A back-of-the-envelope calculation is a rough calculation, typically jotted down on any available scrap of paper such as an envelope. It is more than a guess but less than an accurate calculation or mathematical proof. The defining characteristic of back-of-the-envelope calculations is the use of simplified assumptions. A similar phrase in the U.S. is "back of a napkin", also used in the business world to describe sketching out a quick, rough idea of a business or product. In British English, a similar idiom is "back of a fag packet".
In the natural sciences, back-of-the-envelope calculation is often associated with physicist Enrico Fermi, who was well known for emphasizing ways that complex scientific equations could be approximated within an order of magnitude using simple calculations. He went on to develop a series of sample calculations, which are called "Fermi Questions" or "Back-of-the-Envelope Calculations" and used to solve Fermi problems.
Fermi was known for getting quick and accurate answers to problems that would stump other people. The most famous instance came during the first atomic bomb test in New Mexico on 16 July 1945. As the blast wave reached him, Fermi dropped bits of paper. By measuring the distance they were blown, he could compare to a previously computed table and thus estimate the bomb energy yield. He estimated 10 kilotons of TNT; the measured result was 18.6.
Perhaps the most influential example of such a calculation was carried out over a period of a few hours by Arnold Wilkins after being asked to consider a problem by Robert Watson Watt. Watt had learned that the Germans claimed to have invented a radio-based death ray, but Wilkins' one-page calculations demonstrated that such a thing was almost certainly impossible. When Watt asked what role radio might play, Wilkins replied that it might be useful for detection at long range, a suggestion that led to the rapid development of radar and the Chain Home system.
Another example is Victor Weisskopf's pamphlet Modern Physics from an Elementary Point of View. In these notes Weisskopf used back-of-the-envelope calculations to calculate the size of a hydrogen atom, a star, and a mountain, all using elementary physics.
Nobel laureate Charles Townes describes in a video interview for the University of California, Berkeley on the 50th anniversary of the laser, how he pulled an envelope from his pocket while sitting in a park and wrote down calculations during his initial insight into lasers.
The Bailey bridge is a type of portable, pre-fabricated, truss bridge and was extensively used by British, Canadian and US military engineering units. Donald Bailey drew the original design for the bridge on the back of an envelope.
The Laffer Curve, which claims to show the relationship between tax cuts and government income, was drawn by Arthur Laffer in 1974 on a bar napkin to show an aide to President Gerald R. Ford why the federal government should cut taxes. 
Upon hearing that the S-IV 2nd Stage of the Saturn I would need transport from California to Florida for launch as part of the Apollo program, Jack Conroy sketched the cavernous cargo airplane, the Pregnant Guppy.
- Buckingham pi theorem, a technique often used in fluid mechanics to obtain order-of-magnitude estimates
- Scientific Wild-Ass Guess
- Order-of-magnitude analysis
- Rule of thumb
- Sanity testing
Notes and references
- Brown, Bob (2011-07-19). "Napkins: Where Ethernet, Compaq and Facebook's cool data center got their starts". Network World. Archived from the original on 15 August 2019. Retrieved 2020-10-06.
Robert Metcalfe's early Ethernet diagrams from his days at Xerox PARC back in the early 1970s might be the most famous napkin sketches in the technology industry.
- Where Fermi stood. - Bulletin of the Atomic Scientists | Encyclopedia.com (Archived)
- Back of the Envelope Calculations
- High School Mathematics at Work: Essays and Examples for the Education of All Students
- Nuclear Weapons Journal, Los Alamos National Laboratory, Issue 2 2005.
- Austin, B.A. (1999). "Precursors To Radar — The Watson-Watt Memorandum And The Daventry Experiment" (PDF). International Journal of Electrical Engineering Education. 36 (4): 365–372. doi:10.7227/IJEEE.36.4.10. S2CID 111153288. Archived from the original (PDF) on 2015-05-25. Retrieved 2016-07-08.
- Lectures given in the 1969 Summer Lecture Programme, CERN (European Organization for Nuclear Research), CERN 70-8, 17 March 1970.
- Video of interview with Charles Townes; envelope mention comes about halfway in
- Timberg, Craig (31 May 2015). "Net of Insecurity; Quick fix for an early Internet problem lives on a quarter-century later". The Washington Post. Archived from the original on 1 June 2015. Retrieved 4 January 2021.
As the prospect of system meltdown loomed, the men began scribbling ideas for a solution onto the back of a ketchup-stained napkin. Then a second. Then a third. The “three-napkins protocol,” as its inventors jokingly dubbed it, would soon revolutionize the Internet. And though there were lingering issues, the engineers saw their creation as a “hack” or “kludge,” slang for a short-term fix to be replaced as soon as a better alternative arrived.
- "Usage Survey of Character Encodings broken down by Ranking". w3techs.com. Retrieved 2018-11-01.
- Email Subject: UTF-8 history, From: "Rob 'Commander' Pike", Date: Wed, 30 Apr 2003..., ...UTF-8 was designed, in front of my eyes, on a placemat in a New Jersey diner one night in September or so 1992...So that night Ken wrote packing and unpacking code and I started tearing into the C and graphics libraries. The next day all the code was done...
- Services, From Times Wire (1985-05-07). "Sir Donald Bailey, WW II Engineer, Dies". Los Angeles Times. ISSN 0458-3035. Retrieved 2018-11-01.
He sketched the original design for the Bailey Bridge on the back of an envelope as he was being driven to a meeting of Royal Engineers to debate the failure of existing portable bridges
- "This is not Arthur Laffer's famous napin" NY Times 13 Oct. 2017
- Bloom, Margy (15 September 2011). "PilotMag Aviation Magazine | The Pregnant Guppy | The Problem: Logistics". Pilot Magazine. Archived from the original on 15 July 2011. Retrieved 25 April 2012.
[Conroy] listened to the conversations around him, then picked up a cocktail napkin and a ballpoint pen. And with the precision he’d learned during the brief months he’d attended engineering school many years before, he drew an airplane that had never been built, to carry a rocket that had never been launched, to take man to a place nobody had ever been before. Jack Conroy had just sketched the airplane that would become the Pregnant Guppy.
- Reimer, Jeremy (18 March 2016). "A history of the Amiga, part 9: The Video Toaster". Ars Technica. Archived from the original on 18 March 2016. Retrieved 4 January 2021.
Montgomery suggested that Jenison meet his friend Brad Carvey, who had been working on projects involving robotic vision. The three of them got together in a pizza restaurant in Topeka and started drawing block diagrams on the placemats.
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