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Blaise Alexander Mansfield – Pappus and Pascal | Elementary Mathematics (K-6) Explained 12 | NJ Wildberger

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Continuing with our introduction to elementary projective geometry, meant for primary school students, we discuss two of the most famous theorems in mathematics: one due to Pappus of Alexandria around 300 A D and one due to Blaise Pascal in the 1600’s. The first result only requires a piece of paper, a pen and a straightedge, or ruler, to appreciate. Pascal’s theorem requires also a circle. Both theorems really ought to be more widely known in primary school mathematics education.

Video Content:
00:00 Introduction
8:44 Pappus’ theorem
13:53 Tips on reading mathematics
22:59 Blaise Pascal (1623-1662)
25:23 Pascal’s theorem

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  1. Playing around with Geogebra revealed some interesting anomalies. 1. If one of the lines with one of the points on crosses over and that point joins the other line then Pappus' third line becomes the same as the second line. 2. If this point is dragged even further below the crossing of the two lines then it is not possible to create the three pappus' points as the connecting join lines no longer cross over or intersect each other. It is as if the shape has been turned inside out.

    [N.B. This anomoly was discovered by a young 10 year old who was testing Pappus's theorem to destruction. As I was at a loss to explain the anomaly I said that hence forth it would be called 'Otty's anomaly'. He has already cottoned on to the fact that even if someone else has discovered a theorem before you; if you are smart (like Pythagoras was) you can still have your name associated with it.]

  2. Pappus' theorem doesn't work if (one of) the constructed lines are parallel, e.g. if AB' is parallel to A'B, then there's no intersection, and no collinear points.

    EDIT: I just realized that the actual label of the points doesn't matter. We can connect any four points as long as the lines intersect each other.

  3. Hi! i wanted to know about Hypatia's problem, actually she did the same thing as Pappus with the difference that she joined two lines out of the ellipse and the points form a triangle. lines were tangents of the six points of the ellipse.

  4. You made me recall my middle school days when I was puzzled by the need of my classmates to always use the same notations and their inability to grasp that all notations are arbitrary and that instead of x, you could use a pictogramme of a monkey dancing around a coconut tree. It does get annoying at times though. E.g. physicists, electrical engineers, and mathematicians use incompatible notations and conventions.

  5. Since my first encounter of this theorem, I cannot find a reason why this is not in our curriculum or syllabus, not until I find out that the focus of modern high school geometry is mostly on the angles and distances in a triangle, and so the basic geometry of lines and points are neglected.

  6. You as a professional mathematician and a high end teacher – you need notation to function in your world – that language of notation is not natural in the sense of obvious and non-arbitrary. These are great exploration tasks and great amazement activities and as such the notation can be a drag on that. The trick is to find a way of notation to come about as natural or necessary.

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