## Sekai π πΊ

Sekai is the kanji for δΈη, meaning “the world”. That’s a great word because of the scale it designates.

Goal Show that
$$ \ln (x) \le x - 1 $$ for all $x > 0$.
Proof by inverse functions Think about their reflection along the line $y = x$ (i.e. their inverse function). Then we get
$$e^x \ge x + 1$$ for all $x \in \Bbb{R}$, which is obviously true.
Proof by definite integrals Here’s a second proof using definite integrals.
Case 1: x > 1 The rectangle $[1,x] \times [0,1]$ contains the region under the graph of the reciprocal $y = 1/x$ in the domain $[1,x]$.
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## A Plot for Euler's Constant

### My first PGF Plot for step functions

I firsted tried with foreach, but that would create so much paths. I found them difficult to operate on later, for example, with PGF plots library fillbetween. const plot is a better solution.
\documentclass{standalone} \usepackage{pgfplots} \pgfplotsset{compat=1.18} \usetikzlibrary{patterns} \usepgfplotslibrary{fillbetween} \begin{document} \newcommand\myN{8} \pgfplotsset{ axis lines=center, legend style={at={(1,1)},anchor=north east,fill=none}, title style={at={(0.5,1.05)}}, every axis x label/.style={ at={(ticklabel* cs:1)}, anchor=west, }, every axis y label/.style={ at={(ticklabel* cs:1)}, anchor=south, }, } \begin{tikzpicture} \begin{axis}[ title={sum of hatched region converges to Euler's constant}, xlabel={$x$}, ylabel={$y$}, xmin=0, xmax={\the\numexpr\myN+2}, ymin=0, ymax=1.
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## Simpler Diagram for Trigonometric Functions With Unit Circle

Background In my previous post about unit circle and trigonometric functions, I included a graph with three trigonometry functions. I’m quite satisfied with my TikZ picture.
Problem Unluckily, a secondary school student found that my diagram was too complicated.
Solution \documentclass[tikz,border=2pt]{standalone} \usepackage{pgfplots} \pgfplotsset{compat=1.18} \usetikzlibrary{calc} \newcommand\mytheta{110} %angle theta \begin{document} \begin{tikzpicture}[scale=2] \coordinate[label=left:$O$] (O) at (0,0); \coordinate[label=above left:${A = (\cos\theta, \sin\theta)}$] (A) at (\mytheta:1); \coordinate[label=below left:${B = (\cos(-\theta), \sin(-\theta))}$] (B) at (-\mytheta:1); \coordinate[label=right:$E$] (E) at (1,0); \draw (O) circle (1); \draw (A) -- (O) node [midway, left] {$1$} -- (E); \draw (B) -- (O); \draw[-stealth] ($(O)!
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## Tikz Illustration for Triangle Inequality

I thought the code would be easy, but it turned out that I spent one hour and half to draw this figure.
\documentclass[border=2pt,preview]{standalone} \usepackage{tikz} \usepackage{caption} \newcommand{\sideA}{2} \newcommand{\sideB}{3} \newcommand{\sideC}{6} \colorlet{myred}{red!40!yellow} \colorlet{mygreen}{green!40} \colorlet{myblue}{blue!20} \begin{document} \begin{figure} \centering \begin{tikzpicture}[thick] \coordinate (A) at (0,0); \coordinate (B) at (\sideC,0); \draw[<->,>=stealth,myred] (A) -- ++ (-\sideA, 0) node [pos=0.5,below] {$\color{myred}{a}$}; \draw[<->,>=stealth,mygreen] (B) -- ++ (\sideB, 0) node [pos=0.5,below] {$\color{mygreen}{b}$}; \draw[<->,>=stealth,myblue] (A) -- (B) node [pos = 0.5, below]{$\color{myblue}{c}$}; \draw[dashed,myred] (A) circle (\sideA); \draw[dashed,mygreen] (B) circle (\sideB); \end{tikzpicture} \captionsetup{labelformat=empty} \caption{triangle inequality ${\color{myred}{a}} + {\color{mygreen}{b}} > {\color{myblue}{c}}$ with $\color{myred}{a = \sideA}$, $\color{mygreen}{b = \sideB}$, $\color{myblue}{c = \sideC}$} \end{figure} \end{document} The colors are customized for the Discord bot TeXit.
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## Git Warning: LF Will Be Replaced by CRLF

I’m using Git Bash while writing this post. I’ve never tested the commands on *nix.
Background Some of the tags and titles were written as “LateX”/“latex” in the source files. I batch corrected them with Git Grep and GNU Sed.
git grep --name-only -IiE ^-\ latex$'\r'? content/{post,page} | \ xargs -n1 sed -i "s/^- latex/- LaTeX/I" I tried to match
a leading - using ^- a whitespace \ escaped by a backslash to avoid wrapping the special character below with double quotes the string “latex” (case-insensitive) the carriage return \r, which is represented by $'\r' in bash, for at most once (?
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## LaTeX Table Background Color for TeXit

Problem In my previous post about LaTeX tables, I used \cellcolor{white} to override the \rowcolor{cyan} in a cell in the leftmost column, which was the column type name.
On Discord, that cell became black because TeXit inverted black and white.
Discussion I first tried blending black and white, and I found that black!27 and black!28 give a good approximation to Discord’s background color.
Thanks to this Discord Color Palette and a bot command channel on a server, I’ve observed that the desired color is RGB(54,57,62).
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## Line Equation in Intercept Form

### LaTeX code for a TikZ figure

This is a first draft of a TikZ picture illustraing this classical formula to be used for math help channels.
Adding \caption{for the picture} without “Figure 1:” requires \usepackage{caption} and wrapping with \begin{figure}. It also possible to use the primitive TeX command \par, but it would be complicated to use that with standalone. In the previous post, the SVG picture from the LaTeX table in an article has too much useless whitespace around the table.
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## LaTeX Multirow Multicol Table Coloring

Inspiration Number of two-digit numbers not containing the digit ‘8’.
Code The standalone class can’t be used with \caption. *{8}{c} means 8 c. >{\columncolor{cyan}}c gives the column color. It \usepackage[table]{xcolor}. \rowcolor{cyan} overrides the column colors. \cellcolor{white} overrides the above two commands. It has to be carried out one-by-one without any fancy package. \multicolumn{n}{<alignment>}{content} defines a multi-column cell. \multirow{n}{*}{content} defines a multi-row cell. It \usepackage{multirow}. It can’t contain \multicolumn. \cline{i-j} draws a horizontal line above the row from column i to column j.
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## Upgrade Font Awesome in Beautiful Hugo Theme

### Technical adjustment to Font Awesome's official guide

Background Beautiful Hugo comes with Font Awesome icons. However, it seems that it’s no longer maintained.
Goal To upgrade Font Awesome to the current stable version (6.1.1).
Problem The official guide for self-hosting Font Awesome Web Fonts + CSS suggests users to place the following files
filepath role /fontawesome/webfonts/ Free Web Fonts /fontawesome/css/all.min.css CSS file for using Free Web Fonts into somewhere in my theme. The left column of the diagram features the folder assets/.
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## Second Homothety between Nine-Point Circle and Circumcircle

### Proof of Euler line by h(G, β1/2)

Motivation I saw someone illustrating his/her solution with a “superior triangle”.
This reminds me the homothety about the centroid of factor β1/2.
The above picture \usetikzlibrary{calc} for computing coordinates from those of existing points. (A)!.25!(B) means $(A)+.25[(B)-(A)]$.
\begin{tikzpicture}[scale=2] \coordinate (D) at (-0.7,1); \coordinate (E) at (-1,0); \coordinate (F) at (1,0); \coordinate (A) at ($(E)!.5!(F)$); \coordinate (B) at ($(F)!.5!(D)$); \coordinate (C) at ($(D)!.5!(E)$); \coordinate (G) at ($(D)!.5!(E)!1/3!(F)$); \draw (A) -- (B) -- (C) -- cycle; \draw (D) -- (E) -- (F) -- cycle; \begin{scriptsize} \fill (G) circle (0.
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