Visualizing cycles in row-major transposition encodings

2021-01-23, post № 239

mathematics, programming, c++, shell, #matrix, #encoding, #permutation, #rainbow

A matrix A\in \mathcal{D}^{h\times w} of discretely representable entries \mathcal{D} may be linearly layed out in memory using row-major order, concatenating successive rows into a contiguous (h\cdot w \cdot\texttt{sizeof}\,\mathcal{D})-bytes long array. Such a representation, however, is disruptive to the matrix’ two-dimensional nature: whilst horizontally neighboring elements remain neighbors, vertically neighboring entries are torn apart by insertion of w non-neighboring elements. As such, on matrices naturally defined operations get distorted by this encoding.
One such inherently two-dimensional operation is matrix transposition. In the realm of matrices, \mathcal{D}^{h\times w}\neq\mathcal{D}^{w\times h} are for nonsquare dimensions semantically different, being mapped to one another by transposition. Projecting onto their encoding, this semantic is lost and one is left with a permutation on memory \bullet^\top\in\mathrm{Sym}(\{1,\dots,hw\}).
To visualize this permutation, its cycle decomposition is computed of which each cycle is given a color of the rainbow dyeing this cycle’s corresponding two-dimensional pixels when interpreting its path on the underlying array in the semantics of the original matrix.

Initial transposition cycles

Above listed are all visualizations for 1\leq h\leq 8,1\leq w\leq 16, shuffled. Whilst some behave extremely regularly — for example square matrices’ transposition permutations decompose into transpositions —, others are wildly intricate. Each of them adheres to a rotational symmetry; the top left and bottom right are fixed points.

Factoids #2

2020-12-26, post № 238

mathematics, #bijection, #calculus, #Lipschitz, #naturals

VII) Cardinality coercion: \mathbb{R}\hookleftarrow\{\mathbb{N}\to\mathbb{N}\}\twoheadrightarrow\mathbb{R}

Claim. There exist both \iota:\{\mathbb{N}\to\mathbb{N}\}\hookrightarrow\mathbb{R} together with \pi:\{\mathbb{N}\to\mathbb{N}\}\twoheadrightarrow\mathbb{R}.
Proof. Iota. Define \iota:\{\mathbb{N}\to\mathbb{N}\}\hookrightarrow\mathbb{R} via
    \iota(a):=\quad&\sum_{j=1}^\infty 2^{\left(1-\sum_{i=1}^{j}(1+a(i))\right)}\cdot(2^{a(j)}-1)\\
    =\quad&\left(0.\underbrace{1111\dots11}_{\times a(1)}0\underbrace{111\dots11111}_{\times a(2)}0\dots\right)_2
and observe any sequence’s reconstructibility by dyadic expansion.
Pi. Define \pi:\{\mathbb{N}\to\mathbb{N}\}\twoheadrightarrow\mathbb{R} via
    \pi(a):=\quad&a(1)+\sum_{j=2}^\infty 2^{1-j}\cdot\delta_{a(j)\in 2\mathbb{Z}}\\
    =\quad&a(1)+\left(0.\delta_{a(2)\in 2\mathbb{Z}}\delta_{a(3)\in 2\mathbb{Z}}\delta_{a(4)\in 2\mathbb{Z}}\dots\right)_2
and observe any real’s constructibility by dyadic expansion.

Thus, \{\mathbb{N}\to\mathbb{N}\}\cong\mathbb{R} in \mathbf{(SET)} is shown.

A Month of Pining

2020-11-28, post № 237

hardware review, #opinion, #diary, #freedom

Precisely one month ago, I opened a long awaited parcel shipped directly from Hong Kong: my very own Pinebook Pro had arrived. Eager to further my grasp on both software and hardware freedom, I unwrapped my mail and began to incorporate pine64’s flagship laptop into my workflow. Accompanying this journey, I wrote a diary which I share in this post.

Whilst a Pinebook may not boast with specs of performance, novel input methods or included licenses, it is part of a niche computing sector which values and upholds principal human rights even in the alien realm of microprocessor-based technology. Following my steadily growing discomfort regarding my computing equipment over the past months, I hoped for the Pinebook Pro to introduce me to a world of productive, in part minimalist and most importantly truly free computing experience. After a month of usage, I can confidently proclaim it to have delivered on this front more satisfactory than any other of my multitude of computing devices I have amassed over the years.

Especially in current times, it is calming to interact with an intricately mingled system of physical machinery and abstract commandments in which I can at least to a certain extent trust, lifting an ever-present veil of fear cloaking most boxes of bits. A Pinebook comes with software switches to turn off the built-in camera, microphone and WiFi card as well as Manjaro KDE Plasma running out of the box. Whilst I would prefer not having a camera or microphone built into my device at all and have a real physical switch to turn both off, as well as a more minimalist desktop environment, I am capable to entrust it my presence infront of it; in contrast to e. g. a proprietary phone lying on a tabletop nearby.

I am unaffiliated with pine64 [1] and purchased my Pinebook Pro myself. I wrote the entirety of this blog post including the diary seen below using my Pinebook [2].

A detailed account of my experience with my Pinebook is given by a diary I wrote over the course of November 2020: