今天我们来看看一种基于闵可夫斯基空间构造的一系列世界,并介绍一些微分几何学的相关概念。许多奇特世界与其都有一定的联系。本文更多涉及到这些空间的几何性质,后续我们再讨论这些世界里的天文学、居民生活等是什么样的。提前剧透:闵可夫斯基世界对我们来说既陌生又熟悉。
特色内容
- 双曲函数与双曲角
- 闵氏空间中的物理学
- 德西特与反德西特时空几何
- 内蕴几何与内禀曲率
- 黎曼曲率张量的几何意义
// Note: This article involves 4D planetary direction positioning based on Hopf polar coordinates. Please first read Four-Dimensional World (I): Day, Night and Seasons on Planets to understand the most basic directional terminology on 4D planets.
Previously in “Four-Dimensional World (VII): Electromagnetism,” I mentioned how to use the geomagnetic field of a 4D planet for navigation, but didn’t specifically analyze how to design and use compasses. This article will supplement this content, providing a correct directional identification guide for explorers on 4D planets using magnetic fields, Coriolis force, and the starry sky.
/// Note: This article only provides an intuitive understanding of homology-related concepts, without rigorous derivations. Corrections for any errors are welcome.
The previous article discussed Homotopy Groups. This time we’ll look at Homology Groups. Homology is somewhat more abstract, but in some ways simpler than homotopy. Returning to the initial idea of determining holes through paths, homotopy studies the process of continuous contraction of circles, which is direct but difficult to compute. Upon careful observation, we find that those shapes that can be contracted to a point all have closed regions inside. If we assume that whenever a closed path is the boundary of some region in the shape, then it hasn’t looped around or crossed any holes, we call such loops “nullhomologous.” Note that nullhomologous paths here are similar to contractible paths in homotopy, but nullhomologous paths may not actually be contractible - we’ll see that homology conditions are weaker than homotopy. For example, in the left figure below, the boundary of the cyan region contains the hole’s boundary and the red loop. The red loop alone is not the boundary of any shape, so we know it encloses a hole. In the right figure, the red loop is the boundary of the blue region, so it doesn’t contain any holes and is “nullhomologous.”
(** 注:本文为一篇好几年前的旧文,是作者对于对偶空间这一数学概念的形象化的想象,因下一篇文章涉及对偶空间概念,故在此发出(然而与其它文章并无因果联系,本文也毫无逻辑可言,看一乐就行,仅在文末给出了一些概念之间的对应关系) **)
(藤瑟先生被邀请来参观无限大养猪场。工作人员负责向他介绍这里的管理模式。)
工作人员:欢迎您前来参观无限大养猪场。这是个无限大的养猪场,我们养了无数头猪,有无数个饲养员,我们的管理模式很特别。每个饲养员都对猪编号,方便识别它们,但每个饲养员都很有个性,他们对猪的编号系统都完全不一样。
藤瑟先生:你们不统一编号不会导致管理混乱吗?
工作人员:不会的,无限大养猪场中的饲养员的个性丰富程度也是无穷的,我们必须要充分尊重饲养员们的个性,这自有解决方法。
由于4DViewer代码混乱且计算截面在CPU端性能低下,今年7月底我决定尝试使用新技术WebGPU API重新实现四维渲染引擎,因为WebGPU的计算着色器可以让截面计算也放在GPU中,彻底解决性能问题,于是新项目——Tesserxel诞生了。Tesserxel取的是单词Tesseract(四维立方体的拉丁词根tessera“四”)和Pixel(像素)。
Tesserxel自带的示例库截图
目前Tesserxel实现了以下功能:
下面就让我们进入Tesserxel构建的四维世界。这里是示例场景库链接(注意要启用WebGPU才打得开):
https://wxyhly.github.io/tesserxel/examples/
请参考玩Tesserxel后续系列文章中的教程深入了解Tesserxel的玩法哦~
启用WebGPU方法: WebGPU是一个实验性的API,是WebGL的未来“接班人”,它的标准还处于W3C的草稿阶段,未正式发布。目前据说仅Windows下Chrome提供较好支持,而且想开启这项功能还有点麻烦,首先你需要下载Canary版本的Chrome浏览器(谷歌官网,或自行找下载资源),添加–enable-unsafe-webgpu参数启动浏览器,打开chrome://flags/,将WebGPU Developer Features打开(选Enabled)就可以启用WebGPU。
目前的Tesserxel只是一个早期版本,后续会补充Tesserxel说明手册,继续开发更多物理解算、高级材质、离线渲染,以及基于Tesserxel引擎的4D游戏等。(但愿不鸽~)
How do you play piano using a computer keyboard? Actually, there are many computer piano software options available, such as EveryOnePiano which I’ve used before, and the more famous FreePiano. Of course, we can also create something similar ourselves. My online piano doesn’t have a specific name, so I simply used the abbreviation EOP (EveryOnePiano) for the Github repository name.
The basic usage instructions can be found in the “?” menu after opening (not very detailed, it’s better to check the examples given later in this article). Click the keyboard button and a virtual computer keyboard will appear on screen, showing the note names, solfege syllables, or functions for each key (holding Ctrl, Shift, Alt will show corresponding function shortcut combinations). This article also includes some usage instructions, but focuses more on recording my thoughts and implementation process.
By the way, if you’re a mobile user, clicking the keyboard can simulate a computer keyboard, and clicking again can simulate a piano keyboard, similar to mobile apps like “Perfect Piano”!
Note: This article does not require readers to have a professional background in topology. The term “algebraic” might give the impression of something abstract and difficult to understand, so this article aims to provide an intuitive understanding of homotopy and homology in algebraic topology. Specific formal definitions and technical details can be found in any algebraic topology textbook.
# This article is a sequel to 《Four-Dimensional Space (X): Knots and Links》. This content requires high spatial imagination and may be difficult to understand in some places, but the entire article contains no formulas and only involves geometry, not algebra, making it suitable for challenging spatial imagination!
# I’ve discovered that not only me, but everyone online is mixing up “knot” terminology, so I won’t bother to correct it either.
In the previous article of this series, we learned about knotting phenomena of two-dimensional surfaces in four-dimensional space and various types of holes, links, etc. The introduction to two-dimensional surface knots wasn’t actually very detailed last time - we only gave some non-trivial tubular knots as examples. Today we want to analyze knots from another perspective. The main content of this article comes from this YouTube video and this paper, which introduces how to construct knots homeomorphic to spheres by rotating trefoil knots (i.e., knots that can be restored to spheres if self-intersections are allowed), and proves that some are truly impossible to unknot while others can be unknotted into spheres through certain steps. Let’s try to unknot some knots in high-dimensional space!