Category Archives: abstract

Quadrilaterals – a Christmas journey – part 3

So what else do stretches and shears do?
A stretch will turn a square into a rectangle if it acts in the direction of one of the sides.
A stretch will turn a square into a rhombus if it acts in the direction of a diagonal.
A shear will turn a square or a rectangle into a parallelogram.
Try it out !

Now to continue the journey – the mathematician now thinks “Is that it ? Are these the only transformations of the plane that map straight lines to straight lines ?”. His answer, with a Eureka moment, is “No ! What about those artists, with their perspective drawings ? Not only do straight lines in reality go to straight lines in their pictures, but when the lines are parallel in reality they go to convergent straight lines in the picture. “This is projection !”, cries the mathematician, and pursues the matter further and further…..

old master picArchitect_drawing_conceptual_2012

If I fix a sheet of glass or acrylic, sit still with marker pen in hand, and copy onto the glass exactly what I see through the glass I get a point projection of reality on the glass. Doing this when reality is a flat wall creates a projection from the reality plane to my glass plane. This is a new type of line preserving transformation of the points on a plane.

What is really nice about projection is that all the transformations we have seen so far can be described by projections.

First we have to classify projections, from a source plane to a target plane:
1. Point projection. The projection lines all pass through a fixed point (that’s the point where your eye was earlier). then each projection line passes through a point on the source plane, and where it hits the target plane is the transformed or projected point.
2. Line projection. The projection lines all start on a fixed line and are at right angles to that line. Imagine a spout brush or a very hairy caterpillar.
3. Parallel projection, in which the projection lines are all parallel to each other, as in rainfall or a bundle of spaghetti.

Here are translations, reflections, stretches and shears as projections:

Projection 0 Projection 1 Projection 2 Projection 3

What about rotations, you ask. Simply, a rotation can be seen as a sequence of two reflections.

But in general projection we lose the preservation of ratio. For example, the midpoint of a line segment is not projected to the mid point of the projected line segment:
A slice will do …..

midpoint 1midpoint 2It gets worse ! In the lower picture, as the point C is moved to the right along the line its image H moves further and further along, and, when C reaches D, H disappears altogether.
So if we look at the source plane from above, and we have a pair of lines that meet at point D, their projections must be parallel lines. (There could now follow a digression on the meaning of the word “infinity”, but it ain’t happenning).

What else ? Well, we have seen that ratios of distances are altered by projection, but when we take four points A, B, C and D on a line we can take two distance ratios (involving all four points) and take the ratio of these ratios. THIS quantity is NOT destroyed by projection. It is called the “cross-ratio” of the four points, and its value is (AB/AD)/(CB/CD), easier as (AB.CD)/(AD.CB)
(proof later)
A quick check:xratio testCopy the diagram to scale, but you can put the target line where you like.
Then measure EF, FG and GH, and calculate the cross-ratio. Should be the same, if you kept to the point matching.

Finally, and back to the quadrilateral:
arect6
Each diagonal has four points on it, two vertices and two intersection points with the other two diagonals.
If we join D to H, the point of intersection of the two diagonals BC and EF, we can see that the two sets of four points are connected by a projection from D, as H to H, B to F, G to I and C to E.
Consequently they have the same cross-ratio.

Not only that, it can be shown (one day !) that the value of this cross-ratio is -1, for all quadrilaterals, all the time.
Also, it is possible to map any quadrilateral to any other by a sequence of projections.

This is my introduction to projective geometry, a very interesting and underexposed branch of geometry. There may be a part 4 eventually !

And two more 3D gifs :

gif cube 3gif torus 2

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Quadrilaterals – a Christmas journey – part 2

It is a popular activity to join the opposite vertices of a quadrilateral (the diagonals), as in special cases they
have interesting properties. Unfortunately, for the general quadrilateral this does nothing of interest.
However, with the extended or complete quadrilateral we have six vertices to go at, and so we get another “diagonal”:

arect4

Things get more interesting when we extend the diagonals and find their points of intersection.

arect6

The three points shown circled are the points of intersection of the three pairs of diagonals.

Observe that each diagonal has four points on it, the two vertices interlaced with two points of intersection.

Now we started with “any” quadrilateral so it might be thought that nothing much can be said about measurements and
quadrilaterals in general – not so! To go any futher we need to go back to rigid motions of the plane, and their
effect on plane figures.
A rigid motion of a geometrical object just moves it to a new position, its shape and size are unchanged.
Rotations, reflections and translations (shifts would have been a simpler term)are rigid motions.
Basically what is not changed is distances between points.
The next level of transformation of plane figures adds dilations,stretches and shears. The figures change their
shapes, but one thing remains: relative distances of collinear points. Rigid no longer.
arect7arect8
Dilations can be “the same in all directions”, as in the example, and these preserve the shape of a figure but not the
size, or different in the y direction from the x direction, these are the stretches. These turns circles into ellipses !
Shears turn circles into ellipses anyway.
Notice that in both cases the mid point of the transformed line segment is the image of the midpoint of the original
line segment, and it is easy to see that ratios of distances in the same direction are preserved.
It is not quite as easy to see that any triangle can be transformed into any other triangle, with the help of these
extra transformations.
Notice that dilations have a fixed point and shears have a fixed line.

The next three pictures show the effects of
1: A stretch up and down 2: A stretch to the right and left
3. A shear horizontally

gif001gif002gif003

What use is all this, you ask. well, here is a gif showing that the medians of a triangle are concurrent, and this is preserved under stretch and shear. This means that you only have to prove it for an equilateral triangle. (which is obvious!)

gif medians2

More soon…….

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Quadrilaterals – a Christmas journey – part 1

It all started with an aside in a blog post in which the author said how
some students have a real problem with statements such as “A square is a
rhombus”.

From early years kids naturally like exclusive definitions, and have to be weaned off this. This would be easier if we were more careful with the word “is”. Even to me the statement ” a square is a rhombus” sounds weird, if not actually wrong. It would be better to be less brutal, and say “a square is also a rhombus” (and all the other such statements).
Even better, and quite mathematical, is the phrasing “a square is a special case of a rhombus”, as the idea of special cases is very important, and usually overlooked.
It is odd that the classification of triangles is done entirely with adjectives and the difficulty is thus avoided (but see later).
After fighting with a Venn Diagram I did a tree diagram to show the relationships:

quadrilaterals 1 pic

I then got thinking about the words “triangle”, “quadrilateral”, “pentagon” etcetera.

“triangle” : three angles
“quadrilateral” : four sides
“pentagon” and the rest : … angles
The odd one out is the quadrilateral.
Take a look:
arect2
It consists of four line segments, AB, BD, DC and CA

Let us see what the full extended lines look like:
arect3
Let ab be the name for the full line through A and B
Likewise ac, bd and dc
Then we can see that the quadrilateral is determined by the points of intersection of the two pairs of lines ab,cd and ac,bd.

ab and cd meet at point E; ac and bd meet at point F
But if we consider the four lines then there are three ways of pairing them up. The two others are ab,ac with bd,cd and ab,bd with ac,cd.
This gives us two more quadrilaterals, and they all have the property that each side falls on one only of the four lines.
The three quadrilaterals are ABCD, FCEB and FDEA
ABCD is convex, FCEB is twisted and FDEA is not convex (concave at A)
Not only that, but also the first two are fitted together to give the third one.
This arrangement is called the “complete quadrilateral”, and has four lines and six points.

More next time.

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Calculus without limits 5: log and exp

The derivative of the log function can be investigated informally, as log(x) is seen as the inverse of the exponential function, written here as exp(x). The exponential function appears naturally from numbers raised to varying powers, but formal definitions of the exponential function are difficult to achieve. For example, what exactly is the meaning of exp(pi) or exp(root(2)).
So we look at the log function:-
calculus5 text
calculus5 pic

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Fractions as parts of a whole, again !

This was found on “talking math with your kids” as an example of the “strange” stuff that kids bring home and cause mystification in their parents.

“The whole is 8. One part is 8. What is the other part ?”.

Just what exactly is this supposed to mean?
That the whole always consists of two parts?
Since when did numbers have parts?
What is the definition of “part”?
Even if we are talking about 8 things, then the natural AND logical answer is “There isn’t another part”.
If I want to see ways of creating 8, using adding, then what is wrong with
8=1+7 8=2+6 8=3+5 … 8=7+1 and 8=8+0 for completeness’ sake.

To call zero a part of 8 is going to lay the groundwork for a feeling that math hasn’t got a lot to do with real life, which is a crying shame. This feeling can arise at any stage, we should give reality a chance at this level. To conclude “What a stupid question!”.

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Common sense versus logic and math: Congruence again

I thought I would write a computer routine to check if two figures were congruent by the CCSS definition (rigid motions). One day I will post it.

The most important thing was to be specific as to what is a geometrical figure. You can read the CCSS document from front to back, back to front, upside down and more, but NO DEFINITION of a geometrical figure. For the computer program I decided that a geometrical figure was simply a set of points. My diagram may show some of them joined, but any two points describe a line segment (or a line). So a line segment “exists” for any pair of points.

The question is “Are the two figures shown below congruent or not?

congruent or not

I rest my case…..

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Commutative, associative, distributive – These are THE LAWS

Idly passing the time this morning I thought of a – b = a + (-b).
Fair enough, it is the interpretation of subtraction in the extended positive/negative number system.

I then thought of a – (b + c)
Sticking to the rules I got a + (-(b + c))
To proceed further I had to guess that -(b + c) = (-b) + (-c)
and then, quite ok, a – (b + c) = a – b – c

But -(b + c) = (-b) + (-c) is guesswork.
I cannot see a rule to apply to this situation.

The only way forward is to use -1 as a multiplier:
So a – b = a + (-1)b = a + (-b),
and then -(b + c) = (-1)(b + c) = (-1)b + (-1)c = (-b) + (-c)
by the distributive law.

It’s not surprising that kids have trouble with negative numbers!

Do we just assert that the distributive law applies everywhere, even when it is only defined with ++’s ?

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Infinity, a place beyond.

That most strange place, infinity,
Is somewhere I don’t want to be.
I’d rather stay with Brouwer
In his ivory tower.

 

and for something lighter try Heavy Man

 

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Read this!

This is a plug for Jose Vilson, in particular this post:

http://thejosevilson.com/another-reason-dont-like-foil-math/

The comments at the end of this post about the futile nature of current math education by  Ted Dintersmith  should be posted on the side of the Empire State Building.

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Geometry test. Maybe simple ! Try it.

Three maybe parallel lines, AD BE CF
Two transversals, AC DF
question how many parallel
Given that AB is equal to BC and DE is not equal to EF
(“equal” = “congruent” if you like)
show that at most two of the maybe parallel lines can be parallel.
(proof by contradiction is a last resort)

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