Chasing Infinity – 3D Version

The frequency of my blog posts have been diminishing more and more, or should I say, the wave length has been getting longer.  I attribute this mostly to my accession as math department chair.  I just haven’t had the time and/or mental energy to do many math explorations even in the summer. I am also playing less golf.  End of excuses.

Four years ago, I explored what happens at “infinity” for plane curves.   This summer I was curious about what happens at infinity for three dimensional surfaces.  The original idea was to project a curve onto a sphere by placing a plane containing the curve tangent to a sphere, projecting a line from a point on the curve through the center of the sphere and noting where the line intersected the surface of the sphere, effectively sketching the curve on the surface of the sphere like so,

Project a Parabola onto a 3D sphere

Project a Parabola onto a 3D sphere

The sketch on the sphere looks like this.  Note the two symmetric curves.  This was just to make the later 2D projection easier.

Parabola Sketched on 3D Sphere

Parabola Sketched on 3D Sphere

Now project the curve on the sphere onto a plane.

Parabola Sketched on 3D Sphere Projected onto the Projective Plane

Parabola Sketched on 3D Sphere Projected onto the Projective Plane

The disk is called the projective plane and opposite points on the edge are identified – are the same. To get a good look at what happens at infinity just rotate the sphere and project.

Parabola on 3D Sphere Rotated and Projected on Projective Plane

Parabola on 3D Sphere Rotated and Projected on Projective Plane

So why not try this one dimension up.  Take a 3D figure, place it tangent to a 4D sphere, “sketch” the figure on the surface of a 4D sphere, and then project it in 3D. In theory this is not hard – just add a dimension.  So I wrote a Mathematica  script, built a crude 4D rotater and took a look.  Thus a paraboloid in 3D.

Paraboloid in 3D

Paraboloid in 3D

Now projected onto the surface of a 4D sphere and projected in 3D.

Paraboloid Projected on 4D Sphere

Paraboloid Projected on 4D Sphere

This is what I expected.  Since any cross-section of the paraboliod is parabola, all the cross-sections should “go” to the same point at infinity as shown. In this case, no rotation was needed.  Note the distortion of the grid lines. They will help later to figure out what is going on.

A natural next shape to explore is the hyperbolic paraboloid or saddle.

Hyperbolic Paraboloid in 3D

Hyperbolic Paraboloid in 3D

Unrotated it looks like this.

Hyperbolic Paraboloid Projected on 4D Sphere

Hyperbolic Paraboloid Projected on 4D Sphere

Those two points at infinity should be able to be identified. Remember a line’s “ends” are the same point at infinity in the projective plane. The figure should be able to be rotated so they are (look like) the same point.  This took some experimenting but here it is.

Hyperbolic Paraboloid Projected on 4D Sphere Rotated

Hyperbolic Paraboloid Projected on 4D Sphere Rotated

That is cool.  A cone is next.

Cone in 3D

Cone in 3D

This looks a little odd.  The important idea is that the surface extends to infinity along radial lines extending to different points of infinity.

Cone Projected on 4D Sphere

Cone Projected on 4D Sphere

The gird lines can be seen bunching up on the rim at all the different points of infinity.  No matter how I try I shouldn’t be able to rotate the figure so that any of the points identify.  I couldn’t.  Here is an attempt.

Cone Projected on 4D Sphere Rotated

Cone Projected on 4D Sphere Rotated

I explored a few other surfaces to test my understanding.  Here is a plane.

Plane in 3D

Plane in 3D

Unrotated, all the points at infinity can be seen on the rim.

Plane Projected on 4D Sphere

Rotating just resulted in distortion.

Plane Projected on 4D Sphere Rotated

Plane Projected on 4D Sphere Rotated

A parabolic surface looks like this in 3D.

Parabolic Surface in 3D

Parabolic Surface in 3D

Projected in 4D like this.

Parabolic Surface Projected on 4D Sphere

Parabolic Surface Projected on 4D Sphere

The infinity point shown is where all the parabolas converge, but all the horizontal grid lines should also converge.  I managed to rotate the figure to show this.  The point indicated is the “horizontal” infinity.

Parabolic Surface Projected on 4D Sphere Rotated

Parabolic Surface Projected on 4D Sphere Rotated

Lastly, here is a cubic surface.

Cubic Surface in 3D

Cubic Surface in 3D

Unrotated in 4D it looks like this.

Cubic Surface projected on 4D sphere

Cubic Surface projected on 4D sphere

Those points should be able to be identified.

Cubic Surface Projected on 4D Sphere Rotated

Cubic Surface Projected on 4D Sphere Rotated

At the end of this exercise I was just beginning to get a feel for rotating objects in four dimensions.  The shapes are aesthetically interesting and sometimes unexpected.

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Outcome Switching – What Golfers Know

Last Friday a rare event occurred at our golf course.  My foursome all sank long putts for birdies on the second hole.  What a great feeling and I could go on and on in detail but I won’t.  Anyway we get back to the clubhouse eager to relate our story and all we hear is  a collective, tepid “That’s interesting.”  What is up with that?

My theory is that golfers understand that rare events happen often.  They can’t predict which ones but they know they happen – a line drive tee shot hits the flagpole and drops in, a drive headed out-of-bounds hits a tree and lands in the middle of the fairway, or a ball hits a sprinkler head and bounds over the green.  They have seen it all.  Rare events happen.  Andrew Gelman points us to an article on Vox that discusses outcome switching in medical research.  The idea is,  you design a study to test if a drug works.  It doesn’t according to the statistical measures you set up, so you look for other outcomes.  Maybe it worked on a sub-population or patients lost weight or something and then you publish results for that outcome instead.  You switched outcomes after the fact.  Of course you are going to find something.  Rare events happen.  Golfers know that and don’t get too excited.  Apparently some medical researchers don’t and do.

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The Future of Math Textbooks – Lower Division Variety

Our department is looking at some open-source textbooks for our  statistics and precalculus classes.  This process elicited the following thoughts expressed in list form.

Facts on the Ground.

  1. Modern textbooks from mainline publishers are very expensive, $200 and more.
  2. Many students do not buy the textbook, even a used copy. They find a bootlegged copy online or try to get by with a library copy or without one.
  3. Reading a textbook is often a last resort. If a student wants to know about some algorithm, they check their notes and then turn to google.
  4. Explicit reading assignments with consequences – quizzes over the material or journal entries – are not commonly made by math instructors.
  5. Math textbooks are often chosen by instructors for their problems – number, difficulty and variety.
  6. Math textbooks are often chosen for their ancillaries – online homework systems, test banks, detailed instructor outlines, etc.
  7. Math textbooks have many pages of explanations for many varieties of problems. This is a result of trying to be all things to all people (instructors).
  8. Lower level math textbooks have not materially changed in fifty or more years. They cover the same material in substantially the same way.  Instructors learned the material that way and are comfortable teaching it in the same way.

What Instructors Get from a Commercial Textbook

  1. A carefully written and edited reference for students and themselves.
  2. Carefully sequenced explanations.
  3. A source of professional quality diagrams.
  4. A source of problems with answers – answers to odd problems for the students, all answers for the instructor.
  5. Extra help for their students.
  6. An outline for the material to be covered.
  7. A pace – one section per day though not always.

What Students Get from a Commercial Textbook

  1. See what instructors get.
  2. Several pounds of paper to lug around in a sagging backpack.
  3. A $200 expense partially recovered by selling the text back to the bookstore.

Conclusion:

Marketplace forces have constructed a system of textbooks with lots of bells and whistles which students can’t afford.   The upshot is that students avoid math classes, use bootleg copies and answer keys, and select instructors based on the cost of the textbook.  State legislatures are considering incentives to encourage cheaper texts. In response math departments are looking at open-source textbooks.

The future:

The math textbook will become obsolete going the way of the LP and CD in the music industry.  Smartphone apps for writing math are developing. YouTube is teeming with math videos.  Free complete math courses can be found online. Professors are posting their notes with extensive hyperlinks. People are building mathematical images, graphing tools, and math concept demonstrators.  Math instructors will become curators picking and choosing from the myriad of internet math artifacts.  Master curators will develop – the DJ’s of the math world.

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Thinking with Units

I sign up at the golf course as a single and am paired with a couple of strangers.  On the second tee the inevitable question arises, “What do you do?”  I return, “I teach math at the university,” expecting to hear, “I was never good at math but my niece (daughter, grandson) got straight A’s” and such. Instead Norm says, “I know forty divided by one-half plus fifteen is ninety-five because the computer says it is, but how can eighty halves plus fifteen wholes add up to ninety-five.  The units don’t work. How can you explain that to a kid with candy bars?”  My jaw (mentally) dropped.  I told them I love arguments about units and that I might be quiet for a while but I would explain it before we got to the eighteenth hole. [I am sure this is some internet controversy somewhere.]

We tee off and while we are waiting to hit the next shot, I explain, “You get into a taxi.  Just for sitting there they charge you fifteen dollars.  They charge one dollar for each half mine. You go forty miles.  What do you pay?”  Both said, “Ninety-five dollars,” immediately.  “Yes,” Norm says, “But you can’t explain taxi rides to a kid.  What about the candy bars?”

I walk to the third hole, some two hundred yards away.  There is time to think since those two drive ahead in a cart.  As we are waiting to tee off, I say, “Little Joey is having a birthday party.  Lots of kids are coming.  He wants to give each kid a half of a candy bar.  So he sits at the kitchen table cutting candy bars in half.  He cuts eight bars in half and eats of the halves.  He has forty bars left to cut in half.  How many guests did Joey invite to his party?”  Both golfers calculate ninety-five half bars and therefore ninety-five guests.  Norm is still skeptical but we have a pleasant game for the rest of the round, most of the conversation of the “good shot”, “nice putt”, “you’re away” variety.

Those two old guys knew how units worked, I am sure, based on life experiences.  The key to a suitable explanation is noting that the fifteen counts half bars the same as forty divided by one-half counts how many half bars are in forty whole bars.  The units of the one half is whole bars per half bars: one whole bar per two half bars.  Now let’s play some more golf.

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Making Myself Sad

Written this morning.

I am making myself sad today.  Giving a test.  While they take it instead of catching up on my reports, I am exploring a list of topics (it goes without saying) on the internet.  Most were from MAA Focus.  I looked at formative assessment lessons, IBM quantum computing, and Boy’s surface.  There is so much interesting stuff out there that I want know and try and so little time.

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Infinity

Infinity

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Too Old for This?

I was going over a written assignment: “Using a calculator, graph f(x)=(x-1)(x-2) and g(x) = |(x-1)(x-2)|.  Describe and explain the difference.”  Several students said there was no difference and produced these graphs copied from their calculators.

Graphs From Afar

Graphs From Afar

To demonstrate the error of their ways, I stated that Rock Hudson and I look the same from a distance (a very far distance) but, when you zoom in, we are different.  What?!  How would they know anything about Rock Hudson, a  1950’s movie star. So I changed it to Brad Pitt who probably is still just some old man to them.  I went back to my point:  different functions (algebraically)  => different graphs. So look closely and explain the difference: particular character of the function expression (in this case, absolute value) => particular look of the graph (in this case, all on or above the x-axis, negative values reflected about the x-axis.)

Graphs Closer

Graphs Closer

I will be introducing the standard equation of a circle today asking, “What is a circle?” Usually some student will say it is round.   I hope I can refrain from singing, “Round thing, you make my heart sing, ” with the appropriate phrasing and a slight twang.  I think I have become quaint.

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