Science outreach

7 years ago when I first learned projective geometry, I thought it could be a nice math popularization topic for secondary school students. Finally I realized this idea thanks to the atelier de diffusion of the summer school JC2A!

The projective plane, as the name implies, may be related to the projector? (A projector was placed obliquely, projecting a circle on a piece of paper.) We turn the paper and find that the circle becomes an ellipse, a parabola, a hyperbola, with some points getting infinitely further away and eventually disappearing. However, what comes out of the projector remains the same: that is — (tapping chalk dust over the projection, so that the rays emitted by the projector can meet the cloud of dust and thus embodies) — a cone. A point on the paper corresponds to a line of light from the projector; a line on the paper thus corresponds to a plane of light; and the various conic sections correspond to the very same cone. There is no longer any distinction between parallel and intersecting lines: parallel lines on the paper are in fact images of intersecting planes of light captured by the paper in a particular position, and the point of intersection cannot be seen simply because that line of light is parallel to the paper and "runs off to infinity". All the beams projected from the source of light form the projective plane. Many theorems about intersections become uniform and simple. (Projecting figures of different cases of Pappus's hexagon theorem and Pascal's theorem) These seem to be complicated intersections of lines at random, and also with conic sections. Nevertheless, we only need to adjust the angle of the paper to get a very simple and special image of the projection with many lines parallel to each other and the conic curve becoming a circle. However, we know that this is exactly equivalent to the previous intersection case, because what remains unchanged is always (again, tapping chalk dust over the rays) the projective model in front of our eyes.

7年前，当我第一次接触到射影平面时，我就觉得这会是一个适合讲给中学生的数学科普话题。感谢这次暑期学校JC2A数学科普推广研讨会让我终于实现了这个计划！

射影(projective)平面，顾名思义，可能和投影仪(projector)有关？（斜放的投影仪在纸上投出一个圆的影像）我们转动纸，发现圆变成了椭圆，抛物线，双曲线，有的点逐渐跑向无限远、直至消失。然而，投影仪投出来的东西一直没变，那就是（往投影前拍粉笔灰，投影线束在粉笔灰烟雾上照出）—— 一个圆锥。纸上的一个点对应投影仪的一条光线，直线则对应一个平面的线束，各种不同的圆锥曲线统一对应成圆锥，直线不再有平行或相交的区分：纸上平行的直线实际是摆放在特殊位置的纸截到的相交的平面线束的图像，而看不到的交点只是因为这条光线和纸平行了，从而跑到了无限远去。投影出的这些线束就构成了射影平面。一系列的相交定理都变得统一和简单。（投出帕普斯定理，帕斯卡定理的各种不同情形的图形的图形）看似复杂的直线随意相交，又和各种圆锥曲线相交；不过，只需要转动纸张角度就可以截取到一个非常简单的有很多直线平行，圆锥曲线变成圆的情况。但我们知道这完全与之前的相交情形是完全等价的，因为不变的总是（再次往投影光线前拍粉笔灰烟雾）这个在我们眼前的射影模型。

Last time with the Fête de la Science, I demonstrated the mathematics in soap bubbles and building blocks (minimal surfaces and Euler's formula) to children in Joseph de Maistre Elementary School, and also took charge of a stand and presented game theory to the general public through a series of pizza-cutting riddles. This time, I worked with the association Science Ouvertes to tutor high school students who participated in a training course in cryptography during their holidays.

I was as delighted as them when I saw how excited they were to work in small groups and to finally work out the keys to the math riddles with our hints. Some of them discussed their plans for the university with me, seeking my advice, and also asked about my educational experience. By the was, I was amazed to see that the standard calculator for high school students nowadays has a color screen with Python programming...

I always think popularizing math is important and meaningful, as we can lead the general public to recognize its beauty and better comprehend how math research works. The association Science Ouverte targets children from underprivileged neighborhoods, aiming to broaden their horizon, keep them informed, and guide them to the same opportunities in higher education as those accessible to elite families. Hopefully, we are getting one step closer to equity in education.

Photo credit to François Gaudel

上次Fête da la Science科学文化节时，我去Joseph de Maistre小学给孩子们展示了吹泡泡、搭积木中的数学（极小曲面、欧拉公式），并且负责了一个展台向公众介绍分比萨游戏的博弈论。这次，我又和公益组织Science Ouverte一道迎来了愿意利用假期来参加密码学入门课程的高中生们。

看着大家小组时高昂的兴致、逐步获得提示启发后找到答案的喜悦，我真的同他们一样开心。有的小朋友很愿意和我聊他们之后大学的兴趣理想，希望得到一些建议，也很好奇我的求学经历，而我看到现在高中生人手一个的标配计算器都是彩屏带Python编程功能的，也大开眼界...

我总是以为向大众科普数学，让大家真正了解到数学的美、认识到数学研究工作是怎样进行的是非常重要和有意义的。另外，Science Ouverte公益组织面向的是家庭条件并不优渥的孩子们，旨在打开他们的眼界、扩展他们的信息渠道、并且让他们了解到之后可以得到和精英家庭出生的孩子无二的的求学机会。但愿我们能离教育公平性的理想更近一小步吧。