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I am trying to understand this new way of multiplying in projective geometry.

enter image description here

Why is it defined like this? Also does this have anything to do with multiplication using a slide ruler? (The picture in the link shows that $4 \cdot 4 = 16$ and $ 4 \cdot 2 =8$. Every unit is a power of 2. Slide rulers were commonly used in the old days way before the use of a calculator.)

enter image description here

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    One method computes$a$product, the other a sum. One can be computed by a ruler &straight edge, the other can't (ie, construct the Naperian log using a ruler & straight edge).2012-04-20

1 Answers 1

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Error in image

First off, your image looks wrong to me. In my opinion, you'd want the following elements (too lazy to draw an image just now):

  • a line $g$ connecting $0$, $1$, $a$ and $b$
  • a different line $h$ through $0$
  • a point $p$ on $h$, perhaps directly above $1$
  • a triangle $t$ connecting $1$, $a$ and $p$
  • a similar triangle $b$, $a\cdot b$ and $p'$

In other words, the construction would be:

  1. choose $p$ arbitrarily
  2. $p'$ is the point where (the connection of $0$ and $p$) intersects (the line parallel to $1\vee p$ through $b$)
  3. $a\cdot b$ is the point where (the parallel to $a\vee p$ through $p'$) intersects (the line $0\vee 1$)

In that case, your image would represent the Euclidean version of multiplication, which is a special case of the projective version. In your image, the triangle over $1$ and $b$ looks similar to the one over $b$ and $a\cdot b$, so it appears that you'd have constructed $b^2$ instead of $a\cdot b$.

http://www-m10.ma.tum.de/bin/view/MatheVital/GeoCal/GeoCal4x2a has applets of both the Euclidean and the projective view of these situations, although the text is in German.

First question

For your first question, I guess the simplest answer would be “because it works”. So what is the objective? Given a projective scale along a line, i.e. the points $0$, $1$, $a$, $b$ and $\infty$, one wants to construct the point $a\cdot b$ using only tools from projective incidence geometry, i.e. joins (lines connecting points) and meets (intersection points of lines). There aren't many configurations which can accomplish this, but the von-Staudt construction your immage suggests can accomplish this, as you can check for the euclidean case and generalize as all operations can be interpreted projectively.

Second question

I see only a slight connection to the slide ruler. Both use some isomorphism to make multiplication accessible in a geometric way. One uses logarithms to translate multiplication into addition, whereas the other transforms it to the algebra of projective geometry. Apart from that, I see no connection.