Just like for the finite general linear group, the definition of special also works for finite fields, where 1 is the multiplicative identity!
Note that the definition of orthogonal group may not have such a clear finite analogue on the other hand.
The group of all transformations that preserve some bilinear form, notable examples:
- orthogonal group preserves the inner product
- unitary group preserves a Hermitian form
- Lorentz group preserves the Minkowski inner product
The Horrible Horrendous Terrible Tremendous Mining Pool inscribed a few cute Coinbase messages during their operation in 2012-2013.
Many of their messages also mention
SockThing, which was part of their mining infrastructure:Starting from their very first ASCII transaction on block 197602 (2012-09-07), there is what seems to be a poem spread across several transactions. Some of the lines are repeated, presumably because they didn't update the current line to a new line and so mined the same thing multiple times:The sentences are not very coherent together, perhaps this is because lines were chosen by different miners one at a time.
I am a pretty princess
covered in mud and blood
water with stuff in it
like everything else that wiggles or jiggles
screaming might not be your waY
see no reason to operate otherwise since
came into the world naked, wet and screaming
but silence will never be mine
until I am dead
but the smell will also give that away
gather all my things
load them in a big boat
airlift that to Kansas
and light it on fire
drop it from 7,000 feet
then railgun my corpse straight down
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.
What happens to the definition of the orthogonal group if we choose other types of symmetric bilinear forms by 
Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16We looking at the definition the orthogonal group is the group of all matrices that preserve the dot product, we notice that the dot product is one example of positive definite symmetric bilinear form, which in turn can also be represented by a matrix as shown at: Section "Matrix representation of a symmetric bilinear form".
By looking at this more general point of view, we could ask ourselves what happens to the group if instead of the dot product we took a more general bilinear form, e.g.:The answers to those questions are given by the Sylvester's law of inertia at Section "All indefinite orthogonal groups of matrices of equal metric signature are isomorphic".
- : another positive definite symmetric bilinear form such as ?
- what if we drop the positive definite requirement, e.g. ?
Elements of the orthogonal group have determinant plus or minus one by Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16 The orthogonal group is the group of all matrices with orthonormal rows and orthonormal columns by Ciro Santilli 37 Updated 2025-07-16
Ciro Santilli 37 Updated 2025-07-16Or equivalently, the set of rows is orthonormal, and so is the set of columns. TODO proof that it is equivalent to the orthogonal group is the group of all matrices that preserve the dot product.
For now we are going to keep this site porn-free and only link to prevent bad things from happening, as it might violate GitHub porn policy depending on how it is hosted, and Google may dislike it. Video "What is more obscene: sex or war? scene from The People vs. Larry Flynt".
If illegal porn were to ever be found, we would be unable to acknowledge that, and would just have to silently remove it. Of course, the reproducible nature of Bitcoin Inscription Indexer means that anyone who regenerates the data would immediately see such entries in the diff.
Another issue is that it can be quite hard to determine if porn is legal or illegal, e.g. it can be hard to distinguish legal an illegal ages or revenge vs consensual porn, especially at the low resolutions that you may expect to find embedded in the blockchain. We are generally going to be quite strict about this, and in case of uncertainty on porn legality we will censor first.
OK the list:
- tx 4c903a377addab7c1e35a685d3dabc664199e406374b1e5ce2fc59e78fb5b754 (2016-07-09) contains an animated GIF of a woman pole dancing. She is seen from behind, in revealing blue clothes that show her buttocks. Reproduced at: twitter.com/cirosantilli/status/1755378949117370668. The file contains the following strings embedded into it:and:
This GIF file was assembled with GIF Construction Set from:
Alchemy Mindworks Inc.
This comment block will not appear in files created with a registered version of GIF Construction SetCryptograffiti.info messages now cheaper.
HD 101584 is a star located in the constellation of Centaurus, approximately 1,140 light-years away from Earth. It is classified as a post-AGB (Asymptotic Giant Branch) star, which indicates that it is in a late stage of stellar evolution. In this phase, a star has exhausted the hydrogen in its core and has moved beyond the red giant phase, potentially leading towards becoming a white dwarf.
Group of the unitary matrices.
Complex analogue of the orthogonal group.
One notable difference from the orthogonal group however is that the unitary group is connected "because" its determinant is not fixed to two disconnected values 1/-1, but rather goes around in a continuous unit circle. is the unit circle.
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:
All our software is open source and hosted at: github.com/ourbigbook/ourbigbook
Further documentation can be found at: docs.ourbigbook.com
Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact