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

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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.)

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    Not sure if the your second picture is relevant in this context...Is there a reason why you would want us to show a slide rule?2012-04-20
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    No, slide rules use logarithms. The multiplication above just relies on ratios being the same.2012-04-20
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    This isn't a complete answer, but worth considering: The reason we use a product with four inputs (cross-ratio) is that there's no meaningful product with just 3 respected by linear fractional transformations. Any three points in projective space can be mapped to any other three via a linear fractional transformation, so we need at least 4 inputs to define any kind of measurement that doesn't depend on your model of projective space.2012-04-20
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    Let me clarify what I'm seeing here: Look at the first diagram. Let a = 2 and b = 3 then ab = 6. Align 1 from the top so that it matches with b at the bottom and align a from the top so that it matches with ab at the bottom. This is the same as using the slide ruler that I've drawn.2012-04-20
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    Not that it matters, but I used slide rules in school. The slide rule just adds the logs of numbers. The geometry method actually produces a length that is the product of $a$ and $b$, whereas the slide rule produces a length that is the sum of $\log a$ and $\log b$. So, you need a lot more room to multiple with the geometric method.2012-04-20
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    The ruler that I've drawn is rudimentary. Of course you could use it for logs. For example, what is the power of 2 that gives you 3? Without using the log button on the calculator you will find that it is ~1.585 which means that 3 is 1 unit and 0.585 units away from the number marked 1.The length of 6 is the same adding up the length of 2 and the length of 3. Slide rulers can also be used for division and multiplication. That was the whole point of my question.2012-04-20
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    Using a slide ruler a person who is interested in 16 times 32 can move the one below the 16 from the top. S/he can then read the value from the top above 32 which is 512.2012-04-20
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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

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