New articles - OurBigBook.com

Ciro Santilli 40 Updated 2025-07-16

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Group homomorphism by

Ciro Santilli 40 Updated 2025-07-16

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Like isomorphism, but does not have to be one-to-one: multiple different inputs can have the same output.

The image is as for any function smaller or equal in size as the domain of course.

This brings us to the key intuition about group homomorphisms: they are a way to split out a larger group into smaller groups that retains a subset of the original structure.

As shown by the fundamental theorem on homomorphisms, each group homomorphism is fully characterized by a normal subgroup of the domain.

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ELF Hello World Tutorial / Specified file formats by

Ciro Santilli 40 Updated 2025-07-16

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The ELF standard specifies multiple file formats:

Object files (.o).
Intermediate step to generating executables and other formats:
```
Source code

    |
    | Compilation
    |
    v

Object file

    |
    | Linking
    |
    v

Executable
```
Object files exist to make compilation faster: with make, we only have to recompile the modified source files based on timestamps.
We have to do the linking step every time, but it is much less expensive.

Executable files (no standard Linux extension).
This is what the Linux kernel can actually run.

Archive files (.a).
Libraries meant to be embedded into executables during the Linking step.

Shared object files (.so).
Libraries meant to be loaded when the executable starts running.

Core dumps.
Such files may be generated by the Linux kernel when the program does naughty things, e.g. segfault.
They exist to help debugging the program.

In this tutorial, we consider only object and executable files.

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Orthogonal matrix by

Ciro Santilli 40 Updated 2025-07-16

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Members of the orthogonal group.

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Astera Institute by

Ciro Santilli 40 Updated 2025-07-16

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astera.org/agi/

By the rich founder of Mt. Gox and Ripple, Jed McCaleb.

Obelisk is the Artificial General Intelligence laboratory at Astera. We are focused on the following problems: How does an agent continuously adapt to a changing environment and incorporate new information? In a complicated stochastic environment with sparse rewards, how does an agent associate rewards with the correct set of actions that led to those rewards? How does higher level planning arise?

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Unitary matrix by

Ciro Santilli 40 Updated 2025-07-16

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Complex analogue of orthogonal matrix.

Applications:

in quantum computers programming basically comes down to creating one big unitary matrix as explained at: quantum computing is just matrix multiplication

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Finitely generated group by

Ciro Santilli 40 Updated 2025-07-16

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Symmetric matrix by

Ciro Santilli 40 Updated 2025-07-16

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A matrix that equals its transpose:

M = M^{T}

(1)

Can represent a symmetric bilinear form as shown at matrix representation of a symmetric bilinear form, or a quadratic form.

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Cycle graph (algebra) by

Ciro Santilli 40 Updated 2025-07-16

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How to build it: math.stackexchange.com/questions/3137319/how-in-general-does-one-construct-a-cycle-graph-for-a-group/3162746#3162746 good answer with ASCII art. You basically just pick each element, and repeatedly apply it, and remove any path that has a longer version.

Immediately gives the generating set of a group by looking at elements adjacent to the origin, and more generally the order of each element.

TODO uniqueness: can two different groups have the same cycle graph? It does not seem to tell us how every element interact with every other element, only with itself. This is in contrast with the Cayley graph, which more accurately describes group structure (but does not give the order of elements as directly), so feels like it won't be unique.

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ELF Hello World Tutorial / Implementations by

Ciro Santilli 40 Updated 2025-07-16

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Compiler toolchains generate and read ELF files.
Sane compilers should use a separate standalone library to do the dirty work. E.g., Binutils uses BFD (in-tree and canonical source).

Operating systems read and run ELF files.
Kernels cannot link to a library nor use the C stlib, so they are more likely to implement it themselves.
This is the case of the Linux kernel 4.2 which implements it in th file fs/binfmt_elf.c.

Specialized libraries. Examples:
- github.com/eliben/pyelftools. By a hardcore Googler: plus.google.com/+EliBenderskyGplus/posts
- sourceforge.net/projects/elftoolchain

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ELF Hello World Tutorial / Minimal ELF file by

Ciro Santilli 40 Updated 2025-07-16

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It is non-trivial to determine what is the smallest legal ELF file, or the smaller one that will do something trivial in Linux.

Some impressive attempts:

In this example we will consider a saner hello world example that will better capture real life cases.

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ELF Hello World Tutorial / Generate the example by

Ciro Santilli 40 Updated 2025-07-16

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Let's break down a minimal runnable Linux x86-64 example:

hello_world.asm

section .data
    hello_world db "Hello world!", 10
    hello_world_len  equ $ - hello_world
section .text
    global _start
    _start:
        mov rax, 1
        mov rdi, 1
        mov rsi, hello_world
        mov rdx, hello_world_len
        syscall
        mov rax, 60
        mov rdi, 0
        syscall

Compiled with:

nasm -w+all -f elf64 -o 'hello_world.o' 'hello_world.asm'
ld -o 'hello_world.out' 'hello_world.o'

TODO: use a minimal linker script with -T to be more precise and minimal.

Versions:

NASM 2.10.09
Binutils version 2.24 (contains ld)
Ubuntu 14.04

We don't use a C program as that would complicate the analysis, that will be level 2 :-)

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ELF Hello World Tutorial / Object hd by

Ciro Santilli 40 Updated 2025-07-16

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Running:

hd hello_world.o

gives:

00000000  7f 45 4c 46 02 01 01 00  00 00 00 00 00 00 00 00  |.ELF............|
00000010  01 00 3e 00 01 00 00 00  00 00 00 00 00 00 00 00  |..>.............|
00000020  00 00 00 00 00 00 00 00  40 00 00 00 00 00 00 00  |........@.......|
00000030  00 00 00 00 40 00 00 00  00 00 40 00 07 00 03 00  |....@.....@.....|
00000040  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
*
00000080  01 00 00 00 01 00 00 00  03 00 00 00 00 00 00 00  |................|
00000090  00 00 00 00 00 00 00 00  00 02 00 00 00 00 00 00  |................|
000000a0  0d 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000000b0  04 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000000c0  07 00 00 00 01 00 00 00  06 00 00 00 00 00 00 00  |................|
000000d0  00 00 00 00 00 00 00 00  10 02 00 00 00 00 00 00  |................|
000000e0  27 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |'...............|
000000f0  10 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000100  0d 00 00 00 03 00 00 00  00 00 00 00 00 00 00 00  |................|
00000110  00 00 00 00 00 00 00 00  40 02 00 00 00 00 00 00  |........@.......|
00000120  32 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |2...............|
00000130  01 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000140  17 00 00 00 02 00 00 00  00 00 00 00 00 00 00 00  |................|
00000150  00 00 00 00 00 00 00 00  80 02 00 00 00 00 00 00  |................|
00000160  a8 00 00 00 00 00 00 00  05 00 00 00 06 00 00 00  |................|
00000170  04 00 00 00 00 00 00 00  18 00 00 00 00 00 00 00  |................|
00000180  1f 00 00 00 03 00 00 00  00 00 00 00 00 00 00 00  |................|
00000190  00 00 00 00 00 00 00 00  30 03 00 00 00 00 00 00  |........0.......|
000001a0  34 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |4...............|
000001b0  01 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000001c0  27 00 00 00 04 00 00 00  00 00 00 00 00 00 00 00  |'...............|
000001d0  00 00 00 00 00 00 00 00  70 03 00 00 00 00 00 00  |........p.......|
000001e0  18 00 00 00 00 00 00 00  04 00 00 00 02 00 00 00  |................|
000001f0  04 00 00 00 00 00 00 00  18 00 00 00 00 00 00 00  |................|
00000200  48 65 6c 6c 6f 20 77 6f  72 6c 64 21 0a 00 00 00  |Hello world!....|
00000210  b8 01 00 00 00 bf 01 00  00 00 48 be 00 00 00 00  |..........H.....|
00000220  00 00 00 00 ba 0d 00 00  00 0f 05 b8 3c 00 00 00  |............<...|
00000230  bf 00 00 00 00 0f 05 00  00 00 00 00 00 00 00 00  |................|
00000240  00 2e 64 61 74 61 00 2e  74 65 78 74 00 2e 73 68  |..data..text..sh|
00000250  73 74 72 74 61 62 00 2e  73 79 6d 74 61 62 00 2e  |strtab..symtab..|
00000260  73 74 72 74 61 62 00 2e  72 65 6c 61 2e 74 65 78  |strtab..rela.tex|
00000270  74 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |t...............|
00000280  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000290  00 00 00 00 00 00 00 00  01 00 00 00 04 00 f1 ff  |................|
000002a0  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000002b0  00 00 00 00 03 00 01 00  00 00 00 00 00 00 00 00  |................|
000002c0  00 00 00 00 00 00 00 00  00 00 00 00 03 00 02 00  |................|
000002d0  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000002e0  11 00 00 00 00 00 01 00  00 00 00 00 00 00 00 00  |................|
000002f0  00 00 00 00 00 00 00 00  1d 00 00 00 00 00 f1 ff  |................|
00000300  0d 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000310  2d 00 00 00 10 00 02 00  00 00 00 00 00 00 00 00  |-...............|
00000320  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000330  00 68 65 6c 6c 6f 5f 77  6f 72 6c 64 2e 61 73 6d  |.hello_world.asm|
00000340  00 68 65 6c 6c 6f 5f 77  6f 72 6c 64 00 68 65 6c  |.hello_world.hel|
00000350  6c 6f 5f 77 6f 72 6c 64  5f 6c 65 6e 00 5f 73 74  |lo_world_len._st|
00000360  61 72 74 00 00 00 00 00  00 00 00 00 00 00 00 00  |art.............|
00000370  0c 00 00 00 00 00 00 00  01 00 00 00 02 00 00 00  |................|
00000380  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000390

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100,000 Genomes Project by

Ciro Santilli 40 Updated 2025-07-16

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Astera Institute person by

Ciro Santilli 40 Updated 2025-07-16

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ELF Hello World Tutorial / Executable hd by

Ciro Santilli 40 Updated 2025-07-16

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Running:

hd hello_world.out

gives:

00000000  7f 45 4c 46 02 01 01 00  00 00 00 00 00 00 00 00  |.ELF............|
00000010  02 00 3e 00 01 00 00 00  b0 00 40 00 00 00 00 00  |..>.......@.....|
00000020  40 00 00 00 00 00 00 00  10 01 00 00 00 00 00 00  |@...............|
00000030  00 00 00 00 40 00 38 00  02 00 40 00 06 00 03 00  |....@.8...@.....|
00000040  01 00 00 00 05 00 00 00  00 00 00 00 00 00 00 00  |................|
00000050  00 00 40 00 00 00 00 00  00 00 40 00 00 00 00 00  |..@.......@.....|
00000060  d7 00 00 00 00 00 00 00  d7 00 00 00 00 00 00 00  |................|
00000070  00 00 20 00 00 00 00 00  01 00 00 00 06 00 00 00  |.. .............|
00000080  d8 00 00 00 00 00 00 00  d8 00 60 00 00 00 00 00  |..........`.....|
00000090  d8 00 60 00 00 00 00 00  0d 00 00 00 00 00 00 00  |..`.............|
000000a0  0d 00 00 00 00 00 00 00  00 00 20 00 00 00 00 00  |.......... .....|
000000b0  b8 01 00 00 00 bf 01 00  00 00 48 be d8 00 60 00  |..........H...`.|
000000c0  00 00 00 00 ba 0d 00 00  00 0f 05 b8 3c 00 00 00  |............<...|
000000d0  bf 00 00 00 00 0f 05 00  48 65 6c 6c 6f 20 77 6f  |........Hello wo|
000000e0  72 6c 64 21 0a 00 2e 73  79 6d 74 61 62 00 2e 73  |rld!...symtab..s|
000000f0  74 72 74 61 62 00 2e 73  68 73 74 72 74 61 62 00  |trtab..shstrtab.|
00000100  2e 74 65 78 74 00 2e 64  61 74 61 00 00 00 00 00  |.text..data.....|
00000110  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
*
00000150  1b 00 00 00 01 00 00 00  06 00 00 00 00 00 00 00  |................|
00000160  b0 00 40 00 00 00 00 00  b0 00 00 00 00 00 00 00  |..@.............|
00000170  27 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |'...............|
00000180  10 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000190  21 00 00 00 01 00 00 00  03 00 00 00 00 00 00 00  |!...............|
000001a0  d8 00 60 00 00 00 00 00  d8 00 00 00 00 00 00 00  |..`.............|
000001b0  0d 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000001c0  04 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000001d0  11 00 00 00 03 00 00 00  00 00 00 00 00 00 00 00  |................|
000001e0  00 00 00 00 00 00 00 00  e5 00 00 00 00 00 00 00  |................|
000001f0  27 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |'...............|
00000200  01 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000210  01 00 00 00 02 00 00 00  00 00 00 00 00 00 00 00  |................|
00000220  00 00 00 00 00 00 00 00  90 02 00 00 00 00 00 00  |................|
00000230  08 01 00 00 00 00 00 00  05 00 00 00 07 00 00 00  |................|
00000240  08 00 00 00 00 00 00 00  18 00 00 00 00 00 00 00  |................|
00000250  09 00 00 00 03 00 00 00  00 00 00 00 00 00 00 00  |................|
00000260  00 00 00 00 00 00 00 00  98 03 00 00 00 00 00 00  |................|
00000270  4c 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |L...............|
00000280  01 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000290  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000002a0  00 00 00 00 00 00 00 00  00 00 00 00 03 00 01 00  |................|
000002b0  b0 00 40 00 00 00 00 00  00 00 00 00 00 00 00 00  |..@.............|
000002c0  00 00 00 00 03 00 02 00  d8 00 60 00 00 00 00 00  |..........`.....|
000002d0  00 00 00 00 00 00 00 00  01 00 00 00 04 00 f1 ff  |................|
000002e0  00 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
000002f0  11 00 00 00 00 00 02 00  d8 00 60 00 00 00 00 00  |..........`.....|
00000300  00 00 00 00 00 00 00 00  1d 00 00 00 00 00 f1 ff  |................|
00000310  0d 00 00 00 00 00 00 00  00 00 00 00 00 00 00 00  |................|
00000320  00 00 00 00 04 00 f1 ff  00 00 00 00 00 00 00 00  |................|
00000330  00 00 00 00 00 00 00 00  2d 00 00 00 10 00 01 00  |........-.......|
00000340  b0 00 40 00 00 00 00 00  00 00 00 00 00 00 00 00  |..@.............|
00000350  34 00 00 00 10 00 02 00  e5 00 60 00 00 00 00 00  |4.........`.....|
00000360  00 00 00 00 00 00 00 00  40 00 00 00 10 00 02 00  |........@.......|
00000370  e5 00 60 00 00 00 00 00  00 00 00 00 00 00 00 00  |..`.............|
00000380  47 00 00 00 10 00 02 00  e8 00 60 00 00 00 00 00  |G.........`.....|
00000390  00 00 00 00 00 00 00 00  00 68 65 6c 6c 6f 5f 77  |.........hello_w|
000003a0  6f 72 6c 64 2e 61 73 6d  00 68 65 6c 6c 6f 5f 77  |orld.asm.hello_w|
000003b0  6f 72 6c 64 00 68 65 6c  6c 6f 5f 77 6f 72 6c 64  |orld.hello_world|
000003c0  5f 6c 65 6e 00 5f 73 74  61 72 74 00 5f 5f 62 73  |_len._start.__bs|
000003d0  73 5f 73 74 61 72 74 00  5f 65 64 61 74 61 00 5f  |s_start._edata._|
000003e0  65 6e 64 00                                       |end.|
000003e4

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ELF Hello World Tutorial / Global file structure by

Ciro Santilli 40 Updated 2025-07-16

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An ELF file contains the following parts:

ELF header. Points to the position of the section header table and the program header table.
Section header table (optional on executable). Each has e_shnum section headers, each pointing to the position of a section.
N sections, with N <= e_shnum (optional on executable)
Program header table (only on executable). Each has e_phnum program headers, each pointing to the position of a segment.
N segments, with N <= e_phnum (only on executable)

The order of those parts is not fixed: the only fixed thing is the ELF header that must be the first thing on the file: Generic docs say:

Although the figure shows the program header table immediately after the ELF header, and the section header table following the sections, actual files may differ. Moreover, sections and segments have no specified order. Only the ELF header has a fixed position in the file.

In pictures: sample object file with three sections:

            +-------------------+
            | ELF header        |---+
+---------> +-------------------+   | e_shoff
|           |                   |<--+
| Section   | Section header 0  |
|           |                   |---+ sh_offset
| Header    +-------------------+   |
|           | Section header 1  |---|--+ sh_offset
| Table     +-------------------+   |  |
|           | Section header 2  |---|--|--+
+---------> +-------------------+   |  |  |
            | Section 0         |<--+  |  |
            +-------------------+      |  | sh_offset
            | Section 1         |<-----+  |
            +-------------------+         |
            | Section 2         |<--------+
            +-------------------+

But nothing (except sanity) prevents the following topology:

            +-------------------+
            | ELF header        |---+ e_shoff
            +-------------------+   |
            | Section 1         |<--|--+
+---------> +-------------------+   |  |
|           |                   |<--+  | sh_offset
| Section   | Section header 0  |      |
|           |                   |------|---------+
| Header    +-------------------+      |         |
|           | Section header 1  |------+         |
| Table     +-------------------+                |
|           | Section header 2  |---+            | sh_offset
+---------> +-------------------+   | sh_offset  |
            | Section 2         |<--+            |
            +-------------------+                |
            | Section 0         |<---------------+
            +-------------------+

But some newbies may prefer PNGs :-)

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Human protein by

Ciro Santilli 40 Updated 2025-07-16

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ELF Hello World Tutorial / Section vs segment by

Ciro Santilli 40 Updated 2025-07-16

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We will get into more detail later, but it is good to have it in mind now:

section: exists before linking, in object files.
One ore more sections will be put inside a single segment by the linker.
Major information sections contain for the linker: is this section:
- raw data to be loaded into memory, e.g. .data, .text, etc.
- or metadata about other sections, that will be used by the linker, but disappear at runtime e.g. .symtab, .srttab, .rela.text
segment: exists after linking, in the executable file.
Contains information about how each segment should be loaded into memory by the OS, notably location and permissions.

 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either https://OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade