We can almost reach the Lie algebra of any isometry group in a single go. For every in the Lie algebra we must have:because has to be in the isometry group by definition as shown at Section "Lie algebra of a matrix Lie group".
Bibliography:
Mathematical definition that most directly represents this: the orthogonal group is the group of all matrices that preserve the dot product.
The orthogonal group is the group of all matrices that preserve the dot product by 
Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16 Elements of the orthogonal group have determinant plus or minus one by Ciro Santilli 40 Updated 2025-07-16
Ciro Santilli 40 Updated 2025-07-16Although transistors were revolutionary, it is fun to note that they were just "way cheaper and more reliable and smaller" versions of exactly the main functions that a vacuum tube could achieve
The orthogonal group has 2 connected components:
- one with determinant +1, which is itself a subgroup known as the special orthogonal group. These are pure rotations without a reflection.
- the other with determinant -1. This is not a subgroup as it does not contain the origin. It represents rotations with a reflection.
It is instructive to visualize how the looks like in :
- you take the first basis vector and move it to any other. You have therefore two angular parameters.
- you take the second one, and move it to be orthogonal to the first new vector. (you can choose a circle around the first new vector, and so you have another angular parameter.
- at last, for the last one, there are only two choices that are orthogonal to both previous ones, one in each direction. It is this directio, relative to the others, that determines the "has a reflection or not" thing
As a result it is isomorphic to the direct product of the special orthogonal group by the cyclic group of order 2:
A low dimensional example:because you can only do two things: to flip or not to flip the line around zero.
Note that having the determinant plus or minus 1 is not a definition: there are non-orthogonal groups with determinant plus or minus 1. This is just a property. E.g.:has determinant 1, but:so is not orthogonal.
Ciro Santilli started taking some notes at: github.com/cirosantilli/awesome-whole-cell-simulation. but they are going to be all migrated here.
It is interesting to note how one talks about single cell analysis, in contrast to whole cell simulation: experimentally it is hard to analyse a single cell. But theoretically, it is hard to simulate a single cell. This mismatch is perhaps the ultimate frontier of molecular biology.
By Andy Haas, an experimental particle physics professor: as.nyu.edu/content/nyu-as/as/faculty/andy-haas.html What an awesome dude!
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!
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-calculusArticles 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/derivativeVideo 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: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.
- 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
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. 
- Infinitely deep tables of contents:
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Further documentation can be found at: docs.ourbigbook.com
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