Math formula that when graphed, produces an image of the formula itself

Tupper's self-referential formula is a self-referential formula defined by Jeff Tupper that, when graphed in two dimensions, can visually reproduce the formula itself. It is used in various math and computer science courses as an exercise in graphing formulae.
The formula was first published in his 2001 SIGGRAPH paper that discusses methods related to the GrafEq formula-graphing program he developed.
The formula is an inequality defined by:
{1\over 2} < \left\lfloor \mathrm{mod}\left(\left\lfloor {y \over 17} \right\rfloor 2^{-17 \lfloor x \rfloor - \mathrm{mod}(\lfloor y\rfloor, 17)},2\right)\right\rfloor 
If one graphs the set of points (x,y-k) with 0 \le x \le 106 and k \le y \le k + 17 such that they satisfy the inequality given above, the resulting graph looks like this:

Tupper's self referential formula plot.png

This formula itself is a general purpose method of decoding a bitmap stored in the constant k, so it could actually be used to draw any other image. When applied to the unbounded positive range 0 \le y, the formula tiles a vertical swath of the plane with a pattern that contains all possible 17 pixel tall bitmaps. (from: wikipedia)

56 comments:

  1. AnonymousNovember 14, 2012 at 7:50 AM

    Yes! Finally someone writes about business & career.
    Feel free to visit my web-site work at home jobs

    ReplyDelete
  2. AnonymousNovember 20, 2012 at 1:58 PM

    Hello just wanted to give you a quick heads up. The text in
    your content seem to be running off the screen in Internet explorer.
    I'm not sure if this is a formatting issue or something to do with internet browser compatibility but I figured I'd
    post to let you know. The layout look great though!
    Hope you get the issue solved soon. Kudos
    Here is my webpage - where to buy electronic cigarettes

    ReplyDelete
  3. AnonymousNovember 20, 2012 at 2:10 PM

    Excellent beat ! I would like to apprentice while you amend your website, how
    could i subscribe for a blog web site? The account helped me a acceptable deal.
    I had been tiny bit acquainted of this your broadcast
    offered bright clear concept
    Take a look at my blog post ... Cash Advance ca

    ReplyDelete
  4. AnonymousNovember 20, 2012 at 2:10 PM

    Hello there, just became alert to your blog through Google, and
    found that it is really informative. I am going to watch out
    for brussels. I'll be grateful if you continue this in future. Numerous people will be benefited from your writing. Cheers!
    Here is my web page :: quit smoking hypnosis free

    ReplyDelete
  5. AnonymousNovember 20, 2012 at 2:10 PM

    This comment has been removed by a blog administrator.

    ReplyDelete
  6. AnonymousNovember 21, 2012 at 2:09 AM

    Hi there! Quick question that's completely off topic. Do you know how to make your site mobile friendly? My blog looks weird when browsing from my iphone4. I'm trying to
    find a template or plugin that might be able to fix this
    issue. If you have any suggestions, please share.
    With thanks!
    Review my weblog :: i want to invest money

    ReplyDelete
  7. AnonymousNovember 21, 2012 at 2:42 AM

    Hmm is anyone else experiencing problems with the pictures on this blog loading?
    I'm trying to find out if its a problem on my end or if it's the blog.

    Any suggestions would be greatly appreciated.
    Also visit my web-site ; options trading game

    ReplyDelete
  8. AnonymousNovember 21, 2012 at 2:44 AM

    hello!,I really like your writing very a lot! proportion we keep up a correspondence extra
    about your article on AOL? I need a specialist in this area to resolve my problem.

    Maybe that's you! Taking a look forward to look you.
    Look at my web blog - make money online now free

    ReplyDelete
  9. AnonymousNovember 21, 2012 at 2:53 AM

    Hi, I check your blog regularly. Your writing style is witty,
    keep it up!
    My blog : affiliate

    ReplyDelete
  10. AnonymousNovember 21, 2012 at 2:56 AM

    Very nice write-up. I absolutely appreciate this website.
    Keep it up!
    My weblog ... geld verdienen via internet

    ReplyDelete
  11. AnonymousNovember 21, 2012 at 3:05 AM

    It is the best time to make some plans for the future and it's time to be happy. I've learn this publish and if
    I could I want to suggest you some attention-grabbing issues or suggestions.
    Perhaps you could write next articles referring to this article.
    I wish to read more issues approximately it!
    Also visit my weblog forex iqd

    ReplyDelete
  12. AnonymousNovember 21, 2012 at 3:19 AM

    What's up to all, it's really a pleasant for me to visit this web site,
    it consists of useful Information.
    Here is my web site :: medicaid options

    ReplyDelete
  13. AnonymousNovember 21, 2012 at 3:20 AM

    Paragraph writing is also a fun, if you know afterward you can write or else it is complicated to write.
    Also visit my web site ... free online investing

    ReplyDelete
  14. AnonymousNovember 21, 2012 at 3:33 AM

    I feel that is among the so much significant info for me.
    And i'm glad studying your article. However wanna statement on few general things, The web site taste is great, the articles is in point of fact excellent : D. Just right process, cheers
    Also visit my web page Online Work From Home Jobs

    ReplyDelete
  15. AnonymousNovember 21, 2012 at 4:03 AM

    Thank you for the auspicious writeup. It in fact was a amusement account it.
    Look advanced to far added agreeable from you! By the way, how could we
    communicate?
    Feel free to visit my web page ; payday loans in nc

    ReplyDelete
  16. AnonymousNovember 21, 2012 at 4:35 AM

    First of all I want to say excellent blog! I had a quick question
    which I'd like to ask if you do not mind. I was interested to know how you center yourself and clear your head prior to writing. I have had a tough time clearing my thoughts in getting my thoughts out there. I truly do take pleasure in writing however it just seems like the first 10 to 15 minutes are generally wasted just trying to figure out how to begin. Any suggestions or tips? Many thanks!
    Here is my blog ... Day trading options

    ReplyDelete
  17. AnonymousNovember 21, 2012 at 4:50 AM

    I absolutely love your website.. Great colors & theme.
    Did you build this amazing site yourself? Please reply back
    as I'm trying to create my own personal website and would love to know where you got this from or what the theme is called. Appreciate it!
    Here is my web site : technical indicators for day trading

    ReplyDelete
  18. AnonymousNovember 21, 2012 at 6:25 AM

    Hi there i am kavin, its my first occasion to commenting anyplace, when i read this piece of writing i thought i could also make comment due to this good paragraph.
    Have a look at my homepage :: instant payday loans no brokers

    ReplyDelete
  19. AnonymousNovember 23, 2012 at 3:50 AM

    Simply want to say your article is as astonishing. The clearness in your post is just
    excellent and i could assume you are an expert on this
    subject. Fine with your permission allow me to grab
    your RSS feed to keep updated with forthcoming post. Thanks a million and please carry on the enjoyable work.
    Also visit my site ; cash advance no fax

    ReplyDelete
  20. AnonymousNovember 23, 2012 at 4:02 AM

    Someone essentially lend a hand to make seriously posts I might
    state. This is the very first time I frequented your web page and up
    to now? I surprised with the analysis you made to make this actual post extraordinary.

    Great activity!
    My web page - lets make lots of money

    ReplyDelete
  21. AnonymousNovember 23, 2012 at 4:06 AM

    I have read so many posts about the blogger lovers but this post is
    in fact a good article, keep it up.
    My homepage - mayday loans no credit check no phone calls

    ReplyDelete
  22. AnonymousNovember 23, 2012 at 4:22 AM

    I want to to thank you for this very good read!! I definitely loved every little bit of it.
    I have got you saved as a favorite to check
    out new stuff you post…
    Feel free to visit my weblog : stock market chart

    ReplyDelete
  23. AnonymousNovember 23, 2012 at 5:13 AM

    This piece of writing presents clear idea in support of the new viewers of blogging,
    that really how to do blogging and site-building.
    Also visit my homepage :: cash loan network

    ReplyDelete
  24. AnonymousNovember 23, 2012 at 5:25 AM

    In fact no matter if someone doesn't know after that its up to other viewers that they will assist, so here it happens.
    Also visit my blog : online silver trading

    ReplyDelete
  25. AnonymousNovember 23, 2012 at 5:28 AM

    It's going to be ending of mine day, but before finish I am reading this impressive piece of writing to improve my knowledge.
    Also visit my homepage : cedarfinance scam

    ReplyDelete
  26. AnonymousNovember 23, 2012 at 5:29 AM

    Right now it seems like BlogEngine is the top blogging platform available
    right now. (from what I've read) Is that what you're using on your blog?
    Also see my webpage > long term forex signals

    ReplyDelete
  27. AnonymousNovember 23, 2012 at 5:32 AM

    I think the admin of this site is genuinely working hard
    in support of his web page, because here every stuff is
    quality based data.
    Here is my webpage ; work from home assembly jobs

    ReplyDelete
  28. AnonymousNovember 23, 2012 at 5:33 AM

    WOW just what I was looking for. Came here by searching for instant approval payday loans
    My blog post instant approval cash advance

    ReplyDelete
  29. AnonymousNovember 23, 2012 at 6:16 AM

    Hi, I read your blog like every week. Your humoristic style is awesome, keep doing what you're doing!
    Here is my webpage ; day trading training

    ReplyDelete
  30. AnonymousNovember 23, 2012 at 7:56 AM

    Wow, incredible blog layout! How long have you been blogging for?
    you make blogging look easy. The overall look of your web site is great,
    as well as the content!
    Feel free to surf my webpage - Way to make money online

    ReplyDelete
  31. AnonymousNovember 23, 2012 at 8:13 AM

    Hello there, I found your blog by way of Google whilst searching for a comparable topic, your website
    came up, it appears to be like good. I've bookmarked it in my google bookmarks.
    Hello there, just became aware of your blog through Google, and located that it's truly informative.

    I am going to watch out for brussels. I will appreciate in case you
    continue this in future. Many other folks will likely be benefited out of your writing.
    Cheers!
    Also visit my site : trade penny stocks

    ReplyDelete
  32. AnonymousNovember 27, 2012 at 5:19 PM

    Hi there, You have done a fantastic job. I will certainly digg it and personally recommend to my friends.
    I am sure they'll be benefited from this web site.
    Here is my weblog : extra make money online

    ReplyDelete
  33. AnonymousNovember 27, 2012 at 5:22 PM

    I every time emailed this blog post page to all
    my contacts, as if like to read it then my links will too.
    Look at my webpage ways to earn extra money online

    ReplyDelete
  34. AnonymousNovember 27, 2012 at 7:21 PM

    I do trust all of the ideas you have presented
    for your post. They're really convincing and will definitely work. Nonetheless, the posts are very quick for novices. May just you please extend them a bit from next time? Thanks for the post.
    Also visit my blog - way to make money online

    ReplyDelete
  35. AnonymousNovember 27, 2012 at 7:49 PM

    With havin so much content and articles do you ever run into any problems of plagorism or copyright infringement?
    My website has a lot of unique content I've either written myself or outsourced but it appears a lot of it is popping it up all over the web without my agreement. Do you know any ways to help reduce content from being ripped off? I'd truly appreciate
    it.
    Feel free to visit my blog : easy ways to make money quick

    ReplyDelete
  36. AnonymousNovember 27, 2012 at 8:05 PM

    Does your site have a contact page? I'm having a tough time locating it but, I'd like to send you an e-mail.
    I've got some ideas for your blog you might be interested in hearing. Either way, great blog and I look forward to seeing it grow over time.
    Here is my weblog :: how to Earn more money online

    ReplyDelete
  37. AnonymousNovember 27, 2012 at 9:13 PM

    Please let me know if you're looking for a writer for your blog. You have some really good posts and I feel I would be a good asset. If you ever want to take some of the load off, I'd
    absolutely love to write some articles for your blog
    in exchange for a link back to mine. Please blast me an
    e-mail if interested. Many thanks!
    Take a look at my weblog ... best ways to make money online

    ReplyDelete
  38. AnonymousNovember 28, 2012 at 4:17 AM

    Yesterday, while I was at work, my sister stole my iphone and tested to see
    if it can survive a 40 foot drop, just so she can be a youtube sensation.

    My iPad is now destroyed and she has 83 views. I know this
    is entirely off topic but I had to share it with someone!
    Have a look at my webpage :: earn money online fast

    ReplyDelete
  39. AnonymousNovember 29, 2012 at 1:05 AM

    I truly love your site.. Great colors & theme. Did you make this site yourself?
    Please reply back as I'm wanting to create my own blog and would love to learn where you got this from or just what the theme is called. Appreciate it!
    Here is my weblog ; forex

    ReplyDelete
  40. AnonymousNovember 29, 2012 at 11:37 PM

    If some one needs to be updated with latest technologies afterward he must be pay a quick visit this site and
    be up to date all the time.
    Here is my web site ideas on how to make money from home

    ReplyDelete
  41. AnonymousNovember 30, 2012 at 3:44 PM

    If you desire to get much from this paragraph then you have to apply these techniques to
    your won webpage.
    My web site legitimate work at home business

    ReplyDelete
  42. AnonymousNovember 30, 2012 at 7:58 PM

    I am not sure where you're getting your info, but good topic. I needs to spend some time learning more or understanding more. Thanks for great info I was looking for this info for my mission.
    Also visit my webpage - online job work From home

    ReplyDelete
  43. AnonymousNovember 30, 2012 at 9:05 PM

    At this time I am ready to do my breakfast, afterward having my
    breakfast coming yet again to read more news.
    Feel free to visit my blog get money fast

    ReplyDelete
  44. AnonymousNovember 30, 2012 at 10:36 PM

    Great article, just what I needed.
    Also visit my site ; how to make money working online

    ReplyDelete
  45. AnonymousNovember 30, 2012 at 11:56 PM

    great issues altogether, you just received a new reader.
    What may you recommend in regards to your post that you just made some days
    in the past? Any sure?
    Feel free to visit my site - how to earn money online fast

    ReplyDelete
  46. AnonymousDecember 1, 2012 at 11:14 AM

    Asking questions are actually good thing if you are not understanding anything entirely, but this post offers fastidious
    understanding even.
    Feel free to surf my blog payday loans in one hour

    ReplyDelete
  47. AnonymousDecember 1, 2012 at 1:13 PM

    Terrific post however I was wondering if you could write a litte more
    on this subject? I'd be very grateful if you could elaborate a little bit further. Thank you!
    Feel free to visit my webpage : how to make money online fast

    ReplyDelete
  48. AnonymousDecember 2, 2012 at 12:32 PM

    This is a topic that is near to my heart... Best wishes!
    Where are your contact details though?
    Feel free to surf my page stock market penny stocks

    ReplyDelete
  49. AnonymousDecember 2, 2012 at 2:45 PM

    It is in reality a nice and helpful piece of information.

    I'm glad that you simply shared this helpful info with us. Please stay us informed like this. Thanks for sharing.
    Check out my page ; http://www.tdameritrade.com/tradingtools/quotescope.html

    ReplyDelete
  50. AnonymousJanuary 12, 2013 at 3:23 AM

    Write more, thats all I have to say. Literally, it seems as though you relied on the video to make your point.
    You definitely know what youre talking about, why throw
    away your intelligence on just posting videos to your weblog when you could be giving
    us something enlightening to read?
    Here is my web site : free casino slot games

    ReplyDelete
  51. Stock MarketMarch 1, 2013 at 3:04 AM

    I think the things you covered through the post are quiet impressive, good job and great efforts. I found it very interesting and enjoyed reading all of it...keep it up, lovely job..

    ReplyDelete
  52. Equity AdvisoryMarch 11, 2013 at 4:45 AM

    This is the first time i am reading your post and admire that you posted article which gives users lot of information regarding particular topic thnaks for this share.

    ReplyDelete
  53. Stock RecommendationMay 23, 2013 at 5:16 AM

    Great article ...Thanks for your great information, the contents are quiet interesting. I will be waiting for your next post.

    ReplyDelete
  54. Commodity tipsOctober 17, 2014 at 12:08 AM

    Thanks so much for this. This is awesome post I ever seen on internet. This is rare to find that’s why difficult to understand. Anyway, you are definitely someone that has something to say that people need to hear. Keep up the good work. Keep on inspiring the people.

    ReplyDelete
  55. GraphMarch 28, 2015 at 9:18 AM

    That's cool

    ReplyDelete
  56. NCDEX TipsFebruary 23, 2017 at 3:48 AM

    Really no matter if someone doesn’t understand after that its up to other visitors that they will help, so here it occurs.

    ReplyDelete