Tag Archives: abstraction

Duality, fundamental and profound, but here’s a starter for you.

Duality, how things are connected in unexpected ways. The simplest case is that of the five regular Platonic solids, the tetrahedron, the cube, the octahedron, the dodecahedron and the icosahedron. They all look rather different, BUT…..

take any one of them and find the mid point of each of the faces, join these points up, and you get one of the five regular Platonic solids. Do it to this new one and you get back to the original one. Calling the operation “Doit” we get

tetrahedron –Doit–> tetrahedron –Doit–> tetrahedron
cube –Doit–> octahedron –Doit–> cube
dodecahedron –Doit–> icosahedron –Doit–> dodecahedron

The sizes may change, but we are only interested in the shapes.

This is called a Duality relationship, in which the tetrahedron is the dual of itself, the cube and octahedron are duals of each other, and the dodecahedron and icosahedron are also duals of each other.

Now we will look at lines and points in the x-y plane.

3x – 2y = 4 and y = (3/2)x + 2 and 3x – 2y – 4 = 0 are different ways of describing the same line, but there are many more. We can multiply every coefficient, including the constant, by any number not 0 and the result describes the same line, for example 6x – 4y = 8, or 0.75x – 0.5y = 1, or -0.75x + 0.5y + 1 = 0

This means that a line can be described entirely by two numbers, the x and the y coefficients found when the line equation is written in the last of the forms given above. Generally this is ax + by + 1 = 0

Now any point in the plane needs two numbers to specify it, the x and the y coordinates, for example (2,3)

So if a line needs two numbers and a point needs two numbers then given two numbers p and q I can choose to use then to describe a point or a line. So the numbers p and q can be the point (p,q) or the line px + qy + 1 = 0

The word “dual” is used in this situation. The point (p,q) is the dual of the line px + qy + 1 = 0, and vice versa.

dual of a rotating line cleaned up1

The line joining the points C and D is dual to the point K, in red.  The line equation is 2x + y = 3, and we rewrite it in the “standard” form as  -0.67x – 0.33y +1 = 0  so we get  (-0.67, -0.33) for the coordinates of the dual point K.

A quick calculation (using the well known formula) shows that the distance of the line from the origin multiplied by the distance of the point from the origin is a constant (in this case 1).

The second picture shows the construction of the dual point.

dual of a rotating line construction1

What happens as we move the line about ? Parallel to itself, the dual point moves out and in.

More interesting is what happens when we rotate the line around a fixed point on the line:

gif duality rotating line

The line passes through the fixed point C.  The dual point traces out a straight line, shown in green.

This can be interpreted as “A point can be seen as a set of concurrent lines”, just as a line can be seen as a set of collinear points (we have fewer problems with the latter).

It gets more interesting when we consider a curve. There are two ways of looking at a curve, one as a (fairly nicely) organized set of points ( a locus), and the other as a set of (fairly nicely) arranged lines (an envelope).

A circle is a set of points equidistant from a central point, but it is also the envelope of a set of lines equidistant from a central point (the tangent lines).

So what happens when we look for the dual of a circle? We can either find the line dual to each point on the circle, or find the point dual to each tangent line to the circle. Here’s both:

dual of a circle4

In this case the circle being dualled is the one with center C, and the result is a hyperbola, shown in green.  The result can be deduced analytically, but it is a pain to do so.

dual of a circle3

The hyperbola again.  It doesn’t look quite perfect, probably due to rounding errors.

The question remains – If I do the dualling operation on the hyperbola, will I get back to the circle ?

Also, why a hyperbola and not an ellipse ? Looking at what is going on suggests that if the circle to be dualled has the origin inside then we will get an ellipse. This argument can be made more believable with a little care !

If you get this far and want more, try this very heavy article:

http://en.wikipedia.org/wiki/Duality_(mathematics)

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The mean ? Which mean ? With interesting ratios.

Playing around with the Harmonic Mean of two numbers I stumbled on an interesting ratio, and looked at the others as well.

Here are the definitions, for numbers a and b

means

If we use m for the mean, then

for the arithmetic mean we have the ratio (b-m)/(m-a) = 1

for the geometric mean we have b/m = m/a

for the harmonic mean we have (b-m)/(m-a) = b/a

and for the RMS mean we have (b2 – m2)/( m2 – a2) = 1

I am quite sure that there is a way of seeing these which ties them all together, perhaps Mr. Joseph Nebus can find it !

The harmonic mean can be used to explain the harmonic tuning of a keyboard instrument (as opposed to equal temper tuning). I am planning a post on this for later.

The formula I gave for the harmonic mean is not the usual one – use a bit of algebra ! – but it is easier to calculate with.

The RMS mean is used extensively in Statistics, Rigid Body Dynamics and Electrical Engineering. The well known 110 volts in your house electric system is the RMS mean of the alternating voltage actually supplied. The Standard Deviation is the RMS average of the distances of the data values from the arithmetic mean value.

A non formal view of these means (the first three) is that the arithmetic mean is about the positions of the two numbers, the geometric mean is about the absolute sizes of the numbers and the harmonic mean is about the relative sizes of the numbers.

if we take the zero, the two numbers, and the harmonic mean the four values have a cross ratio of -1 (see part 3 of the Christmas Tale)

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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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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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The abstract approach to the abstraction which is “Negative Numbers”

An approach to the formal definition of negative numbers, using the ideas of abstract algebra.

Section 1 – is background. Skip it if you like.

What is a negative number?

1: It’s a number with a “-” in front.

2: It’s the opposite of a positive number.

Well, 1 is very poor, and 2 is no good, as there are no positive numbers until we have  negative numbers, they are just numbers (referred to later as the original numbers).

There is a need to compare numbers, and one way is to ask “What is the difference  between this number and that number. This is easy – the difference between 3 and 7 is  4, and we all learn to write 7 – 3 = 4.

Everything is fine for a while until someone says “But what about 3 – 7 ?”.

“Cannot be calculated. Has no meaning. You cannot take 7 things away from 3 things.  You cannot cut a 7 inch piece of wood off a 3 inch piece.” are the answers.

These original numbers are usable for counting and measurement of quantity, but numbers can also be used to measure position, leading to questions of the form “How far is it  from this number to that number?”. Temperature is the most obvious situation. “How  much warmer is it today, compared to yesterday?”. With numbers we can ask “How far  is it from 3 to 7 ?” and get a response ” 4 “, but we can also ask the question “How far  is it from 7 to 3 ?”. The response is the same, with the extra “but in the opposite  direction”.

Thus there arises a need for numbers capable of dealing fully with this new situation , the  measurement of changes in position. So negative numbers are born (or created), and  we hope they obey the same rules as the original numbers. Playing around seems to  support this position, with a few mysteries, such as (-1) times (-1) equals 1, and two  negatives make a positive.

However, in math we should not be satisfied by “Well, it seems to work OK”.

Section 2

What follows is a formal definition of an extended number system, in which every number has an “opposite”, or an “additive inverse”, and in which every number not equal to zero has a multiplicative inverse, and in which the “properties of operations” are still valid.

The definition only uses brackets (parentheses), commas, and pairs of original numbers. It does NOT use the negative sign, and subtraction  b – a, is only applied where it makes sense, that is when  b>a.

An extended number (or “thing”), written  ( a , b ), is defined as the distance from a to b.

It is immediately obvious that  ( a , b ) = ( a + 1 , b + 1 ) = ( a + 2 , b + 2 ) and so on.

So we can write  ( a , b ) as ( 0 , b – a )  when  b>a, and as  ( b – a , 0 ) )  when  b<a,  using subtraction only  with the original numbers.

 

Addition needs to match the original number addition, so

(definition)              ( a , b ) +  ( p , q ) =  ( a +p , b + q )                    check it

Zero is the “thing” which when added to anything has no effect, so the zero “thing” is ( p , p ) for any p.

Now we can have the additive inverse of a “thing”, the one which when added to the “thing” gives zero.

(definition)   Additive inverse of   ( a , b )  is   ( b , a )  since   ( a , b )  +   ( b , a )  =  ( a + b , a + b )

Subtraction for “things” can now be defined as addition of the additive inverse.

We can define multiplication of “things” by looking at the product of two differences.

(original number definition)  ( b – a )( d – c ) = bd + ac – ( ad + bc ) , so we have

(definition)              ( a , b ) X ( c , d ) =  ( ad + bc , bd + ac )

For multiplication we need a unit or identity “thing”, and the obvious choice is ( 0 , 1 ), or anything where the second number is one bigger than the first, for example  ( 12 , 13 ).  Using the “multiply” definition we have                                           ( 0 , 1 ) X ( 0 , 1 ) = ( 0 , 1 ),

and                          ( 0 , 1 ) X ( 12 , 13 ) = ( 12 , 13 ) = ( 0 , 1 ),

and                          ( 0 , 1 ) X (3, 7 ) = (3, 7 )

Division is defined as the inverse of multiplication, so to divide a “thing” by another “thing” we multiply the “thing” by the multiplicative inverse of the “other thing”, which we now define.

(original number definition)       multiplicative inverse of  ( b – a ) is 1/(b-a)

(definition)           Multiplicative inverse of  ( a , b ) is ( 0 , 1/(b-a))  if  b>a  and  ( 1/(a-b) , 0 )  if  b<a

If we multiply a “thing” by its inverse we should get the unit or identity ( 0 , 1 ), and so we do:

( a , b ) X ( 0 , 1/(b-a)) = ( a/(b-a) , b/(b-a) ) = (  0 , b/(b-a) – a/(b-a)  ) = ( 0 , 1 )

We have enough here to show that the new operations of + and X have the same properties as add and multiply in the original numbers. Go on, show it !!!!

Now what has this got to do with negative numbers ?  Well, the first thing is that ( 0 , 1 ) has an additive inverse, namely  ( 1 , 0 ), or any of its other representations, say ( 5 , 4 ) for example.

The second thing is that  ( 1 , 0 ) X ( 1 , 0 ) =  ( 0 , 1 ) .

The third, and most important thing is that we have an arithmetic for the “How far is it from A to B” quantities which incorporates direction.  When A<B the direction is one way. When A>B the direction is the other way. These directions are coventionally called “the positive direction” and “the negative direction”.

So, finally, we identify distances in the positive direction with the original numbers, and distances in the negative direction with new numbers, each of which is the “opposite” or “additive inverse” of one of the original mumbers.

Using the minus sign for “the additive inverse of” makes it quicker to write, at a cost of some possible confusion.  We see now that for example  ( 3 , 7) is identified with the original number  4 and  ( 7 , 3 ) is identified with the new number  -4.  Also  ( 0 , 1 ) is identified with the original number  1 and  ( 1 , 0 ) is identified with the new number  -1.

So since we have  ( 1 , 0 ) X ( 1 , 0 ) =  ( 0 , 1 ) it follows that  -1 x -1 = 1, and there is no mystery about it!

 

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Negative numbers: truth, existence, reality, abstraction.

There are things with names which I can pick up, see, feel, or if too radioactive then at least observe and measure…..this is the reality we have (though some philosophers, and neuroscientists would have us believe otherwise). But what with numbers?

I can see three coconuts, I can count them, and match the count to the quantity, but the number THREE is not real, it is an abstraction from all the occurrences of three object that I have seen or can visualize.

Well, if THREE is not real then MINUS THREE hasn’t got a chance at being real, it is in fact a second stage abstraction, as negative numbers were invented by humans to deal with situations not adequately described with “ordinary” numbers. It gets worse, as complex numbers were invented to get over the difficulties with “real” numbers (the positive and negative numbers). It is a shame about the use of the word “real” in this situation (see above). They should have been called “simple numbers”.

Abstraction is also the basis of geometry. Euclid says “A line is that with extent but no breadth”, which does make it difficult to see! 

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