Newtown Connections

Route and design consultation on the Newtown Connections cycleway project is now underway on the WCC website. I don’t think most of my submission is of particular interest (I ended up supporting option C but with a note that I think two-way cycleways need particular attention to design at the ends or more experienced cyclists will ignore them and stay on the road), but I do want to briefly talk about Mt Albert.

The two full options (option B was barebones) included cycleways up Russell Tce, ending around the hockey stadium. I think that it would be great if Mt Albert Rd, or the various footpaths paralleling it, were upgraded to allow people to cycle easily up to the lookout at the top of the hill.

IMG_20160418_115222

On this image (from Hawkins Hill), the proposed cycleway on Russell Tce is shown in red. The slope up to Mt Albert hill is reasonable (i.e. not any worse than Mt Vic, and probably doable for the majority of people on a road bike) – a cycleway could either follow the road, or go up one of the walking tracks that already exists (these are quite a bit steeper, though). The area at the top has been recently redone (the council refurbished the water storage tanks) and features wide expanses of gravel and a very nice lookout over Cook Strait – the tracks are paved and wide enough for a shared path, and are a very nice grade (they were built for trucks, after all).

One could even upgrade Houghton Bay Rd, and there would be a nice safe scenic route from Newtown to the sea down that way, creating a loop together with the Island Bay cycleway.

Anyway, that’s just an idea.

Advertisements

Scout Halls in Upper Hutt

Our regular programming will resume shortly. Please bear with us as we post a post that should preferably have been posted in the past.

I have a couple of things to post; first of all, a set of activities that I wanted to run for scouts and/or venturers but never got around to it which I post here verbatim with minimal editing in the hope someone else finds it useful:-

  • Regional capture-the-flag (one flag on Cannon Point, the other at Hawkin’s Hill/Seddon Memorial/comparable location). This would take place over a weekend, and would require some organisation!
  • Hillbilly Peak to Climie hike (2-3 day?) – as far as I know there are only two people who know the tracks up to Hillbilly Peak and through to Brookfield, and the author is the only one who knows the tracks beyond the start of the water catchment area fence. 

    This slideshow requires JavaScript.

  • Night spy game. Two groups of spies in CBD, one group trying to catch the other carrying secret documents from (say) the Beehive to MBIE (no, that’s too close:- Beehive to Malaysian high commission?)
  • Related note. Race between every embassy with country-related bases at each? (Race between every ___ with ___-related bases at each? :P)
  • “Civil defense drill” (i.e. different teams, “water”, “electricity”, “communications”, “medical aid”, etc. who start at a central point in the middle of nowhere (AKA Miramar) and have to work out how to get their given utility back to that base as quickly as possible in some creative way: communications, for example, might have to get a voice link between base and the govt precinct without using electronics).
  • Southern crossing to Akatarawa saddle.
  • CHEESECAKE TRAMP!
  • PL camp for bonding and teamwork games/activities. Kaitoke?
  • Bridges at Silverstream? (Although we might loose a small one or two).
  • Explore the Western hills.
  • Pencarrow cycling trip (easy); full river trail from Te Marua to Seaview? (I have done that, it was a fun day.)
  • Zombie game at midnight in a cemetery.
  • More generally, midnight hike. (Not necessarily through a cemetery.)
  • City-wide scavenger hunt.
  • Codes and ciphers, but fun. (editor’s note: Says the mathematician :D.)
  • Akatarawas, cross-range hike.
  • Group raid, Wellington.
  • Group raid, Porirua (but the nice part).
  • Kaitoke airfield?
  • Mapping an area (Witako, perhaps).
  • Building a hydro generator? (Should be fairly straightforward, it only needs Y12 physics.)
  • Building a battery  (even easier, just needs lemon juice, salt, some nails, and a lot of bush chemistry.) (editor: See experiment 6.1 on p 59 of these notes).
  • Explore the Kawekas and/or central Hawke’s Bay.

Secondly, as I have promised, I am posting my collection of photos from 2014 of the then-extant scout halls in Upper Hutt. This was prompted by the recent (in the last couple of years) demolition of the old Silverstream scout hall in Dunns Park. Apologies, it is just a copy-paste of a word document I wrote when I was actively doing research on this subject in relation to badge- and scarf-collecting in 2013-4.

If anyone is interested, I have a large collection of scouting-related files (mainly patrol leader training stuff, as well as badge workbooks and so forth) that are currently just sitting on my google drive because I am no longer actively involved in scouting (temporarily?). The author has made literally no effort to verify whether information written below is still valid, but is happy to put readers in touch with people who are still actively involved in the area.

FUN EXCITING FACT: The author suggested the name “Akatarawa” to the former SPL of Rimutaka (shoutout to Jason) and was rather surprised to have it accepted. Bonus fun fact: the original plan was for the merged group to be called Pakuratahi, after one of the groups which merged to form it. However, those from the area will be aware that the Pakuratahi River valley is actually the one which SH2 crosses at Kaitoke Regional Park (the big bridge just before the Kiwi Ranch Rd turnoff), and is nowhere near the Hoggard Den on the Emerald Hill bend in the Hutt River.

Former Scout groups in Upper Hutt

Name

Founded

Closed

Scarf colours

Location

Cannon Point

19571

2005 merged to form Heretaunga

Cannon Point den, Savage Reserve

Kaitoke

Pakuratahi

Kaitoke Hall2

Pinehaven

1999 merged to form Silverpine

Solid red

Pinehaven den, Pinehaven Reserve

Red Shield3

Rimutaka

2013 merged to form Akatarawa

Brown with white trim

Haggard den, Black Beech Street

Silverstream

1999 merged to form Silverpine

Grey with green trim

Silverstream den, Dunns Park

St John’s

August 1919, disbanded 1945, reformed May 1961

2005 merged to form Heretaunga

St John’s Church, Moonshine Road (hall now known as the Molly Newman room)

Te Marua

19604

Plateau Road?

1st Totara Park

2013 merged to form Akatarawa

Blue with yellow trim

Totara Park den, California Park

1st Upper Hutt

The Blockhouse den, McHardie Street

1Upper Hutt Leader 7 November 1957 – http://newspaperarchives.uhcc.govt.nz/cgi-bin/upperhutt?a=d&d=UpperHuttLeader19571107.2.12 (retrieved 28 November 2014)

2Upper Hutt Leader 28 June 1962 – http://newspaperarchives.uhcc.govt.nz/cgi-bin/upperhutt?a=d&d=UpperHuttLeader19620628.2.22 (retrieved 28 November 2014)

3Upper Hutt Leader 14 May 1964 – http://newspaperarchives.uhcc.govt.nz/cgi-bin/upperhutt?a=d&d=UpperHuttLeader19640514.2.43 (retrieved 28 November 2014)

4Upper Hutt Leader 14 July 1960 – http://newspaperarchives.uhcc.govt.nz/cgi-bin/upperhutt?a=d&d=UpperHuttLeader19600714.2.21 (retrieved 19 October 2014)

Current Scout groups in Upper Hutt

Name

Founded

Scarf colours

Location

Akatarawa

2013 from Rimutaka and Totara Park

Fluorescent green with black trim

Haggard Memorial den, Black Beech Street

Silverpine

1999 from Silverstream and Pinehaven1

Grey and red (half and half)

Pinehaven den, Pinehaven Reserve

St Josephs

19362

Black with white trim

St Joseph’s Scout and Guide Den, Royal Street

Heretaunga

2005 from Cannon Point and St John’s3

Orange with black trim

Heretaunga den, McLeod Street

Fell Rovers (currently exists on hiatus)

Silverstream den, Dunns Park

1http://uhcl.recollect.co.nz/nodes/view/14293 (retrieved 19 October 2014) says that the group was formed in March.

2http://www.stuff.co.nz/dominion-post/news/local-papers/upper-hutt-leader/5080105/Events-mark-parish-century (retrieved 19 October 2014) says that 2011 is the 75th anniversary of the foundation.

3http://www.stjohnstrentham.org.nz/html/history/html/childyouth.html (retrieved 19 October 2014)

Groups

The Fell Rovers

The Fell Rovers was never a particularly strong Rover Crew – most Venturers from Upper Hutt moved on to study at Victoria University in Wellington and the Vic Rovers. In 2014, the then-Zone Leader decided to close the Crew as it only had one or two motivated members. It was decided to try again in a year and see if any success could be met.

1st Upper Hutt Group

The Blockhouse den is currently used to store costumes for the Hutt Valley Gang Show.

Group mergers

Akatarawa Scout Group merger 2013

The merger in 2013 occurred because of the very small number of members at both Rimutaka and Totara Park groups (12 cubs and 12 scouts from both), as well as a lack of leaders.1 The Totara Park group had recently (2012) restarted their Scout troop after a leader moved there from Silverpine Scouts, although they did not attract many members (3-5 at scout age were enrolled).

Heretaunga Scout Group merger 2005

Cannon Point and St John’s Groups (both church groups – managed by St Hilda’s and St John’s respectively) merged in 2005 after dropping numbers. The newly formed Heretaunga Group took on the name and badge of the former Heretaunga Region (which had merged with Upper Hutt Region over a decade earlier).

Some members of the Cannon Point Group joined St Josephs Group as it was much closer (just across the road).

The Heretaunga Den was chosen as the hall for the new group. The author was unable to find any information about which group was previously occupying it, however it is likely that it was St John’s as they may have decided to move out of the St John’s church hall.

This badge (with some modifications) was also used when Silverpine and Heretaunga formed a contingent (Silvertaunga) to the 2010-2011 New Zealand Jamboree in Mystery Creek (Adventure Jam).

Of course, in 2013-4 Silverpine went with Lansdowne and the author was quite impressed with their single patrol (this was the contingent for which a friend of the author’s drew the Silver Dragon badge); in 2016-7, Silverpine and Heretaunga went together once again and the draft contingent name was the “Hairy Pineapples”. Something a little more mundane was eventually chosen, but it escapes the author as they had a management role in the YST rather than staying with their group.

Silverpine Scout Group merger 1999

Silverstream and Pinehaven Groups merged in March 1999. This was due to dropping numbers in both groups, and the Guide groups merged several years later (originally in St Mary’s church by Gloucester St and then in the Pinehaven community hall).

The newly formed Silverpine Group took on a half-and-half scarf of red and grey – the red for the old Pinehaven scarf, and the grey for the old Silverstream scarf. Between 2008 and 2016, the grey used changed to become darker.

It should be noted that the Pinehaven scarf previously had a pine tree badge on the peak. This was not carried over. The Silverstream scarf had no badge.

In 2015-6, a new group badge was drawn and is now on the peak of the group scarves.

13th New Zealand Scout Jamboree

uhj

Illustration 1: The badge for the Upper Hutt jamboree

The 13th New Zealand Scout Jamboree was held at Trentham Memorial Park in 1993. My understanding is that the river flooded its banks. Bases were held at Kaitoke and in Whiteman’s/Mangaroa Vlys to my knowledge.

Venturers in Upper Hutt

For many years, the only Venturer unit in Upper Hutt was 1st Upper Hutt Group at the Blockhouse den. However, at the time of writing (2014), the primary Venturer unit is at St Josephs Group. Heretaunga Group also has a Venturer unit, although it has been experiencing trouble in recent months, with numbers dropping down to two or three Venturers.

Rimutaka Zone

rimutaka

Rimutaka Zone was formed in 2007 when the new Region/Zone system was introduced. The Zone was a merger of the Upper Hutt and Wairarapa Regions. The new Zone badge featured the Rimutaka Ranges and the upper Hutt Valley, with the railway, river and road prominent.

The Zone was formed from the Wairarapa Region primarily because it was unable to manage itself – even as a Region, it was managed by Palmerston North. This part of the region has always struggled to stay afloat, although recently the Greytown Group Venturer Unit has been growing in size.

Gang Shows

International Gang Show, Trentham Jamboree: 1966
Upper Hutt District: 1968

Former regions

Upper Hutt

upperhutt

http://newspaperarchives.uhcc.govt.nz/cgi-bin/upperhutt?a=d&d=UpperHuttLeader19540826.2.27

Heretaunga

heretaunga

There was also a short-lived Heretaunga region.

1http://www.stuff.co.nz/dominion-post/news/local-papers/upper-hutt-leader/8685237/Scout-groups-face-merger-or-closure (retrieved 19 October 2014)

Photographs from 2014

Pinehaven:

 

Silverstream (since demolished) – was actually home to Pinehaven until the latter split into Pinehaven and Silverstream, if I recall my reading correctly:

 

Blockhouse den (Heretaunga College):

 

Cannon Point:

 

St Joseph’s (original den burned down, arson):

 

Totara Park:

 

Birchville (Hoggard): Fun fact. This den hosted cub 9 when the author did it many years ago. At the time, cub 10 was run at Mt Holdsworth; now all of them are held at Brookfield at Smith’s camp, and cub 8 is held at Te Runga. Now where did I do Cosgrove? I remember doing Sandford at Foxton, but I have a sneaking suspicion I did Cosgrove here too. You couldn’t do that now! (The old Picassa photo pages have been taken offline, but I did it in 2010 if anyone who knows me wants to go digging.)

 Edit: the author has remembered.(!) They did Cosgrove at Rimutaka, and Cub 9 at the Masterton air scouts.

Heretaunga den (AKA the enemy):

 

 

Mathematical Diary (#008)

Courses for next year

I am planning to enrol in the following mathematics courses: 341, 334, 315. (I will also be doing PHYSICS 201, 203, 244, 245, and 333; and STATS 125.)

I have a bad habit of looking at course reviews online for courses I’ve taken (not that I take any notice, but I find them amusing to some extent). Here are some of the highlights (my notes are in italic):

  • “If you take this course… the questions asked are actually pretty easy once you know how to do them, the hard part just comes from learning how to do them”
  • “when I die I want people I’ve done group projects with to lower me into my grave so they can let me down one last time”
  • “Easiest paper of my life! The prerequisites are a joke, if you passed Maths 108 this’ll be a breeze — you might not even need that. None of the other maths papers here give such an intuitive and clear explanation of basic concepts, such as continuity, limits and the chain rule.” (on MATHS 333)

Calculus books

It has recently come to my attention that I have a reputation for disliking every calculus book ever written. I would like to spend some time dispelling this rumour, by listing my favourite calculus texts.

Introductory non-rigorous book. Calculus Made Easy by Silvanus Thompson [515 THO]. This is the book I originally learned calculus from (in year 11, or around 2013-4). It is a little old fashioned, but the intuition in here is very clear and the author has a sense of humour! This is perhaps the book I would recommend to the Y12 student or the less able Y13 student if they want some reading material. It covers various integration tricks (or ‘dodges’, as Thompson calls them), as well as some geometry (lengths of curves, and curvature) and gives reasonable intuitive justifications for the rules.

Thompson is unsuitable for even first year university, because it is far from rigorous (it doesn’t mention limits, although the recent editions have a foreword and initial chapters by Martin Gardner which do cover them to some extent) and is too informal to really be a good introduction to mathematical thinking. Indeed, my favourite introductory grown-up calculus book is, as you can probably guess, the One True Calculus Book:

Introductory university book. Calculus, by Michael Spivak [515 S76]. Do I really need to say any more? Indeed, this book is more of an introductory real analysis book than a ‘standard’ calculus book. I don’t believe any university in NZ actually uses it as a course text in first year (or at all), but that is no excuse for students not to use it (there always seems to be at least one copy out of the general library despite not being an official text).

22780348_10156926288418228_5225309319067129173_n
(Relevant pic from MMM)

Highlights include a very readable motivation for completeness of the reals and \varepsilon\delta proofs; most exercises are also very interesting. I would avoid the complex analysis chapters at the end, but beyond that there are no real faults with this book. (The University of Toronto is one university that uses this book for their flagship first-year course, which has a reputation for being a trial-by-fire for new mathematics students.) One final bonus: it is cheap (well, not really, but cheaper than the shiny calculus books that the book shop sells) and concise (again, it is a doorstop, but much more concise than the texts that try to include absolutely everything from biology examples to Stoke’s theorem). [I also find Lang’s book quite all right.]

Baby Spivak doesn’t include multi-variable calculus; the “standard” sequel to Spivak in this regard is Daddy Spivak (Calculus on Manifolds [515.84 S76]). However, I find this concise introduction a little too concise (and that it introduces technicalities in all the wrong places). Munkres’ Analysis on Manifolds [515.84 M96] is an expanded and updated book which is (anecdotally) based on Spivak; but I have never liked it that much. My favourite multi-dimensional calculus book, is, in fact:

Advanced calculus and manifolds. Advanced Calculus, by Lynn Loomis and Shlomo Sternberg [515.8 L86]. This book started off as notes for the legendary Math 55 at Harvard; it starts off with a recap of linear algebra, and ends up around 500 pages later with a very nice approach to differential forms. This is my favourite calculus text of all time, but it is probably a little too much for students in first year! (Or even students who have completed 253, to be honest… perhaps 255 is enough preparation? The preface recommends baby Spivak as prep.) Highlights include a rigorous (and clear) treatment of infinitesimal functions and differentials, and a final chapter on applications to theoretical physics. Sample exercise from the first section: ‘Show that a mapping T from a vector space V to a vector space W is linear iff the graph of T is a subspace of V + W ‘.

If L&S is a little too sadistic for you, I would actually recommend a less well-known vector calculus book: Vector Calculus, by Peter Baxandall and Hans Liebeck [515.84 B35]. It is very geometric, contains proofs similar in level and style to those in 253, requires the same level of linear algebra, and covers Gauss’ and Stokes’ theorems after Green’s theorem (useful for physics – and in fact it uses electromag and fluid flow as motivational examples throughout). It even ends up stating the real Stokes’ theorem, the one involving differential forms:- however, it does so in  around twenty pages right at the end, only does it in three dimensions, and doesn’t really motivate it that well. Oh well, at least it’s there…! (It really is a very good book though, apart from that.)

Gearing up to LGWM report?

News this week that NZTA has confirmed the Peka Peka-Otaki section of the Kapiti expressway add to my suspicions that that MoT is stepping up to an announcement of the LGWM working group findings. The report was supposed to be released last month but was shifted back: “The LGWM Governance Group is currently engaging with central government, and is targeting to approve the recommended programme for release in October.” There are only six days left until the end of the month, so hopefully we hear something soon.

I am also watching the Dom Post’s current proliferation of stories on the bus network with something approaching interest, but I don’t currently feel a need to add more noise into what is basically at this stage a whole bunch of people shouting the same things at each other over and over again.

Conic sections and projective geometry

This was originally going to be the first part of my new conic section notes, but on reflection it’s a little difficult for Y13 students. Thus, because I do think it is interesting, I post it here. (See also, the previous post.)

Quadratic Equations (Again)

You know, for a mathematician, he did not have enough imagination. But he has become a poet and now he is fine. — Hilbert (on an ex-student)

We will study in this topic precisely equations of the form

\displaystyle ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0. \ \ \ \ \ (1)

 

This class of equations is the class of quadratic equations (in two variables). Note that, without loss of generality, we can assume { c = 1 } (otherwise, we just divide through by { c }).

The set of all points { (x,y) } satisfying a quadratic equation is called a curve of the second degree.

Our first theorem is a generalisation of the statement from algebra that every quadratic equation in one variable (i.e. every equation { x^2 + px + q = 0 }) has precisely two solutions (if you count correctly).

Theorem. Every line meets a curve of the second degree in precisely two points.

This theorem is very nice, but (like the proof for quadratics in one variable) we do have to make a trade: we have to take our coordinate system to be over the complex numbers. Thus, for the remainder of these notes, we will assume implicitly that rather than living in the real plane { \mathbb{R}^2 } we are in fact living in the complex plane { \mathbb{C}^2 }. In some sense, then, we are doing geometry in four dimensions. On the other hand, in many places we will only be interested in real coordinates. For example, the graph of a four-dimensional object is a little tricky to draw if we only have three dimensions available!

Our first proof attempt looks all right, and indeed it is almost correct;

Proof: Let { ax + by = 1 } be our line. At least one of { a } or { b } is non-zero; assume that { a } is non-zero (but essentially the same proof will work if { a = 0 } but { b } is non-zero). Then we can substitute { x = \frac{1 - by}{a} } into our curve of the second degree to obtain a quadratic equation in the variable { y }; by the fundamental theorem of algebra, this equation has two solutions; and each of these corresponds with a point of intersection. \Box

The problem with this proof is illustrated by the following example:- Consider the quadratic { xy + y + x^2 = 1 }. This describes a perfectly good curve in two dimensions, graphed here.
Screenshot_20181021_211052
If we intersect this curve with the line { y = 2 - x }, then we obtain a linear equation { x + 2 = 0 } instead of a quadratic equation — and so even counting multiplicities and moving up to { \mathbb{C} }, we only have one solution.

The reader might be tempted to discount this example as uninteresting, because it’s simply the intersection of a pair of two lines with a third line and thus isn’t really a problem that affects our programme of studying curves of the second degree.

The next example, though, is both more fundamental and more worrying. If we intersect the parabola { x^2 = y } with the line { x = 0 }, we obtain a single point of intersection, even up to multiplicity.

Does this mean that we have to abandon the beautiful intersection theorem?

It turns out that, like how we expanded our number system to make the fundamental theorem of algebra work nicely, we can expand our plane { \mathbb{C}^2 } further to make our intersection theorem work.

We will not make this precise, but the method we will use is some kind of limiting process; the result, and the system we will work in from now on, is the projective plane over { \mathbb{C} }. Let us see how we climb to it.

Euclidean Geometry

It turns out that in order to approach some kind of resolution, we need to mention the elephant in the room: the foundation of all the geometry which we have been doing thus far in our mathematical lives. (This might seem at first glance completely unrelated to the algebraic problem we’re trying to solve, but bear with me.)

In ancient Greece, in about 300~BCE, Euclid of Alexandria (in Egypt) wrote his book Elements, one of the most influential mathematical texts of all time. In it, he set down the set of basic results which can be proved true about circles and lines on a plane: while modern geometry has gone much further than Euclid (in considering geometry in more than two dimensions, and on surfaces much more complicated than the plane), the basic results which are taught in any introductory geometry course are usually treated in the Elements.

Euclid was influential because he was the first author (or, more likely, the first author whose work survives) to attempt to rigorously prove all his results based on a set of `universal truths’: a set of simple statements (called axioms) whose truth is self-evident. In some sense, the goal of mathematics is to deduce the behaviour of the most complicated systems possible from the simplest axioms possible. (The axiom system which most of modern mathematics is done within is called ZFC (Zermelo-Fraenkel with Choice, named after two mathematicians working on the area in the early 1900s).) Euclid’s axioms were as follows:

  1. A straight line segment can be drawn joining any two points.
  2. Any straight line segment can be extended indefinitely in a straight line.
  3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
  4. All right angles are congruent.
  5. Given any straight line and a point not on it, there exists one and only one straight line which passes through that point and never intersects the first line, no matter how far they are extended.

Putting aside the fact that these axioms are in fact insufficient for the purpose of constructing plane geometry, it is clear that the fifth axiom is both far less elegant than the others and far more complicated. Indeed, Euclid himself avoided using it in his proofs except where necessary (he avoids it completely for the first 28 propositions, for example) and for many years mathematicians believed that it was in fact a theorem: that it could be proved from the previous four axioms.

It turns out that this is false: there exist geometries for which the fifth axiom is false (for example, on a sphere).

Projective Geometry

The mathematical world which we are interested in is known as projective geometry (the name will only make sense towards the end of these notes). We obtain this geometry by replacing the final axiom with the following axiom: Any two distinct lines in a projective geometry meet at exactly one point. This is clearly not true in usual Euclidean geometry. On the other hand, we didn’t let this kind of thing stop us in the past: the number { i } is absolutely useless for counting sheep, for example, but that doesn’t stop the complex numbers from being an interesting (and useful) mathematical space.

The complex numbers became much clearer once we stopped thinking about numbers as a line and started drawing them as a plane. There is an analogous simple way to construct and think about a projective plane by extending the Euclidean plane (and then extending our setting, the complex plane, will be done in the same way).

When we think about it, we find that the only reason that Euclidean geometry doesn’t satisfy this axiom is that parallel lines exist. We will therefore define them not to exist!

More precisely, given every set of parallel lines with slope { m }, we will take our plane and add on a new point { \infty_m } called a point at infinity, and then define this new point to be on all the lines with slope { m }. Thus, these parallel lines now intersect at exactly one point (the point at infinity related to their slope) and all other lines intersect in exactly one point (the same place they did before).

This can be made to make some sense physically, and indeed the first place which projective geometry appeared was when Renaissance artists like Leonardo da Vinci began to experiment with perspective drawing — if you look down two parallel lines in real life, they appear to meet at the horizon, and lines going in the same direction away from you (i.e. lines with the same slope) meet at the same point. We call the line joining all the points at infinity (in this physical case, the horizon) the line at infinity.

Going back to our problem of curves of the second degree, let us consider our second example: { x^2 = y } only intersects the line { x = 0 } at one point.

Screenshot_20181021_210802

But notice, as { x \rightarrow \pm \infty } the curve { x^2 } tends towards being vertical! More precisely, we have that { \lim_{x \rightarrow \infty} \frac{\mathrm{d}}{\mathrm{d}x} x^2 = \infty }; thus, the parabola is eventually parallel to the vertical line, and so the two intersect for a second time at the point at infinity { \infty_\infty }.

This is even more blatant in our first example: we had the following curve (and again, the blue line is the line we intersect with our second degree curve):

Screenshot_20181021_210954

In this case, the curve is not only eventually parallel, but in fact it is actually parallel!

The picture to keep in mind, for conic sections at least, is the following.

infiniteparabola

Note that this picture does not actually reflect what is really going on when we graph second degree equations in the plane — it’s a diagrammatic representation of the fact that we’ve made a fundamental extension to our system of geometry (and to our system of algebra, but this isn’t visible on a real graph).

Philosophically, all we’ve done is decided what property we want our algebro-geometric system to have (that curves of the second degree intersect any line at exactly two places, unless it’s tangent to the line) and then invented a geometry in which this property is true. It turns out that this geometry has a lot of other beautiful properties, most of which we don’t have time to look at — but at this stage, it’s completely synthetic. We simply arbitrarily added a bunch of points (and we’re honestly lucky we didn’t get any contradictions). In fact, eventually we will see that there is an incredibly natural way to construct a projective plane without having to worry about sticking in a bunch more points and cooking it ourselves. For now, we will content ourselves with proving our motivational theorem.

Proof of Theorem

Let { y = mx + n } be our line. Then we can substitute { y } into our curve of the second degree to obtain an equation in the variable { x } which is either of degree 1 or degree 2. (If necessary, let { m } be infinite.)

Case I. If the equation is of degree two, then by the fundamental theorem of algebra it has two solutions; and each of these corresponds with a point of intersection.

Case II. We will be done if we can show the following: if a line { y = mx + n } intersects some quadratic equation { ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 } in exactly one point (i.e. in such a way as to produce a linear equation in { x } rather than a quadratic) then

\displaystyle \lim_{x \rightarrow \infty} y'(x) = m. \ \ \ \ \ (2)

Suppose { \frac{\mathrm{d}y}{\mathrm{d}x} } does not tend to a finite number. If { \frac{\mathrm{d}y}{\mathrm{d}x} } goes to infinity, then in particular it intersects every non-vertical line at two points (draw a picture), so if the line intersects at precisely one point then it is vertical; thus if { m = \infty } then { \frac{\mathrm{d}y}{\mathrm{d}x} \rightarrow \infty }. On the other hand, { \frac{\mathrm{d}y}{\mathrm{d}x} } is either eventually positive or eventually negative (just by inspecting the expression for it above), so it either tends to infinity or to some finite limit.

We will now calculate the limit which it tends to, if it tends to some finite value.

Step 1. We will find the condition on the coefficients of our quadratic for { x^2 } to vanish upon substitution. So let us substitute { y = mx + c } into our quadratic; we obtain ax^2 + 2hx(mx + c) + b(mx + c)^2 + 2gx + 2f(mx + c) + c = 0 .

Expanding, and setting the coefficient of the { x^2 } term to zero, we find that a + 2mh + bm^2 = 0.

Step 2. We will find the derivative { y'(x) }; this is a simple exercise in implicit differentiation and we get that \frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{ax + hy + g}{hx + by + f} . Substituting our condition on the coefficients, we have \frac{\mathrm{d}y}{\mathrm{d}x} = -\frac{-2mhx - bm^2x + hy + g}{hx + by + f}.

Step 3. Taking limits of both sides, we have \lim_{x \rightarrow \infty} \frac{\mathrm{d}y}{\mathrm{d}x} = \lim_{x \rightarrow \infty} -\frac{-2mhx - bm^2x + hy + g}{hx + by + f} .

It is not immediately obvious how we should deal with this right-hand side, because { y } also depends on { x }. Looking at the equation for the quadratic, we note that as { x \rightarrow \infty }, we must have { y \rightarrow \pm\infty } — otherwise the right hand side couldn’t stay at zero. Therefore, both the top and the bottom of the fraction under our limit go to infinity; we can apply the following theorem (which we will not prove here):

Theorem. If { \lim_{x \rightarrow \infty} f(x) = \infty } and { \lim_{x \rightarrow \infty} g(x) = \infty }, and if { \lim_{x\rightarrow\infty} f'(x)/g'(x) = l }, then { \lim_{x\rightarrow\infty} f(x)/g(x) = l }.

Proof: see Spivak, chapter 11, theorem 9 and exercises 36-38. \Box

Let { \lim_{x \rightarrow \infty} \frac{\mathrm{d}y}{\mathrm{d}x} = L }. Then we have

L = \lim_{x \rightarrow \infty} -\dfrac{-2mhx - bm^2x + hy + g}{hx + by + f} = \lim_{x \rightarrow \infty} -\dfrac{-2mh - bm^2 + h\dfrac{\mathrm{d}y}{\mathrm{d}x}}{h + b\frac{\mathrm{d}y}{\mathrm{d}x}} = -\dfrac{-2mh - bm^2 + hL}{h + bL}

and

{ 0 = bL^2 + 2hL - 2mh - bm^2 },

so

L = \dfrac{-2h + \sqrt{4h^2 + 4b(2mh + bm^2)}}{2b} = \dfrac{-2h + 2\sqrt{h^2 + 2bmh + bm^2}}{2b} = \dfrac{-2h + 2(h + bm)}{2b} = m .

Thus we have shown that if the derivative { \frac{\mathrm{d}y}{\mathrm{d}x} } tends to any value, it must tend to { m }.

Q.E.D.

 

Mathematical Diary (#007)

Philosophical notes about lectures

Before I went on exchange, I didn’t think there was such a thing as a stupid question (well, maybe the following).

Screenshot_20181012_143142

But I was in a real analysis lecture this afternoon and the lecturer was talking about how the Euclidean norm on \mathbb{R}^n is the unique norm such that the generalised Pythagorean identity holds for right triangles holds, and someone at the back puts their hand up and said the following:

I don’t see why this is this interesting! Why is the Pythagorean identity so important a thing to satisfy?!

My initial reaction was, of course, if you don’t find this kind of thing interesting then why are you doing a mathematics course?! On the other hand, I think there is something interesting to unpack here: namely, there are two philosophical points which it raises.

  • Why do we prefer some structures (e.g. the Pythagorean identity) over others (say, we could define a norm to be “nice” if it satisfies the Pythagorean identity but replacing powers of two with powers of 22/7 )?
  • Why is the idea of uniqueness so important in mathematics?

In the specific example of the Pythagorean identity, we have a simple answer to the first question: it is the natural generalisation of our concept of distance measurement from one, to two, to three dimensions, so it’s nice for it to hold in n dimensions. More generally, I think that structures tend to be “nice” in the sense that I mean above when they satisfy one of the following properties:-

  • They behave in a particularly ‘simple’ way – for example, linear spaces.
  • They are a natural generalisation of a model of the real world –  for example, Euclidean distance.
  • They transform one notion into another seemingly distinct notion without the loss of information.

The second point there is rather disappointing because it depends on what the ‘real world’ is like, but the other two are quite nice because we can formalise them in terms of category theory.

The other question I want to consider is about uniqueness, and I think it’s so obvious that I don’t have too much to say: if something can be categorised to the point of being unique, then it is well-understood and well-behaved structurally. For example, we have the following uniqueness theorems:-

  • \mathbb{R} is the unique complete ordered field.
  • \mathbb{Z} is the unique infinite cyclic group.
  • Every field F has a unique algebraic closure.
  • There are unique well-behaved ways of defining product, subspace, and quotient topologies based on continuity.
  • There is a unique decomposition of (integers|finitely generated groups) into (primes|cyclic groups and a free group).
  • etc.

Indeed, describing a particular object (or isomorphism class of objects) is precisely the act of either constructing it or providing a list of properties which characterise it uniquely (and then proving that it exists).

Transport news this week

Mainly a lot of trains:

  • Funding has been approved this week by the Beehive for upgrades to the Wairarapa Line, including double-tracking from TREN to UPPE and $50m for “infrastructure upgrades” on the plains side of the tunnel.
  • The Wellington Station throat is also going to be redeveloped, along with line renewals elsewhere. By “new and longer trains”, does Twyford mean more eight car Matangis, or the nine car Wairarapa trains becoming standard, the new rolling stock for rural routes that GWRC asked for funding for in the RLTP, or just futureproofing?
  • Trains have started to move in daylight hours on the Main North Line through Kaikoura.
  • The western footbridge at AVA has reopened after being closed due to earthquake damage.

Just speculatively, I hope that the Transport Minister’s announcement of the specifics of rail funding on the WRL above is the start of the kind of sequence of minor press releases that the Beehive likes to do to ramp up before something big is announced  – hopefully it’s a precursor to the LGWM reports and recommendations being released (my expectation would be that they are close, because the Minister would likely want all the transport funding stuff to be released in a clump).

They really do need to get on with the release though, because according to the June update the report will only be preliminary, with “suggestions for more detailed investigation” – like every other report released since the 60s. If Cabinet wants to get spades in the ground by the next election, they need to speed up a bit. (If the Dom Post’s rumours are correct, the delay might be due to Cabinet wanting to approve funding for construction of some kind of major transport project – I don’t want to say LR – before announcing it. If that’s the case, then we might actually see something built, but if it’s just a normal administrative delay then the report might just end up in the back of a filing cabinet, like the last report on transport in Wellington, and the one before that…)

Incidentally, I’ve noticed a few updates by the Maymorn preservation society about graffiti on their EMU’s, which is quite disappointing – especially given how in-the-middle-of-nowhere Maymorn actually is. On the plus side, apparently the wheels are now in motion to get the Ganz (minus bogies) out of Wellington’s northern yards and up to the Maymorn sidings.