Monthly Archives: January 2015

What’s an algorithm? You ask.

Just thought you all might be interested in this analogy between computer science/mathematics and construction of a novel.

J. Giambrone

All right. Maybe you didn’t, but I’m going to tell you anyway.Thinking about plot, sequencing, cause and effect, that sort of thing, prompted me to recall my many days studying computers. Computers execute instructions, and they do so in sequence. The ordering of the instructions is an algorithm. This is a higher-level concept than the nuts and bolts of ones and zeros. It’s more like a cooking recipe.

In a way characters are like little computer programs, executing their own actions in sequence.

So what sorts of things can be gleaned from computer science, which may cross over to the world of fiction? That’s where I’m at here, and we’re all caught up.

Here’s a little baseball algorithm I drew up just for you guys.

An algorithm combines a sequence of events with a multi-level tree structure (an upside down branching tree is often used). The algorithm goes through a…

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Algebra

But I did use some trig on a spiral staircase once !

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Completing the (four sided) square

completing the square

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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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