velocity vector is a 90° rotation of position vector(速度矢量是位移矢量的90度旋转)
group theory
symmetry(对称)
"Dihedral group" of order 8(八阶二面体群)
heart and soul
how these symmetries play with each other/some way to combine the two actions to get out another one
numbers are actions
additive（successively applying actions作用的相继进行）
additive group of real numbers(实数加法群)
sliding actions on the group of line symmetries
additive group of complex numbers(复数加法群)
sliding actions ont the complex plane
multiplicative
group of stretching /squishing actions(拉伸/压缩群)
fix the number 0 in place while the rest of the number line just does whatever it takes to stay evenly spaced
multiplicative group of positive real numbers(正实数乘法群)
multiplicative group of complex numbers(复数乘法群)
includes not only stretching and squashing actions, but actions that have some rotational component as well
some combination of stretch or squash, an action associated with some point on the positive real number line, followed by a pure rotation, where pure rotations are associated with points on this circle, the one with radius 1.
pure horizontal slide, represented with points on the real number line plus some purely vertical slide, represented with points on that vertical line.
exponentiation(幂)
extend the definition(拓展定义)
presserves the group structure(保持群结构)/"Homomorphism(同态)"
e to the x alone equal to its own derivative
thinking of the inputs to an exponential function as sliding actions and thinking of the outputs as stretching and rotating actions
speacial?e to the x trasforming the plane
first rolling that plane into a cylinder, wrapping all those vertical lines into cricles, and then taking that cylinder and kind of smooshing it onto the plane around 0.(先把平面卷成一个圆筒，把所有竖直的线绕成一个圈，然后在0附近把这个圆筒拍扁到平面上)
For each of those concertric circles spaced-out exponentially correspond with what started off as vertical lines.(这些呈指数间隔的同心圆，对应于最开始竖直的线)
