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Intmath Newsletter: Poly equations, charts, potential, water

By Murray Bourne, 31 Oct 2019

31 Oct 2019

In this Newsletter:

1. New on IntMath: Polynomial equations graph solver
2. Resources: Tilt Brush, charts
3. Math in the news: Lenses, integral equations
4. Math movies: Potential
5. Math puzzle: Square age
6. Final thought: Water

1. New on IntMath: Polynomial equations graph solver

After many reader requests, I rewrote a lot of the Solving Polynomial Equations chapter recently. I'm not a big fan of the way this topic is normally handled (via the Remainder and Factor Theorems) mostly because those techniques only work for low degree polynomials with "nice" numbers and involve an amount of guesswork. But I agree there is some benefit to learning the concepts.

The revised pages are:

How to Factor Polynomials

Roots of a Polynomial Equation

I also added one new interative page:

Polar to rectangular calculator

We can always use numerical approaches to finding the roots of equations (like Newton's Method).

But I've always found (for the last 30 years while we've had such tools) that zooming in on the x-intercepts in a graph application is easier and quicker. We can get whatever accuracy we need by zooming in some more.

See: Roots of Polynomial Equations using Graphs

2. Resources

(a) Tilt Brush: Painting from a new perspective

Google tilt brush

Google's Tilt Brush for Vive, Oculus or Windows Mixed Reality lets you paint in 3D space with virtual reality. There's a lot of scope here for math students and teachers for drawing 3-D functions and vectors.

See: Tilt Brush: Painting from a new perspective

Has anyone played with this? What's your experience been?

(b) Making charts easier to read at a glance

Making charts easier to read

Charts of complex data can be hard to comprehend. This new method from Columbia Engineering and Tufts University aims to develop easier to read data visualizations.

See: New data science method makes charts easier to read at a glance

3. Math in the news

(a) How One Mathematician Solved a 2,000-Year-Old Camera Lens Problem

Mathematician solves 2000 year-old spherical aberration problem

Greek mathematician Diocles first reported on a problem most lenses have - spherical aberration.

Here's a summary of a recent fix from Popular Mechanics:

See: How One Mathematician Solved a 2,000-Year-Old Camera Lens Problem

And here's the actual paper for those interested:

General formula for bi-aspheric singlet lens design free of spherical aberration (link no longer available)

(b) Metamaterials solve integral equations

Metamaterials solve integral equations

Solving integral equations has been a staple of science and engineering for hundreds of years. A new optical approach makes computer solutions significantly quicker.

See: Metamaterials solve integral equations

4. Math Movies - fulfilling our potential

(a) How we can help the "forgotten middle" reach their full potential

How can we empower the forgotten middle to reach their full potential?

Most of us fit in the "forgotten middle" - neither exceptional nor problematic. This talk gives some insights - for both students and teachers - on how such people can reach their full potential.

See: How we can help the "forgotten middle" reach their full potential

(b) The boost students need to overcome obstacles

The boost students need to overcome obstacles

Everyone has a story - a reason why they have not met their own expectations or hopes for their lives. How we approach our obstacles can make a huge difference.

See: The boost students need to overcome obstacles

5. Math puzzles

The puzzle in the last IntMath Newsletter asked about the probability involved in a given hexagon.

There were five attempts at an answer, and all were different. My approach was as follows.

Choose 1 for the length of each side of the octagon. That means the triangles and the square also all have side length 1.

The octagon's area is 2(1 + √2) (a well-known formula, or it can be derived from splitting it up into triangles).

The area of each (equilateral) triangle is √3/4 (half the base times the height) and we have 4 of them, so the total area of the triangles is √3.

The area of the square is 1.

The area of the blue parallelograms (kites) is the hexagon's area minus the area of the trangles and square.

Area blue = 2(1 + √2) − √3 − 1 = 1 + 2 √2 − √3.

So the probability is

(1 + 2 sqrt 2 - sqrt 3)/(2(1+sqrt 2)) ~~ 0.4342

New math puzzle: Square age

Europe 900AD
Europe in 900 CE. [Image credit]

If I said I was x years old in the year x2, it would mean for example, in the year 900 CE (AD), I was 30.

How old would somebody be if they could say that in the 21st century?

When is be the next century when nobody will be able to say the year is the square of their age?

You can leave your response here.

6. Final thought - water

Parts of Australia are undergoing the worst drought in recorded history. Some towns have already effectively passed "day zero" since trucking in water has become necessary. In many places, "day zero" will come sometime next year. Meanwhile, Capetown in South African recently averted a "day zero" situation.

For years scientists have been predicting such scenarios, and for years all we've seen is political football games.

water risk atlas
Extremely high water risk in parts of Western USA. [Image credit]

Yes, the rains will come again. But it's more likely the next drought will be even longer. Many Australian farmers are talking about giving up, and this is in the country that promised to be the "food bowl of Asia".

You can't eat coal, and you can't drink oil.

So how did Capetown do it? They imposed personal water restrictions of 50 liters per person per day and tough industrial restrictions ("Level 6B" water restrictions). By comparison, US citizens consume around 500 liters per day, New Zealanders consume 227 liters a day and in India, it's 120 liters per day.

See the state of your country in this Water Risk Atlas data map.

Until next time, enjoy whatever you learn.

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