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1292: Pi vs. Tau

2026-04-02T07:18:36Z

Geometrist: /* Explanation */

{{comic
| number = 1292
| date = November 18, 2013
| title = Pi vs. Tau
| image = pi vs tau.png
| titletext = Conveniently approximated as e+2, Pau is commonly known as the Devil's Ratio (because in the octal expansion, '666' appears four times in the first 200 digits while no other run of 3+ digits appears more than once.)
}}

==Explanation==
This is yet another of [[Randall]]'s [[:Category:Compromise|compromise comics]]. A few mathematicians argue as to whether to use {{w|pi}}, which is the ratio between a circle's circumference and its diameter, or {{w|Turn (angle)#Proposals for a single letter to represent 2π|tau}}, which is the ratio between a circle's circumference and its radius.

Some consider pi to be the wrong convention and are in favor of using tau as ''the'' circle constant; see the [http://tauday.com/tau-manifesto Tau Manifesto], which was inspired by the article "[http://www.math.utah.edu/~palais/pi.html Pi is wrong!]" by mathematician Robert Palais and [https://www.youtube.com/watch?v=5iUh_CSjaSw publicized by Vi Hart] (video was made private). Others consider proponents of tau to be foolish and remain loyal to pi (see the [https://proper-pi-manifesto.com/ Pi Manifesto]). Of course, regardless of which convention is used, the change is merely in notation — the underlying mathematics remains unaltered. Still, the choice of pi vs. tau can affect the clarity of equations, analogies between different equations, and how easy various subjects are to teach.

Most people know π (pi) by the approximation 3.14, but do not know τ (tau) which, by definition, is twice as large as pi. Randall is suggesting using "pau", which is a {{w|portmanteau}} of "pi" and "tau", as a number situated, appropriately enough, halfway between pi and tau, i.e. 1.5 pi or 0.75 tau. But of course his number would be inconvenient, as this value does not naturally turn up when working with circles or other mathematical constructs, so there are no commonly used formulas that would use pau (4.7123889803 8468985769 3965074919 2543262957 5409906265 8731462416 8884617246 0942931349 7942052238 0131756019).

The title text claims that pau can be approximated by e+2, as both values are roughly 4.71 — a similarity that holds little since it requires another irrational constant, {{w|E (mathematical constant)|e}} (although knowing the value of pau is somewhat more helpful in remembering e to 2 digits.){{Citation needed}} It also attributes the nickname "Devil's Ratio" to pau, due to the sequence {{w|Number of the Beast|666}} supposedly appearing four times in the first 200 digits of pau when expressed in the {{w|octal}} base. However, this is not the case, and was likely due to an error in the computer system used by WolframAlpha; for more details see below.

The 200 digits of pau in base 8 are: 4.5545743763 1441644323 6234514475 0501224254 7157301565 0314763354 5270030431 6771261165 5054674757 0313312523 4035147165 7646433317 2731124310 2010764472 7072362457 3721640220 4376521550 6554422014 3116155742 5156344621, and there is no 666 in it, let alone 4 times.

The tau vs. pi controversy was later mentioned in [[2520: Symbols]].

==Transcript==
:[On the left is a "forbidden"-style slashed circle with the π symbol, captioned "Pi". On the right is a "forbidden"-style slashed circle with 2π, captioned "Tau". Between these is 1.5π, captioned "Pau".]
:[Caption below the panel:]
:A compromise solution to the Pi/Tau dispute

==Math details==
Possibly, [[Randall]] used [http://www.wolframalpha.com/ Wolfram|Alpha] to calculate the result (he uses it a lot, for example ''{{what if|70|The Constant Groundskeeper}}'' or ''{{what if|62|Falling With Helium}}'').
However, when the comic was published, there was a bug in Wolfram|Alpha so that, when getting 200 octal digits from "pau", it just calculates the decimal value rounded to 15 significant digits (this is 4.71238898038469) and expands that as octal digits as far as needed.

This gives a periodically repeating number. In the first 200 digits of the octal expansion, the sequences 666 and 6666 do occur, twice and once, respectively. There are 4 occurrences, however, in the first three hundred and ten (310 in base 8 equals 200 in base 10) digits:
<pre>
4.554574376314416445676661714336617116240444076666510533533077631151350452060436452476274022621206136310000177621674175071262255702044274154476005744176002676623042402346036604733130522524127534777714554305412763636566643022106616734723661726160312772574551366370203115523402704104015532221722772357666</pre>
This expansion does indeed contain 666 (the {{w|Number of the beast|number of the beast}}) four times (with one instance as 6666). It also contains 0000, 222, 444, and 7777, but they only appear once in a run.

In the first 500 digits of the actual octal expansion of pau, we also find that 6666 occurs once, and 666 occurs two other times:
<pre>
4.55457437631441644323623451447505012242547157301565031476335452700304316771261165505467475703133125234035147165764643331727311243102010764472707236245737216402204376521550655442201431161557425156344621363625174410110777026111560241174471252241762037163367420573533032164702576626667446275343255043345060027305171025475041452166612112500275317166412767657355633417212140135534536541060452450664011414377406267077573054507036064406511117752700327100355213521015136220621644573043264505244325316526666260</pre>
(Note that this contains 500 digits after the octal point.) No other run of 3 or more repeated digits (e.g. 111) occurs as many times, although 1111 occurs once, 111 occurs once elsewhere, and 333 and 777 also occur once each. 9 other strings of 3 digits occur 4 times, namely 164, 362, 521, 644, 432, 730, 043, 216, and 450, and only 573 occurs more often, as it occurs 6 times. Therefore, if 6666 is counted as two occurrences of 666, it is actually the joint second most common string of three numbers in the first 500 digits.

{{w|Mathematical coincidence|Coincidentally}}, e+2 is also very similar to 1.5 pi, although only to a few digits.
<pre>
1.5π = 4.71238898038...
e+2 = 4.71828182845...
</pre>

==Trivia==
*For Pi, the sequence '666' occurs for the first time at position 2440. Many more occurrences can be found here: [http://www.angio.net/pi/ The Pi-Search Page].
*In the discussion it has been theorized that Randall used [[356: Nerd Sniping|nerd sniping]], in which case he was aware of the mistake in Wolfram!
*For an entertaining introduction to the concept of tau, see this [https://www.youtube.com/watch?v=jG7vhMMXagQ Vi Hart video].
*In March 2018 the video [https://www.youtube.com/watch?v=bcPTiiiYDs8 How pi was almost 6.283185...] was released on why Pi could just as well have been Tau (6.28), since {{w|Leonhard Euler|Euler}}, who used the letter Pi in his books, used it for both what we call Pi and Tau today... This very comic is also briefly shown in a segment regarding the controversy about these two versions of "Pi".
*Note that "pau" is Catalan for peace, which might make it a good solution for the pi/tau dispute.
*Also, note that "pau" is the Portuguese word for "stick", as well as, in Brazilian Portuguese, a very common slang for "penis". This may add to the humor (although childishly) for Portuguese-speaking readers, though it is fair to presume that it was not Randall's intention to do so.
*Pau is a [https://wildwoodhealth.com/pau-chinese-steam-buns/ Chinese dish], a filled bun which is round and yummy{{Citation needed}}.
*Pau is also {{w|Pau, Pyrénées-Atlantiques|a city of south-western France}}.
*Finally, "pau" means "finished" in Hawaiian.

{{comic discussion}}

[[Category:Comics with color]]
[[Category:Math]]
[[Category:Compromise]]

3174: Bridge Clearance

2025-11-30T11:18:39Z

Geometrist:

{{comic
| number = 3174
| date = November 28, 2025
| title = Bridge Clearance
| image = bridge_clearance_2x.png
| imagesize = 450x233px
| noexpand = true
| titletext = A lot of the highway department's budget goes to adjusting the sign whenever the moon passes directly overhead.
}}

==Explanation==
{{incomplete|This page was created BY A BRIDGE WITH NaN CLEARANCE. Don't remove this notice too soon.}}
Roads passing under bridges (or other overhead structures) often have signs indicating the "clearance" or lowest height of the bridge above the road, primarily to indicate to drivers of taller vehicles the potential hazard of driving underneath. Warnings usually start with signs advising which routes to avoid starting down, so that tall vehicles can find a safe alternate path without having to reverse and obstruct traffic. There may be {{w|Bridge strike#Passive devices|physical warning systems}}, immediately before the actual restrictive feature, to alert inattentive drivers and prevent far more troublesome damage/blockages at the pinch-point itself. In this comic, there is a bridge with one of these signs, preceded by a 'sacrificial' bar, indicating a clearance of 10 feet and 6 inches (3.20 m), well under the height (14 feet, 4.3 m) below which, in the USA, warning signage is typically required.

On a turn-off, leading away to the right to meet the higher-level route that passes over the current one, there's a different sign which says "no bridge", allowing the relevant vehicles to turn off and meet the higher-grade route (potentially to cross straight over it and rejoin the original carriageway via the counterpart on-ramp on the other side of the bridge). It does this while indicating a clearance of (more than) 46 billion light years. This sign indicates a (default) clearance on this stretch of highway that is based upon the approximate radius of the observable universe. This is because there is no bridge or any known obstruction lower than this particularly generous 'limit'. The vertical post for this sign is shown continuing far above the upper edge of the sign and past the upper edge of the panel, implying that the sign has been equipped with a similar sacrificial bar at a ridiculous height above the ground. The tallest human structure at the time of publication is only 2,722 feet (829.8 m), and even far shorter vertical poles require special reinforcement to remain stable.

The title text references the fact that, periodically, the Moon passes over the road (or, from a different point of view, the road passes under the Moon). This implies that the road lies between {{w|Orbit of the Moon#Inclination|latitudes approximately 28.5 degrees N and S}}. The Moon passes over any given longitude almost daily ({{w|Sidereal time|roughly every 24 hours}} but shifted by the Moon's own {{w|Lunar month#Sidereal month|slower motion}}). Whenever this all lines up, the theoretical clearance reduces to the distance between Earth and the Moon (around 225-250 thousand miles), and any vehicle would now need to be shorter than this ''still'' very generous limit, in order to not crash into the Moon.

Due to the transient nature of this situation, it costs a lot of money to adjust the sign on the right before and after this 'danger period'. While the sign might be digital, the implied overhead bar could not move sufficiently without exceeding the speed of light, so one pole would have to remain permanently at the greater height, while a second pole was deployed temporarily at the lower height. And this system apparently doesn't consider that other astronomical bodies, much farther out than the Moon, would frequently (if momentarily) be directly over any particular section of road, not to mention a large number of artificial satellites that will be passing over at relatively far lower altitudes.

There is also a problem of many countries having {{w|Height restriction laws}}. For example FAA restricts a structure height to merely 2000 ft (610 m) so, regardless of clearance or any airplanes present above, you won't normally be allowed to have something that high above the ground (at the point just before it becomes considered a vehicle) higher than that anywhere on US territory.

Of course, all such versions of these 'no bridge' signs ignore the fact that vehicles cannot actually be physically large enough that such extreme clearances would be something necessary to anticipate. The problems with a vehicle (or a sign) tall enough to approach the Moon would be ''much'' worse than for an Earth-based "{{w|space elevator}}", which would be far shorter.

Low bridge heights, such as the 10 feet 6 inches (3.2 m) cited in the comic, are present on highways constructed before the US Interstate Highway system established 14 feet (4.3 m), later 16 feet (4.9 m), as the standard minimum clearance in the late 1950s. These are most common in urban areas in the northeastern portion of the USA. Examples include Memorial (1897) and Storrow (1951) Drives in the Boston MA metropolitan area, where Randall lives, and the Merritt Parkway in Connecticut (late 1930s). The minimum bridge clearance on the Merritt Parkway, when constructed, was 11 feet (3.4 m); these bridges remain, and clearances have tended to decline over time. Clearances as low as 9 feet (2.7 m) are on Memorial Drive. Further examples of low clearances may be found with bridges built in the 19th and early 20th centuries, such as covered bridges over streams and railway bridges over secondary roads.

==Transcript==
{{incomplete transcript|Don't remove this notice too soon.}}
:[There is a road heading into the distance under a bridge, with an off-ramp/sliproad that veers off to the right. There are signs on the shoulders/verges/berms of both routes.]

:[Left/forward-leading sign, on a pole leading up to support a pre-bridge 'hazard bar'. Both the edging of the sign and the overhead bar have diagonal warning stripes. The wording wraps across several lines, the initial line being underscored by a horizontal rule.]
:Bridge
:Clearance
:10 ft 6 in

:[Right-forking sign, on a pole that is seen going upwards off-frame, the sign having a similar style of edging and wording.]
:No Bridge
:Clearance
:46 Billion+
:Light-years
{{comic discussion}}<noinclude>