Inequalities and Mr. Honner’s triangles

Earlier today Patrick Honner posted an interesting piece about a famous triangle:

His post sparked a perhaps odd connection in my mind with a blog post about solving inequalities that Bob Lochel had posted earlier in the week:

The connection had to do with how you describe a triangle using inequalities.  I thought that creating 3D prints of the triangles in Patrick Honner’s blog would be an interesting exercise in inequalities, so I gave it a shot.

First, though, I was pretty surprised by the 2nd picture in the triangle blog, so I used a ruler and compass to draw the triangles (yes, a ruler – I measured out lengths of 13 cm, 14 cm, and 15 cm.  Sorry construction purists!!)

Compass Triangles

Amazing how close the 13-13-13 equilateral triangle is to fitting perfectly into the 13-14-15 triangle – wow!

Next I went to Mathematica to see how tough it would be to describe a 3D triangle.  This is only the third shape hat I’ve created on Mathematica to 3D print, so I am by no means an expert.  I wouldn’t be surprised at all to learn that there is an easier way to do what I did.  Actually, I’d be stunned if there wasn’t, but I wanted to use inequalities specifically for this project even if there was a better way.

Though my code is clumsy, it wasn’t too difficult to create the inequalities that would describe the 3D triangles:


The code above was for the equilateral triangle.  It assumes that the three verticies of the triangle are at points (0,0), (13,0), and (6.5, 13  \sqrt{3} / 2) in the x-y plane.  The 4 variables after the a,b,c, and d are just giving me the slope and y intercept of the 3rd side of the triangle (that I used “xint” for the y intercept says more about the quality of the code than any other words I could speak .. . . ).

The next line defines the triangle and gets to the part about inequalities.  We’ll define this triangle by all of the points satisfying:

(i) the y coordinate is greater than 0.  (y = 0 describes the line segment at the bottom of the triangle.)

(ii) the point is underneath the line y = (a/b) * x.  (This formula describes the line segment on the left side of the triangle.)

(iiI) the point is under the line y = mx + xint.  (This formula describes the line segment on the right hand side of the triangle.)

(iv) and finally, to make the triangle three dimensional, I’ll take the z coordinate be between -0.25 and 0.25.

That’s it for the simple equilateral triangle.  The 13-14-15 triangle and the third triangle I made – the 13-14-2 triangle – are a tiny bit harder since they require a little trigonometry to get the third coordinate (I assumed (0,0) and (13,0) were points in all three triangles).  If you are working through this same exercise with students, I suggest also using (0,0) and (13,0) for two of the points in all three triangles.  The reason is that you have to switch up one of the inequalities to describe the 13-14-2 triangle, and thinking through that switch of inequalities is a nice exercise for students.

Printed Triangles

So, a fun connection between two blog posts and a neat printing exercise which doubles as an interesting inequality exercise.  Made for a fun Sunday afternoon recreational math adventure.  Even got to show my 8 year old how to make an equilateral triangle using a ruler and compass!

As usual, always happy to see and play around with the math ideas that people share on Twitter.

Which reminds me, I’d previously written a post about Mr. Honner’s square, so I guess this post was just a matter of time!!

Terry Tao’s MoMath Talk Part 2: Clocks and Mars

Last week I wrote about finding Terry Tao’s incredible public lecture delivered at the  Museum of Math and how that lecture provides many great examples you can use to talk about math with kids:

Terry Tao’s MoMath Lecture Part 1: The Earth and the Moon

for ease, the direct link to the Terry Tao lecture  is here:

Today I wanted to use a second example from that lecture for a little math talk with the boys.  This topic comes from approximately 42:30 into the video when Tao discusses Copernicus’s calculation of how long it took Mars to orbit the sun.   This calculation is an incredible scientific achievement, especially when you consider that telescopes hadn’t even been invented yet!

In the lecture Tao describes the remarkable story behind the calculation, but does not go into the details of the calculation itself.  To be clear, that’s not a criticism – the point of his lecture was to tell the story not to dive into the details.  Exploring the details of this particular calculation is a great topic to discuss with kids, though.  The only background material required is some basic knowledge about fractions.

We began this morning by watching the (approximately) 5 minute portion of the talk in which Tao describes how Copernicus calculated the time it took for Mars to Orbit the sun.  Following that we went to the whiteboard to talk about what we learned, and to head down the path of understanding the calculation in detail.   The starting point I chose for understanding the calculation is asking questions about the angles formed by the hands of clocks.

I will say at the start that it was a little harder for my kids than I was expecting.  The discussion and the explanations below are not at all flawless and have several false starts.  As I’ve said many times, that’s what learning math (and, in this case, a little physics) looks like.  Watching the films of this discussion prior to publishing this post has reinforced my feeling that Tao’s lecture  is a great spring board to talking math with kids.

Having looked at a few examples of when the angles between the hour hand and minute hand of a clock would be zero, in the next part of the talk we began to try to drill down on the math.  The starting point for the discussion here was the observation by my older son that the minute hand moves 12x faster than the hour hand.    In this video we try to write down some expressions that describe how fast the two hands of the clocks are moving:

The next step was writing down an equation that told us how far the hour and minute hands would move in “t” minutes.  In retrospect I wish I would have made a different choice in the approach here since jumping directly to the algebra made a simple idea a little harder than it needed to be.   If I could do it again I’d probably cover the ideas in this video nearly in reverse (and I’m annoyed with myself for getting frequency and period reversed, too.  Can’t get everything right . . . .)

However, even with the little bit of extra time that introducing the algebra at the wrong moment led to, the discussion here did get us to an equation that looked a lot like the equation Terry Tao had written down in his presentation slides.

At the end of the last video we got to an equation that helps us understand when the hands of a clock are exactly on top of each other – now we solve it!  Solving this equation is a great exercise for kids who have a little familiarity with fractions.  We sort of stumble out of the gates with the solution, but once we get on the right track we actually get to the end in sort of a neat way.

With all of this background out of the way we can return to the equation that Terry Tao had in his presentation.  We being this part by briefly talking about difference between our clock equation and the equation that Copernicus solved..  After that introduction we solve the equation and determine how long it takes for Mars to orbit the Sun!

I’m really excited about using more examples from Terry Tao’s lecture to talk math with kids.  There are so many great things about this lecture – for instance the incredible historical information and the great opportunity to see Terry Tao speak on an accessible topic – but for me the new examples the talk contains for talking  about some basic school math with kids is the best thing about this public lecture.    Who would have thought that calculating the orbit of Mars just boiled down to simple fractions?!?