cyberpunk physicsmaxxing lainpilled sewerslvt femcel twinkhon schizoposter

FUCK NAZIS, FUCK COMMIES

  • 2 Posts
  • 19 Comments
Joined 1 year ago
cake
Cake day: July 27th, 2023

help-circle









  • dyen49k@kbin.socialtoMemes@lemmy.mlBiden on Capitalism
    link
    fedilink
    arrow-up
    7
    arrow-down
    2
    ·
    1 year ago

    动态网自由门 天安門 天安门 法輪功 李洪志 Free Tibet 六四天安門事件 The Tiananmen Square protests of 1989 天安門大屠殺 The Tiananmen Square Massacre 反右派鬥爭 The Anti-Rightist Struggle 大躍進政策 The Great Leap Forward 文化大革命 The Great Proletarian Cultural Revolution 人權 Human Rights 民運 Democratization 自由 Freedom 獨立 Independence 多黨制 Multi-party system 台灣 臺灣 Taiwan Formosa 中華民國 Republic of China 西藏 土伯特 唐古特 Tibet 達賴喇嘛 Dalai Lama 法輪功 Falun Dafa 新疆維吾爾自治區 The Xinjiang Uyghur Autonomous Region 諾貝爾和平獎 Nobel Peace Prize 劉暁波 Liu Xiaobo 民主 言論 思想 反共 反革命 抗議 運動 騷亂 暴亂 騷擾 擾亂 抗暴 平反 維權 示威游行 李洪志 法輪大法 大法弟子 強制斷種 強制堕胎 民族淨化 人體實驗 肅清 胡耀邦 趙紫陽 魏京生 王丹 還政於民 和平演變 激流中國 北京之春 大紀元時報 九評論共産黨 獨裁 專制 壓制 統一 監視 鎮壓 迫害 侵略 掠奪 破壞 拷問 屠殺 活摘器官 誘拐 買賣人口 遊進 走私 毒品 賣淫 春畫 賭博 六合彩 天安門 天安门 法輪功 李洪志 Winnie the Pooh 劉曉波动态网自由门




  • Yep, there’s an alternative, i.e. equivalent mathematical formulation that does everything complex numbers do: the geometric algebra introduced in the article I posted earlier.

    The fundamental object of GA is the “multivector”, which is essentially a sum of scalars, vectors, bivectors and higher grade elements. For instance, you could take the unit x-vector and add it onto some number, say 2, to get the multivector M = 2 + e_x. (To be precise, the space of multivectors is the direct sum over the n-th wedge of the base vector space, n = 0 to dim V).
    Another important concept is k-vectors, which are essentially k-dimensional volume elements. For instance, a bivector is an area with a direction, and a trivector is a volume with a direction (in 3D there is only one possible “direction” for the volume, but in 4D spacetime volumes itself can be oriented like surfaces can be in 3D).

    Then, you introduce the “geometric product” for two vectors a and b:
    ab = a·b + a ∧ b
    where a · b is the normal scalar product between the two vectors, and a ∧ b is the wedge product between them. The wedge product essentially is the plane spanned by the two vectors, and is antisymmetric (a ∧ b = - b ∧ a, because the orientation of the plane is reversed when exchanging the vector). For instance, the unit bivector in the x-y plane is given by
    B_xy = e_x e_y = e_x ∧ e_y
    Notice how the scalar product part of the geometric product is zero, and only the wedge (i.e. bivector part) remains

    In 3D, there are four types (“grades”) of objects: scalars, vectors, bivectors (also known as 3D pseudovectors) and trivectors (or also known as 3D pseudoscalars). It’s already a very rich subject and has many advantages over classical vector calculus, but for replacing complex numbers, we’re mainly concerned with the 2D case.

    In the 2D case, there are three types of objects: Scalars, 2D vectors, and bivectors/2D pseudoscalars. There is only one possible orientation for a 2D plane in 2D, so we just denote a bivector with area A as B = A I, where I = e_x ∧ e_y is the only unit bivector/2D pseudoscalar.

    A nice thing we notice about the I is that it squares to -1 with the geometric product:
    I^2 = (e_x ∧ e_y)^2 = (e_x e_y)^2 = e_x e_y e_x e_y = - e_x e_y e_y e_x = -e_x e_x = -1
    The first step works because the scalar product part between e_x and e_y is zero. The second step is just writing out the square. The third step is e_y e_x = e_y ∧ e_x = - e_x ∧ e_y = -e_x e_y, which again works because e_x · e_y = 0. We see that the 2d pseudoscalar I behaves just like the “classic” imaginary unit i.

    Because the geometric product is associative, and commutative if only scalars and bivectors are involved, the geometric notion of scalars and 2D pseudoscalars can fully replace the notion of complex numbers by making the substitution a + bi -> a + bI.

    If you want to learn more about GA, I can recommend Doran, Lasenby: Geometric Algebra for Physicsists :)