p范数空间中的旋转几何与物理运动

//注:本文于2020年8月大幅改动并加入了大量新内容!
//注:本文于2023年1月修正了最后一小节关于引力轨道计算的错误并做了相应补充!

不一样的距离公式

大家知道勾股定理是直角边两边的平方和等于第三边,这是一个不用怀疑的事实,也有各种方法证明勾股定理,其中最著名的莫过“无字证明”了:

但其实这个证明是有问题的,因为勾股定理是可以不成立的!大家应该听说过非欧几里得几何吧,我前面的一篇文章也讲过一种“双曲几何”。非欧几里得几何本来是从否定平行公设出发推出的一系列新的几何学,但它们的实质是度量不一样,即计算距离的公式不一样。我们熟知的计算两点间距离的公式:$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$也是从勾股定理中来的。如果现在我们“强行”把距离规定成$d=\sqrt[p]{|x_2-x_1|^p+|y_2-y_1|^p}$,即勾股定律变成两边p次方和等于斜边p次方——这种空间我们称为p-范数空间。有一个很棒的短篇科幻小说叫《勾股》,讲述了一个受虫洞影响导致勾股定理中的幂从2偏离到2.013的故事,虽然广义相对论已经能证明弯曲时空勾股定理的次方数还是2,但这并不妨碍它是一篇很好的脑洞很大的小说。如果真实存在这种空间会怎样呢?下面我们就来探究奇妙的p范数空间。

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墨卡托投影

  《维度数学漫步》第一集就讲的是用球极投影来绘制地图。但其实你我都知道,我们熟知的世界地图并不是用这种方法得到的,球极投影只用来绘制极地周围,世界地图用的另一种投影,它得到的地图也不是无限大的,而是矩形,这种投影就叫做墨卡托投影,它和球极投影一样,都是保角的。
墨卡托投影的大概做法就是先拿一个圆柱体使它的轴与地球自转轴重合,我们先把球面上的点投影到圆柱的侧面上,再把圆柱展开就得到长方形的地图了。但具体做法并不是保持z轴不变对应到圆柱上,也不是在球心放一盏灯直接投影到柱面上,为了保持保角性我们能计算出一个特殊的函数,叫古德曼函数,但北极点会投到无穷远,这样的地图是宽度有限、长度无限的,(柱面无限长)我们看到的世界地图把极地附近区域截掉了才得到有限的矩形地图。
来源:英语维基百科
  为什么我们取的圆柱的轴要恰好与地球自转轴重合呢?我们不妨选任意的角度,那将得到你从来没有见过的世界地图!

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趣题:五维立方体截面

本文分两部分,第一部分是关于五维立方体截面谜题,第二部分是对一类四维多胞体性质的探讨。我们要解决五维空间中的问题,但主要还是以四维空间中的讨论为主。

part 1

先来一道只涉及四维(不涉及五维)的题热热身吧:

我们知道,正方体可以被斜着截出正六边形截面,它与正方体所有面都相交;推广到四维,我们希望超立方体也可以被“斜着”截出一种截胞(胞指三维的“面”,下同),它与超立方体所有八个面(立方体胞)都相交,当然符合要求的“斜着”截的方法很多,我们要一种最“对称”的截法,即选垂直于体(最长)对角线的过超立方体体心的截面,这样能满足于所有面都相交的要求吗?会截出什么图形?如果截面不过超立方体体心,我们又可能得到哪些形状的截面?

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Hyperbolic Space - Mathematical Art

From English Wikipedia

Screenshot from "Dimensions: A walk through mathematics" Episode 2

In my previous series of articles explaining “Dimensions: A walk through mathematics”, I mentioned a circular disk pattern that appeared in the film. It was a decoration in the room from Episode 2 “Three-dimensional space” of “Dimensions: A walk through mathematics.” The narrator of this episode is Escher, and both the circular disk pattern and those lizards are his works. The film focuses on telling the story of how one of those two-dimensional lizards escapes from his painting and how it should explain the existence of three-dimensional space to its two-dimensional companions. That circular disk pattern used as decoration only flashes by briefly, but actually this pattern is as fascinating as four-dimensional space: it is hyperbolic tessellation in hyperbolic space.

You can find many of Escher’s paintings online, but the specific mathematical content within them is rarely mentioned. Below we’ll mainly discuss the mathematical meaning of hyperbolic tessellation, and then how we can draw such a painting ourselves - using a computer - online demo here! (including hyperbolic tessellation panda emoji!). Of course, if you’re skilled enough, you can also hand-draw like Escher did!

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用Hutton32玩转数字电路(二):逻辑器件

上篇文章介绍了怎样在Hutton32中搭建与或非门,下面我们就用它们的组合来做一些有趣的事情吧。(比如一个完全平方数计算器!)

异或、同或门

A异或B就是不同为1,相同为0。逻辑表达式为 (A & ~B) | (~A & B)。但注意这里需要两个非门,而非门的体积大和延迟长,所以我们化简一下表达式得到只用一个非门实现异或:(A | B) & ~(A & B);同或也类似:~(A | B) | (A & B)。

有基本的逻辑门我们就能够制作各种组合逻辑、时序逻辑电路了!下面是我做的一些器件:

锁存器

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用Hutton32玩转数字电路(一):逻辑门

  最近看到有人用Minecraft里面的红石电路制作出了计算器,还有一篇神文:《基于Minecraft实现的计算机工程》,视频在此,好像还能算浮点数、三角函数。我对红石不是太了解,那能不能用Hutton32做一个呢?经过不断尝试现在我的成果时能做出一个简单的加减法计算器和Ascii码显示阵列。
加减法计算器 Ascii码显示阵

什么是Hutton32?

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Electron-Deficient Planet vs Electron-Rich Planet

  In the direction of Ursa Major, at the edge of the M82 Cigar Galaxy dozens of light-years from Earth, there exists a lonely star system. Apart from the star at its center, this system contains two planets orbiting the star: one called the Electron-Deficient Planet, which is dark green, and another called the Electron-Rich Planet, which is silvery white. Their orbital paths nearly coincide, and even their orbital periods are almost identical, which causes them to forever remain on opposite sides of the central star, never able to “meet.”

They don’t truly lack electrons, as they are both electrically neutral—the number of protons on these planets equals the number of electrons. However, the chemical substances composing them have an intense desire to lose or gain electrons. These two planets have other names—the Oxidizer Planet and the Reducer Planet.

  Both planets have liquid substances on their surfaces. The Electron-Deficient Planet’s surface is covered by dark green oceans of manganic anhydride, with an atmosphere containing large amounts of fluorine gas, oxygen, dinitrogen pentoxide, sulfur trioxide, neon dioxide, and other gases. The oceans contain small amounts of concentrated sulfuric acid and $MnO_3^+$, with islands of potassium permanganate and potassium dichromate scattered throughout, while the ocean floor contains minerals such as ferrates, perchlorates, and superoxides. The Electron-Rich Planet, on the other hand, has silver oceans of potassium-sodium alloy, containing small amounts of cesium, rubidium, francium, mercury, magnesium, alkyl sodium, and alkyl lithium. The ocean floor is deposited with large amounts of sodium blocks, elemental cesium, charcoal, sodium hydride, and lithium aluminum hydride reefs. The planet’s interior has a core composed of liquid magnesium-aluminum alloy, and the atmosphere consists mainly of argon, elemental ammonium, and trace amounts of hydrogen gas, silanes, and short-chain hydrocarbons.

However, by chance, a massive “light particle” (or “mass point”) flew past the M82 Cigar Galaxy from the direction of the elliptical galaxy M32 near Andromeda at nearly the speed of light. The orbits of the two planets in this star system were slightly affected, their orbital periods developed small deviations and became misaligned, their orbital phase difference decreased, and they slowly approached each other. Finally, after more than twenty million time grains had passed, the two planets entered each other’s gravitational range—they would collide in less than 10 time nodes! This was the inescapable final destiny of the two planets.

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反演地球

  这是一个特殊的世界。在这个世界里你同样头顶着蓝天,脚踏着土地,看起来并没有什么不同——但这个世界却又完全不同。你之所以现在没有察觉出任何异样,是因为这个世界的任何局部都和我们熟知的世界看起来是相同的。

  然而,其实这个世界的每一个局部,甚至每一个分子、每一寸真空的性质都与我们熟知的世界完全不同。因为这个世界中真空都是具有质量的!且真空的密度最大,气体的密度次之,液体密度更小,固体的密度最小,刚好和我们的世界相反。这些古怪的性质并不妨碍你站在这个世界的土地上。因为你仍然受重力,但你所受到的“重力”其实是来自空气的“浮力”——因为其实你在一个星球的内部!

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