how to predict Jupiter’s paths more than 1,000 years before the first telescopes existed, but they were using mathematical techniques that would form the foundations of modern calculus as we now know it.

“This shows just how highly developed this ancient culture was,” historian Matthieu Ossendrijver from Humboldt University in Germany told Maddie Stone at Gizmodo. “I don’t think anybody expected something like this would be discovered in a Babylonian text.”

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