See sections: "Example 1 - N even", "Example 2 - N odd" and "Representation in terms of sines and cosines" of www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal
The transform still has complex numbers.
Summary:Therefore, we only need about half of to represent the signal, as the other half can be derived by conjugation.
- is real
"Representation in terms of sines and cosines" from www.statlect.com/matrix-algebra/discrete-Fourier-transform-of-a-real-signal then gives explicit formulas in terms of .
NumPy for example has "Real FFTs" for this: numpy.org/doc/1.24/reference/routines.fft.html#real-ffts
DFT of with 25 points
. Source at: numpy/fft_plot.py. This plot illustrates how the DFT of a real signal is symmetric around the middle point, and so only half of the transform points are needed to reconstruct the original signal. We also see how the phase of the sinusoids determines if their DFT components are real or imaginary.An important case is the discrete logarithm of the cyclic group in which the group is a cyclic group.
AKA how this GitHub page gets served under the domain: cirosantilli.com
Ciro only touches this very rarely, and always forgets and go into great pain whenever a change needs to done, so it is important to document it.
The last change was of 2019-07-07, when Ciro moved from the www subdomain www.cirosantilli.com to the APEX cirosantilli.com. A redirect is setup from the www subdomain to APEX.
GoDaddy DNS entries:
Type Name Value TTL
A @ 185.199.108.153 1 Hour
A @ 185.199.109.153 1 Hour
A @ 185.199.110.153 1 Hour
A @ 185.199.111.153 1 Hour
CNAME www cirosantilli.github.io 1 HourMoved cirosantilli.com to Porkbun 2022-02, unfortunatly records were not automatically updated and domain went down for a bit, upadded to new entries for IPv6 as well which are not documented by GitHub:
TYPE HOST ANSWER TTL PRIORITY OPTIONS
A cirosantilli.com 185.199.108.153 600
A cirosantilli.com 185.199.109.153 600
A cirosantilli.com 185.199.110.153 600
A cirosantilli.com 185.199.111.153 600
AAAA cirosantilli.com 2606:50c0:8000::153 600
AAAA cirosantilli.com 2606:50c0:8001::153 600
AAAA cirosantilli.com 2606:50c0:8002::153 600
AAAA cirosantilli.com 2606:50c0:8003::153 600
CNAME www.cirosantilli.com cirosantilli.github.io 600where the IPs are obtained from: help.github.com/en/articles/setting-up-an-apex-domain#configuring-a-records-with-your-dns-provider (archive).
- Custom domain:
cirosantilli.com - Enforce HTTPS: checked
And the CNAME file is tracked in this repository: CNAME.
There is only a very fine difference between a very good film, and the best films of all time. Perhaps it is something to do on how epic the subject matter is? It is often very hard to tell, and switches between the categories are also possible.
The Purpose of Harvard is Not to Educate People by Sean Carroll (2008) Updated 2025-06-17 +Created 1970-01-01
Maybe they did try once though: Harvard Project Physics.
Our minimal definition of "electronic money" is the following.
Instead of creating legal tender such as Dollars as banknotes or transactions in some complex obscure banking system, the government offers an official simple centralized API that represents it instead.
Each citizen or legal entity has an account there, and transfers between registered users are just simple API calls.
The most practical/precise volt standard.
It motivated the definition of the ampere in the 2019 redefinition of the SI base units
The wiki page en.wikipedia.org/wiki/Josephson_voltage_standard contains amazing schematics of the device, apparently made by the US Government.
- youtu.be/VoRab8U2eS0?t=354 the desired output voltage is 10V
- youtu.be/VoRab8U2eS0?t=475 lists the three most commonly used 10V implementations currently:
- youtu.be/6pgGNJby1lw?t=296 gives the experimental setup used to compare two different references. Notably it involves a nanovoltmeter
Yet another awk-like domain-specific language to do things from the CLI in a ridiculously short humber of character? Oh yes.
Historian Alan B. Carr:
- www.youtube.com/@AlanBCarr. IMPORTANT NOTE: Although Alan B. Carr is a Los Alamos National Laboratory (LANL) employee, this page has absolutely no formal connection with LANL.
Ah, Ciro Santilli loved this one... games young Ciro Santilli played.
List of similar feeling films: www.youtube.com/watch?v=zwYwFoanrNg 11 Underrated Hard Sci-fi Movies by Marvelous Videos (2021)
Cool data embedded in the Bitcoin blockchain Early AtomSea & EMBII uploads Updated 2025-06-17 +Created 1970-01-01
These are of course likely all made by AtomSea & EMBII themselves while developing/testing their upload system.
They are also artsy peoeple themselves, and as pointed at twitter.com/AllenVandever/status/1563964396656812034 what they were doing was basicaly non-fungible token art, which became much much more popular a few years later around 2021.
The first upload that we could find at github.com/cirosantilli/bitcoin-inscription-indexer/tree/3f53e152ec9bb0d070dbcb8f9249d92f89effa70#atomsea-index was tx 44e80475dc363de2c7ee17b286f8cd49eb146165a79968a62c1c2c4cf80772c9 on block 272573 (2013-12-01) but it does not show on Bitfossil: bitfossil.org/44e80475dc363de2c7ee17b286f8cd49eb146165a79968a62c1c2c4cf80772c9/. This is was due to an upload bug explained by the following entry. By looking at the ASCII data at github.com/cirosantilli/bitcoin-inscription-indexer/blob/master/data/out/0272.txt#L449 that this is meant to contain the same content as the following message: a quote from the Bhagavad Gita, so this is definitely a bugged version of the following one.
The next one is tx c9d1363ea517cd463950f83168ce8242ef917d99cd6518995bd1af927d335828 block 272577 (2013-12-02). It actually shows on bifossil and it reads:followed by:The bug message is definitely a reference to the previous non-visible bugged upload bitfossil.org/4b72a223007eab8a951d43edc171befeabc7b5dca4213770c88e09ba5b936e17/, TODO understand exactly how they fucked up. This illustrates the beauty of the blockchain very well: unlike with version control, you don't just see selected snapshots: you see actual debug logs!!!
He who regards
With an eye that is equal
Friends and comrades,
The foe and the kinsman,
The vile, the wicked,
The men who judge him,
And those who belong
To neither faction:
He is the greatest.
WeAreStarStuff.jpgThe filename is of course a reference to the quote/idea: We Are Made of Star-Stuff that was much popularized by Carl Sagan.
bitfossil.org/fac0b9a4f90414710b806fd286e020aea2404498946845ef3783f305dd4cd3a7 (2024-01-13) contains a cropped version with only AtomSea persent.
HugPuddle.jpgThe fourth AtomSea & EMBII upload, and the second image. Message:
HugPuddle Testing Apertus Disk Drive
And then finally we meet Chiharu, EMBII's partner, with her hair painted blond (she's Japanese): ILoveYouMore.jpg.
Then there are two undecoded ones TODO investigate:
Then Nelson-Mandela.jpg.
Then there's an approximation of pi as ASCII decimal fraction on tx 70fd289901bae0409f27237506c330588d917716944c6359a8711b0ad6b4ce76 from block 273522 (2013-12-07):
3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819326117931051185480744623799627495673518857527248912279381830119491298336733624406566430860213949463952247371907021798609437027705392171762931767523846748184676694051320005681271452635608277857713427577896091736371787214684409012249534301465495853710507922796892589235420199561121290219608640344181598136297747713099605187072113499999983729780499510597317328160963185950244594553469083026425223082533446850352619311881710100031378387528865875332083814206171776691473035982534904287554687311595628638823537875937519577818577805321712268066130019278766111959092164201989
tx b8b9f50a354166c46b69ecd47a0fbd20ee78c3471d2557bf275aff1b4cf4752d (2013-12-07) on bitfossil.org) contains Where the Sidewalk Ends by Shel Silverstein:
There is a place where the sidewalk ends
And before the street begins,
And there the grass grows soft and white,
And there the sun burns crimson bright,
And there the moon-bird rests from his flight
To cool in the peppermint wind.
tx 56768b30dec33bd284223d85c23087975e2360b3391d20d505aa59a5675e5379 (2013-12-13, on bitfossil.org):
Dear Aliens,Hey.Sincerely,
EMBII & AtomSea
tx 415c702759893c63b3a57a7d196b014e51b2a33d2396c74b8e71acfaff6b9360 (2013-12-14) contains a poem by 13th century Persian poet Rumi (TODO find bitfossil.org toplevel), starting with:Reproduced e.g. at: www.abuddhistlibrary.com/Buddhism/H%20-%20World%20Religions%20and%20Poetry/World%20Religions/Islam/Teachers/Rumi/My%20dear%20friend/Rumi%20-%20my%20dear%20friend.htm
My dear friend
never lose hope
when the Beloved
sends you away.
bitfossil.org/73ca50321147bac9010bec43d63f7f76857fe9ede240cc89710e28723fdb242f/ (2013-12-14) has message:and links to 3 .txt files
MULTIFILE SUPPORT TEST
1.txt, 2.txt, 3.txt containing single characters 1, 2, and 3.
CompressedLogo.png
. Source. 2013-12-20. Message:Possibly www.linkedin.com/in/colby-nelson-59b538207/.
Colby Nelson and myself burnt the midnight oils designing the APERTUS imagery last night....Thanks Colby for all your help.
Contains an Apertus logo which is used on bitfossil.org/ itself, presumably they were designing that logo.
Cool data embedded in the Bitcoin blockchain Force of Will Updated 2025-06-17 +Created 1970-01-01
ASCII art of a Force of Will, a famous and powerful Magic: The Gathering card first printed in 1996.
This is Ciro Santilli's personal favorite ASCII art he has found in the blockchain so far. Also Ciro could not find any other previous source of this, so there is some chance it is original. One can dream.
The choice of card is probably linked to the function of the card in the game of Magic: The Gathering. This card essentially prevents the opponent from casting a spell they are about to cast. The presumed intended meaning of this art is further accentuated by the old card type term "interrupt" (late renamed to "instant"), which suggests that "this ASCII art is an interruption to the normal monetary transactions of the blockchain".
One of also reminded of the prayer wars interruption attempts. We could not however identify anything specific that this ASCII art might have tried to interrupt besides the normal flow of monetary transactions.
If one goes full art critic mode, it is also tempting to draw a parallel between the card's "You may pay 1 life" alternative casting cost (as opposed to 5 mana, 3 and two blue, which is a very large cost for most games) as being a reference to the money spent by the uploader of the art to upload it.
TODO understand exactly how it was encoded and why it is so weird. The
UUUU has a slightly weird encoding which we fixed by hand here TODO understand. -------------------------------------
| Force of Will 3 U U |
| --------------------------------- |
| | //////////// | |
| | ////() ()\////\ | |
| | ///_\ (--) \///\ | |
| | ) //// \_____///\\ | |
| | ) \ / / / / | |
| | ) / \ | | / _/ | |
| | ) \ ( ( / / / / \ | |
| | / ) ( ) / ( )/( ) \ | |
| | \(_)/(_)/ /UUUU \ \\\/ | | |
| .---------------------------------. |
| Interrupt |
| ,---------------------------------, |
| | You may pay 1 life and remove a | |
| | blue card in your hand from the | |
| | game instead of paying Force of | |
| | Will's casting cost. Effects | |
| | that prevent or redirect damage | |
| | cannot be used to counter this | |
| | loss of life. | |
| | Counter target spell. | |
| `---------------------------------` |
| l
| Illus. Terese Nelsen |
-------------------------------------
Force of Will Magic: The Gathering card (Alliances)
Source. A high resolution scan of the original card depicted in the ASCII art for comparison.The following two ASCII transactions:suggest this ASCII art might have been uploaded by Figure "Erich Erstu", AKA Hyena, creator of cryptograffiti.info, a service which would have allowed uploading ASCII content to the blockchain.
tx 0f05c47a8caafadecc10d70ba3bf010eaf6bb416b5e1ad7b01cf3445f5fb7a1c
I am. Therefore, I have come to be.
-- Hyena
tx e6d48f6912929a58a2ee30c13768058777d8547215c27109b5cb0724e7abaaba
Erich,
Bro, this looks excellent!!
-DurielThe only other mention of "Duriel" in the blockchain is tx 140562ceb42fc8943fa52ccc0ddbb11ca2d88dae9b5240d7a4b46864538c515a which has some freedom of speech comments and gives the email:paystamper.com was some other blockchain service from circa 2015:
Duriel@paystamper.com = 1HcuhfTAiQCt6KdMG2rZLXsTcKYj9nLDhS Cool data embedded in the Bitcoin blockchain Interesting AtomSea & EMBII uploads Updated 2025-06-17 +Created 1970-01-01
Shiemaa&Vincent.jpgMessage:TODO identify Shiemaa and Vincent.
tx 36d0d77acd760f0aa549b6b314f0c1e9690baa6bcc2d0f07ea9f3167f4a5ec99, block 318836 (2014-09-03)Cost: ~0.002 BTC ~ $0.77 at the time.
Several other interesting uploads were also made around block 318836 (September 2014):
RedRaven.jpgbitfossil.org/e17b83234402d85f3a18207eec11bc5c4397f88aa880aae4fb7d15802806a971/index.htmEarth3Archive.jpgbitfossil.org/ae8d3b46b934bedc363e11abe8c8607171994470957c286274f699a0b3a9bbd7/index.htmSkyEarth5Archive.jpgbitfossil.org/ae8d3b46b934bedc363e11abe8c8607171994470957c286274f699a0b3a9bbd7/index.htm
BikeLady.jpgBike Lady by Allen Lee Vandever. tx 2c4b9497af8c0c0eb9383357b40c3de33dba0b4f481099a32719f2b9036da8e7, block 319927 (2014-09-10)
This seems to be a novel work uploaded by its creator artist Allen Vandever according to EMBII.[ref].
Arecibo_message.svgArecibo message on tx c6d2e535cd2ba4659e954a61198c66fd98c60f6475cf8ff92a404f3fe3a16c4b, block 337874 (2015-01-07)
An "artificially" colored visualization of the Arecibo message ripped from Wikipedia: en.wikipedia.org/wiki/File:Arecibo_message.svg (with attribution).
The cool thing about this image is that it highlights the striking parallels between the encoding of the Arecibo message with crypto graffiti, because in both cases people were creating undocumented new ways of communicating with strangers on a new medium in those early blockchain days.
The associated message contains the Arecibo message as ASCII 0's and 1's. When properly cut at the newlines, they draw the message as ASCII art, as the original Arecibo encoding intends, here's a version with the 0's replaced by spaces to make it more readabale:
1 1 1 1
1 1 1 1 1
1 1 1 1 1 11 1
1 1 1 1 1 1 1 1 1 1
11
11 1
11 1
1 1 1
11111
11 111 11 11
1 11 1
11 1 11 11 11 1
11111 11111 11111 11111
1 1
1 1
11111 11111
11 11 111 11
1 1 1
11 1 11 111 11 1
11111 11111 11111 11111
1 11 1
11
1 11 1
11111 11 11111
11
1 1 1
1 11 1
11 11 1
11 1 11
11 11
11 1 11
11 11 1
1 1 1
1 11 1
1 11 1
1 1 1
1 1 1
1 11
11 11
1 111 1 11
1 1
1 11111
1 1 111 1 1 11 11
1 111 1 111111
1 111 111 11 111
1 1 111 11
1 1 1 111111
1 1 1 11
1 11 11
111 1
111 1 1 1 1 1 1 1 1
111 1 1 1 1
1 1
11111
111111111
111 111
11 11
11 1 1 11
11 11 11 11
1 1 1 1 1 1
1 1 1 1 1 1
1 1 1 1
1 1 1
1 1
1 1 1
1111 11111 1 1111
He sleeps in a temple.jpg
Dr_Craig_Wright.jpg
. Source. Associated message:
The image is present e.g. at: www.kitguru.net/channel/jon-martindale/australian-man-claims-he-is-satoshi-nakamoto-bitcoin-creator/ It was inscribed about two months after Craig publicly claimed that he is Satoshi.
This is a relatively unusual AtomSea & EMBII upload as it does not have the common toplevel transaction, everything, text + image fits into a single transaction. This is perhaps why the image is relatively low resolution to have a smaller size.
YellowRobot.jpgYellow Robot on tx 67b2facfd8160d4fa11b02829b6387d07537b57a7a24f19b029b2a5ae7b81830, block 450516 (2017-01-29)
The associated message reads:This is one of Ciro's favorite AtomSea & EMBII uploads. This is the cutest thing ever, and perfectly encapsules the "medium as an artform" approach to blockchain art. More Chiharu stalking at: ILoveYouMore.jpg.
Chiharu and I found this little yellow robot while exploring Chicago. It will be covered by tar or eventually removed but this tribute will remain. N 41.880778 E -87.629210
At twitter.com/EMBII4U/status/1615389973343268871 EMBII announced that he would be giving off shares of that image on a Bitcoin-based NFT sale system he's making called Sup!?, and in December 2023 gave 2/300 shares to Ciro Santilli. Amen. The transaction list can be seen on the web UI at: p2fk.io/GetObjectByAddress/1KUyhHLrK1ckY8W7Qu31h6gFkXoihWHMzi?mainnet=true&verbose=true It had unfortunately never sold as of 2025, the only activity was EMBII giving off some shares and two listings of 1/300 for 1 BTC. Poor EMBII!
Other possibly novel EMBII street photography:
- bitfossil.com/f2efd446475ad58a3ea808cc0f05a63c55cece9fced70d84799a1ffce5d307e4/index.htm "I will not Stop until the Finite becomes the Infinite."
Audio:
alien.wavblock 318638 bitfossil.org/a3a24d6ea01ce481a50346818b8977220687f3ba385838fe8894ce61c9718bbc/OneGiantLeapForMankind.mp3at tx 4f5b25fa8021c67235423930580e69121aa0d2c2bb779f75139bf442f8dc7297 EMBII-indexed at 743f3286b00fc96c13db4b16d5aead8a1e059fee9ce775b1761be9be5bdc2501 and then indexed at: 0427ec598df38b7d7dc75721316c0bbdec54de4871e11aff8ea64f3717c07efbThe toplevel index does appear on Bitfossil: bitfossil.org/0427ec598df38b7d7dc75721316c0bbdec54de4871e11aff8ea64f3717c07efb/index.htm but the audio is not there as it was for Spock below, maybe a bug on upload/Bitfossil?Spock_Live_Long_And_Prosper.mp3block 345858 bitfossil.org/1bc87dbff1ff5831287f62ac7cf95579794e4386688479bab66174963f9a4a0c/index.htm. Audio of Mr. Spock saying the Vulcan salute.OuterSpace.mp3block 409471 bitfossil.org/c14c1bd862bab6269052bf0a2cda7a35940d7a2d9c3415d4fb8fb8dcb9394fae/ "Outer Space by embii 4MB Large file storage test Apertus 0.3.5-beta" OMG, I don't want to calculate how much it cost to upload this, it will make me sad.At twitter.com/EMBII4U/status/1655969645927563266 EMBII mentions that this inscription, made by him, is the largest inscription he knows of.TODO song composer/performer?- bitfossil.com/c2b170ff450f4529dfbd784e0cf5cdddaca494e67a243dd846c0a9450a5558af/ (2021-03-13) contains
Seikilos.mid, a MIDI file
Interesting text:
- block 273522 bitfossil.org/70fd289901bae0409f27237506c330588d917716944c6359a8711b0ad6b4ce76/index.htm pi to 1000+ decimal digits:
- bitfossil.org/8522787e7e49f3f3b6a9f9e86bc30336d26a3acbaecc93809d2e8b4bb1c4d611/ "Antarctic Ice Cores Revised 800KYr CO2 Data" evidence for global warming
- bitfossil.org/ffa6893a70bcde9b940df9823e0f597f0b6cff964c78473c77db838655e1aeb5/ en.wikipedia.org/wiki/Laudato_si', global warming related
HTML pages:
- block 335290 bitfossil.org/0166db6053f1969c28de8b1f9a8fa4ec890cc4bdfee7602757993b306bb7f295/ JavaScript animated timer clock counting down until the start of the next year
- block 340379 bitfossil.org/062990d54045a9c316110fb713009d1313b2f64c4b216d66891c7284d6c1ca0e/ links to bitfossil.org/062990d54045a9c316110fb713009d1313b2f64c4b216d66891c7284d6c1ca0e/bong-ball.html and has a working JavaScript Pong
- block 328445
tom-signature.jpgbitfossil.org/daa050bf8ac22752e40412c9265b4533f68ab8e6ed26d2db1eeee6710e7d9e4b/index.htm Unrendered HTML of:Likely an obituary for: Thomas L. Magliozzi. Images show fine though.- www.cartalk.com/content/tom-and-rays-bios-photos-2
- www.cartalk.com/content/rant-and-rave-36 "The New Theory of Learning" which agrees perfectly with backward design
- block 401648 bitfossil.com/31c5e5336512568e4a1deb4bbf0e57c3565c32094c0e1a118c48e7929ab49e35/bong-ball.html another one! This one is full-screen, and does not have JavaScript
alerts :-) - block 401657 bitfossil.org/03cb74f270d498302d4dd9cbe82c090d801c8840ab6cb26b71d862489b981db8/ has a JavaScript Pac-Man
A tiny idealized magnet! It is a very good model if you have a small strong magnet interacting with objects that are far away, notably other magnetic dipoles or a constant magnetic field.
The cool thing about this model is that we have simple explicit formulas for the magnetic field it produces, and for how this little magnet is affected by a magnetic field or by another magnetic dipole.
While learning black-hole stuff is not on top of Ciro Santilli's priorities, Hawking's spirit is to be admired.
To never give up even when everything seems lost, and still have a sense of humour is respectable.
An ex-physicist colleague who had met Hawking told an anecdote. Hawking was around in the department one day, they said hi and all. But then Hawking wanted to tell a joke. It took like 5 minutes of typing, and you can imagine that things were pretty awkward and the joke's timing was "a bit off". But Hawking did tell the joke nonetheless.
This is also suggested in the The Theory of Everything (2014) film, and therefore likely the biographies.
Unlisted articles are being shown, click here to show only listed articles.