BWSC 2023

It’s still early days, but here is a list of 41 cars (from 20 countries; 28 Challengers and 13 Cruisers) likely to attend the Bridgestone World Solar Challenge in October this year. There are some familar names as well as some newcomers (aCentauri from Switzerland, Nottingham from the United Kingdom, Ascend from Australia, Solaride from Estonia, DCU from Ireland, SNU from Korea, and AgniRath from India). Check out the new design by JU!

This list is a work in progress, and team numbers are mostly guesswork.

See this page and this blog tag for my past coverage of the event, and the official race social media at        (click on the icons).

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2    University of Michigan Solar Car Team 

Challenger (new car) – I expect them to reveal their car in July.

Previously, Michigan came 9^th at WSC 13; came 4^th at WSC 15; came 2^nd at WSC 17; came 3^rd at WSC 19; won ASC 14; won ASC 16; came 2^nd at ASC 18; and won Abu Dhabi 15. Their team number (2) is a long-standing tradition.

 
Left: credit / Right: Anthony Dekker (click images to zoom – OLD PICS)

3    Brunel Solar Team (Delft) 

Challenger (new car: Nuna12) – they are building their 12^th car for the 2023 World Solar Challenge.

Previously, Delft won WSC 13; won WSC 15; won WSC 17; came 12^th at WSC 19; won SASOL 14; won SASOL 16; won SASOL 18; won SASOL 22; and came 3^rd at SCM 21. Their team number (3) is a long-standing tradition.

 
Left: credit / Right: credit (click images to zoom – OLD PICS)

4    Antakari Solar Team 

Challenger (new car: Intikallpa VI) – they will reveal their car in August.

Previously, Antakari participated in the WSC 13 Adventure class; came 10^th at WSC 17; and came 7^th at WSC 19.

 
Left: credit / Right: credit (click images to zoom – OLD PICS)

6    Top Dutch Solar Racing 

Challenger (Green Lightning) – I am not sure if they are bringing a new car or an existing one.

Previously, Top Dutch came 4^th at WSC 19; came 3^rd at iESC 20; came 5^th at iESC 22; and came 4^th at SCM 21.

 
Left: Anthony Dekker / Right: credit (click images to zoom – OLD PICS)

7    Adelaide University Solar Racing Team 

Asymmetric challenger (Lumen II) – they will be restoring and improving Lumen II for this event.

Previously, Adelaide came 21^st at WSC 15; participated at WSC 17; and came 16^th at WSC 19.

 
Left: credit / Right: credit (click images to zoom)

8    Innoptus Solar Team / KU Leuven 

Challenger (new car) – they have a new name and a new sponsor, and are building a new car (their 10^th).

Previously, Innoptus came 6^th at WSC 13; came 5^th at WSC 15; came 3^rd at WSC 17; won WSC 19; came 3^rd at Abu Dhabi 15; came 2^nd at SASOL 22; came 2^nd at iESC 16; came 6^th at iESC 18; came 1^st and 6^th at iESC 20; won iESC 21; came 6^th at iESC 22; won Carrera Solar Atacama 18; and came 2^nd at SCM 21. Their team number (8) is a long-standing tradition.

 
Left: Anthony Dekker / Right: credit (click images to zoom – OLD PICS)

10    Tokai University 

Monohull challenger (new car: Tokai Challenger) – I am not sure what their plans are, exactly.

Previously, Tokai came 2^nd at WSC 13; came 3^rd at WSC 15; came 4^th at WSC 17; came 2^nd at WSC 19; came 7^th at Abu Dhabi 15; came 2^nd at SASOL 16; and came 2^nd at SASOL 18.

 
Left: credit / Right: credit (click images to zoom – OLD PICS)

12    Cambridge University Eco Racing 

Four-seat cruiser (Helia) – they are hoping to take their cruiser back to Australia.

Previously, Cambridge came 22^nd at WSC 15; participated in the WSC 19 Cruiser class; and came 10^th at iESC 16.

 
Left: Nigel / Right: credit (click images to zoom)

14    Flinders University 

Two-seat cruiser (new car: Investigator IV) – their new car looks like a Cybertruck.

Previously, Flinders participated in the WSC 17 Cruiser class and participated in the WSC 19 Cruiser class.

picture credit (click image to zoom)

15    Western Sydney Solar Team 

Challenger (new car) – I understand that they are building a new car of some kind.

Previously, WS came 11^th at WSC 13; came 10^th at WSC 15; came 6^th at WSC 17; came 20^th at WSC 19; and won ASC 18.

 
Left: Anthony Dekker / Right: Anthony Dekker (click images to zoom – OLD PICS)

18    EcoPhoton Solar Car Team (UiTM) 

Challenger (new car) – I am not sure what their plans are, exactly.

Previously, EcoPhoton came 26^th at WSC 15; participated at WSC 17; and participated at WSC 19.

 
Left: credit / Right: credit (click images to zoom – OLD PICS)

20    Durham University Solar Car 

Challenger (new car) – I am not sure what their plans are, exactly.

Previously, DU Solar came 27^th at WSC 15; participated at WSC 17; and came 14^th at WSC 19.

 
Left: Anthony Dekker / Right: credit (click images to zoom – OLD PICS)

21    Solar Team Twente 

Three-wheel (tadpole) monohull challenger (new car: Red X) – the 2023 World Solar Challenge will see their 10^th car. The design was revealed on March 9. It is the second three-wheeler after RED E.

Previously, Twente came 3^rd at WSC 13; came 2^nd at WSC 15; came 5^th at WSC 17; came 17^th at WSC 19; won iESC 16; came 1^st and 2^nd at iESC 18; came 2^nd and 4^th at iESC 20; came 3^rd at iESC 21; came 3^rd and equal 8^th at iESC 22; and won SCM 21. Their team number (21) is a pun and a wish for success in the race (“Twente-One”).

 
Left: Anthony Dekker / Right: credit (click images to zoom)

22    Illini (University of Illinois) 

Monohull challenger (Brizo) – I am not sure if they are coming to Australia with Brizo, or if they will finish their new Project C before the event.

Previously, Illini participated in the WSC 17 Adventure class; came 7^th at ASC 18; came 4^th at FSGP 19; came 4^th at ASC 21; and came 6^th at ASC 22. Their team number (22) is a tradition since 1995.

 
Left: credit / Right: credit (click images to zoom)

23    Halmstad University Solar Team 

Three-wheel challenger (new car: Heart 4) – In 2021 they revealed a three-fairing car.

Previously, HUST participated at WSC 19 and participated at Swedish Solar Race 21.

 
Left: credit / Right: JU Solar Team (click images to zoom)

30    Team Arrow 

Cruiser – they plan to be at the event, but I am not sure which vehicle they will race.

Previously, Arrow came 7^th at WSC 13; came 8^th at WSC 15; came 3^rd in the WSC 17 Cruiser class; participated in the WSC 19 Adventure class; came 5^th at Abu Dhabi 15; and came 8^th at iESC 18. Their team number (30) is the average age of people on the original team.

picture credit (click image to zoom – OLD PIC)

34    Istanbul Technical University (ITU) 

Challenger (Ariba ZES X) – this car has replaced their older B.O.W. It toured Turkey in 2022.

Previously, ITU came 17^th at WSC 13; participated at WSC 17; came 7^th at iESC 16; came 7^th at iESC 20; came 8^th at iESC 21; and came 4^th at iESC 22. Their team number (34) is the vehicle license plate prefix for Istanbul.

 
Left: Anthony Dekker / Right: credit (click images to zoom)

35    University of Minnesota Solar Vehicle Project 

Four-seat cruiser (Freya) – they are America’s Cruiser class pioneers. I understand that they are returning with Freya.

Previously, Minnesota came 4^th in the WSC 13 Cruiser class; came 5^th in the WSC 15 Cruiser class; participated in the WSC 17 Cruiser class; came 5^th in the WSC 19 Cruiser class; came 2^nd at ASC 14; came equal 10^th at ASC 16; came equal 2^nd in the ASC 18 Cruiser class; came 2^nd in the ASC 21 Cruiser class; and won the ASC 22 Cruiser class. Their team number (35) is derived from the Interstate 35 highway.

 
Left: credit / Right: credit (click images to zoom)

40    Solar Team Eindhoven 

Four-seat cruiser (new car: Stella Terra) – their focus for 2021 was a Self-sustaining House On Wheels (Stella Vita). This year they are building an off-road vehicle. I am not sure if that is actually intended to compete in the Cruiser class.

Previously, Eindhoven won the WSC 13 Cruiser class; won the WSC 15 Cruiser class; won the WSC 17 Cruiser class; won the WSC 19 Cruiser class; came 7^th in the iESC 18 Cruiser class; came 1^st and 2^nd in the iESC 20 Cruiser class; and came 3^rd in the iESC 22 Cruiser class. Their team number (40) is the Eindhoven telephone area code.

 
Left: credit / Right: credit (click images to zoom)

41    Australian National University 

Three-wheel (tadpole) challenger (new car: Solar Car 3) – they have a hot new car.

Previously, ANU participated at WSC 17 and participated at WSC 19.

picture credit (click image to zoom)

45    Lodz Solar Team 

Four-seat cruiser (Eagle Two) – this car is still going strong.

Previously, Lodz participated in the WSC 15 Cruiser class; participated in the WSC 17 Cruiser class; participated in the WSC 19 Cruiser class; came 5^th at SASOL 16; won the iESC 18 Cruiser class; came 2^nd in the iESC 21 Cruiser class; and won the iESC 22 Cruiser class. Their team number (45) is a tradition since 2015.

 
Left: Anthony Dekker / Right: credit (click images to zoom)

46    JU Solar Team 

Three-wheel (tadpole) monohull challenger (new car: Axelight) – they have abandoned the catamaran approach, revealing revealed their new monohull on 22 March.

Previously, JU came 20^th at WSC 13; came 15^th at WSC 15; came 8^th at WSC 17; and came 10^th at WSC 19. Their team number (46) is the Swedish national telephone prefix.

 
Left: Anthony Dekker / Right: credit (click images to zoom)

47    Nagoya Institute of Technology Solar Racing 

Challenger (new car) – I am not sure what their plans are, exactly.

Previously, NITech came 16^th at WSC 15; came 12^th at WSC 17; and came 8^th at WSC 19.

 
Left: credit / Right: credit (click images to zoom – OLD PICS)

49    Siam Technical College 

Cruiser (new car: STC-4) – I am not sure what their plans are, exactly.

Previously, STC came 28^th at WSC 15; participated in the WSC 17 Cruiser class; and participated in the WSC 19 Cruiser class.

 
Left: credit / Right: credit (click images to zoom – OLD PICS)

51    Chalmers Solar Team 

Challenger (new car) – they are building a new vehicle to replace their elegant bullet car.

Previously, Chalmers came 21^st at WSC 19; came 5^th at iESC 21; came 7^th at iESC 22; participated at Swedish Solar Race 21; and came 6^th at SCM 21.

 
Left: credit / Right: credit (click images to zoom – OLD PICS)

66    Sakarya University (SAITEM) 

Three-wheel (tadpole) monohull challenger – after racing in 2009 and 2011, they plan to return to the WSC in 2023. As with other teams they have abandoned the catamaran concept for a bullet car which they hope has higher aerodynamic efficiency.

picture credit (click image to zoom)

70    Sonnenwagen Aachen 

Challenger (new car: Sonnenwagen 4) – they have begun building a new car, which they will reveal on 5 June.

Previously, Aachen participated at WSC 17; came 6^th at WSC 19; came 3^rd at iESC 18; came 5^th and 8^th at iESC 20; came 2^nd and 6^th at iESC 21; came 1^st and 2^nd at iESC 22; won ISC 22; and came 5^th at SCM 21. Their team number (70) is the number they raced with in 2017.

 
Left: Anthony Dekker / Right: credit (click images to zoom – OLD PICS)

75    Sunswift (University of New South Wales) 

Cruiser (new car: Sunswift 7) – they revealed their new car in 2021 and achieved a world record for fastest EV over 1000km on a single charge, doing so in under 12 hours.

Previously, Sunswift came 3^rd in the WSC 13 Cruiser class; came 4^th in the WSC 15 Cruiser class; participated in the WSC 17 Cruiser class; and came 2^nd in the WSC 19 Cruiser class.

 
Left: credit / Right: credit (click images to zoom)

77    Blue Sky Solar Racing (Toronto) 

Monohull challenger (new car: Borealis) – they revealed Borealis in 2022.

Previously, Blue Sky came 8^th at WSC 13; came 12^th at WSC 15; came 11^th at WSC 17; came 11^th at WSC 19; and came 3^rd at ASC 16.

 
Left: credit / Right: credit (click images to zoom)

81    Solar Team Solaris (Dokuz Eylül University) 

Challenger (S11) – they have replaced their old catamaran with a bullet car, which they will be racing again, I believe.

Previously, Solaris participated in the WSC 13 Adventure class; came 25^th at WSC 15; came 18^th at WSC 19; came 9^th at iESC 16; came 7^th at iESC 21; came equal 8^th at iESC 22; came 3^rd at ISC 22; came 2^nd at Albi Eco 18; and came 2^nd at MSRC 19.

 
Left: credit / Right: credit (click images to zoom)

85    aCentauri Solar Racing (ETHZ) 

Three-wheel (tadpole) monohull challenger (new team) – this is the second Swiss team, made up mostly of students from ETH Zurich. They have benefited from the advice of SER, the other Swiss team. Their team number (85) is commemorates the world’s oldest solar car race, the Tour de Sol, which first took place in Switzerland in 1985.

picture credit (click image to zoom)

88    Kogakuin Solar Team 

Challenger (new car) – I am not sure what their plans are, exactly.

Previously, Kogakuin came 14^th at WSC 13; came 2^nd in the WSC 15 Cruiser class; came 7^th at WSC 17; and came 5^th at WSC 19. Their team number (88) is multi-faceted (88 is a lucky number in Japanese kanji; 4 wheels looks like 88; and the team garage is in Hachioji city, with “hachi” meaning “eight”).

 
Left: credit / Right: credit (click images to zoom – OLD PICS)

98    ATN Solar Car Team 

Two-seat cruiser (Priscilla) – I am not sure what their plans are, exactly.

Previously, ATN participated in the WSC 19 Cruiser class.

 
Left: credit / Right: credit (click images to zoom – OLD PICS)

101    Éclipse – Véhicule solaire de l’ÉTS  

Asymmetric challenger (new car: Eclipse XI) – they raced in Australia in 2019 as number 92, finishing 2^nd among North American teams. They revealed their new car in February.

Previously, Eclipse came 18^th at WSC 13; came 9^th at WSC 19; came 10^th at ASC 14; came 8^th at ASC 16; came 4^th at FSGP 17; came 3^rd at ASC 18; and came 5^th at ASC 22.

 
Left: credit / Right: credit (click images to zoom)

230    University of Nottingham Solar Racing Team 

Cruiser (new team) – their rather radical approach is to modify a Renault Twizy to have solar panels, improved electrics, and second life Nissan Leaf batteries.

photo: UoN team (click image to zoom)

333    Institut Teknologi Sepuluh Nopember 

Cruiser (new car) – I am not sure what their plans are, exactly.

Previously, ITS came 21^st at WSC 13 and participated in the WSC 15 Cruiser class.

photo: Anthony Dekker (click image to zoom – OLD PIC)

994    Deakin University / ACCIONA (Ascend) 

Two-seat cruiser (new team with car: Ascend) – this new Australian team has a nice-looking Cruiser.

picture credit (click image to zoom)

995    Solaride 

Two-seat cruiser (new team) – this new team from Estonia has built a good-looking Cruiser. They are based in the city of Tartu.

Previously, Solaride participated in the SCM 21 Adventure class.

picture credit (click image to zoom)

996    DCU Solar Racing (Dublin City University) 

Monohull challenger (new team) – I am not sure what their plans are, exactly.

picture credit (click image to zoom)

998    Seoul National University (SNU SOLO) 

Three-wheel (tadpole) monohull challenger (new team) – this is a new team.

picture credit (click image to zoom)

999    AgniRath (IIT Madras) 

Three-wheel (tadpole) monohull challenger (new team) – this is a new team from IIT Madras.

public domain photo

This page last updated 13:52 on 24 March 2023 AEDT.

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FSGP 2023

The Formula Sun Grand Prix is on again this year at Heartland Motorsports Park in Topeka, Kansas. Scrutineering begins on 27 June.

Here is a list of the 23 teams (19 Challenger/SOV cars and 4 Cruiser/MOV cars) from 2 countries registered for the race. Teams are sorted in team number order and, as always, you can click the social media links, and click images to zoom (I have not yet found images for some of the newer teams). You can also check out the official ASC social media at        (click on the icons).

3    Kentucky 

Symmetric challenger (Gato del Sol VI) – I understand that they are racing their existing car.

Previously, Kentucky came 14^th at FSGP 14; came 10^th at FSGP 15; came 12^th at ASC 16; came 7^th at FSGP 17; came 3^rd at FSGP 19; came 2^nd at ASC 21; and came 3^rd at ASC 22.

 
Left: credit / Right: credit (click images to zoom)

5    University of Florida (Solar Gators) 

Monohull challenger (new car: Sunrider) – they have a beautiful new car.

Previously, Florida came equal 15^th at FSGP 17; came 10^th at FSGP 18; came 8^th at FSGP 19; and came 10^th at FSGP 22.

picture credit (click image to zoom)

6    UC Berkeley Solar Vehicle Team (CalSol) 

Symmetric challenger (Zephyr) – they raced in Australia in 2019 as number 66: read about their Australian adventures here.

Previously, CalSol participated in the WSC 19 Cruiser class; came 15^th at FSGP 14; came 7^th at FSGP 15; came 9^th at ASC 16; won FSGP 17; came 6^th at ASC 18; came 2^nd in the FSGP 19 Cruiser class; came 6^th at ASC 21; and came 4^th at ASC 22.

 
Left: Anthony Dekker / Right: credit (click images to zoom)

8    UT Austin 

Monohull challenger (Lone Star ) – I understand that they are racing their existing car.

Previously, UT came 9^th at ASC 14; came 6^th at FSGP 15; and came 9^th at FSGP 17.

picture credit (click image to zoom)

11    Northwestern 

Symmetric challenger (SC7) – while a new SC8 is planned eventually, they will race the SC7 they sent to FSGP 19.

Previously, Northwestern came 16^th at FSGP 16; came 12^th at FSGP 17; and came 10^th at FSGP 19.

 
Left: credit / Right: credit (click images to zoom)

12    Hot Wheelz (Rochester Institute of Technology) 

Challenger (new team) – this is an all-female team, with experience in Formula SAE.

placeholder photo: Dennis Cantrell / US Navy

17    Illinois State 

Symmetric challenger (Mercury 6) – I understand that they are racing their existing car.

Previously, Illinois St came equal 16^th at FSGP 14; came 3^rd at FSGP 15; came 11^th at ASC 16; came 5^th at FSGP 17; came 12^th at FSGP 18; came 7^th at FSGP 19; came 7^th at ASC 21; came 11^th at FSGP 22; and came 13^th at Abu Dhabi 15.

 
Left: credit / Right: credit (click images to zoom)

19    Bearcat Solar Car (Cincinnati) 

Challenger (new team) – I am not sure of the status of this team.

placeholder photo: Dennis Cantrell / US Navy

21    Kennesaw State

Challenger (new team) – I am not sure of the status of this team.

placeholder photo: Dennis Cantrell / US Navy

32    Principia Solar Car Team 

Asymmetric challenger (Ra XI) – I understand that they are racing their existing car.

Previously, Principia came 17^th at WSC 15; participated at WSC 17; came 5^th at ASC 14; came 5^th at FSGP 15; came 5^th at ASC 16; came 13^th at FSGP 17; came 2^nd at FSGP 19; came 3^rd at ASC 21; came 2^nd at ASC 22; and came 6^th at Abu Dhabi 15.

 
Left: credit / Right: credit (click images to zoom)

42    Missouri S&T 

Asymmetric challenger (Independence) – I understand that they are racing their existing car.

Previously, Missouri S&T came 7^th at FSGP 14; came 4^th at FSGP 15; came 4^th at ASC 16; and came 8^th at FSGP 17. Their team number (42) is a Hitchhiker’s Guide to the Galaxy reference.

 
Left: credit / Right: credit (click images to zoom)

55    Esteban (Poly Montreal) 

Two-seat cruiser (Esteban 10) – they have transitioned to the Cruiser (MOV) class.

Previously, Esteban came 4^th at ASC 14; came 2^nd at FSGP 15; came equal 10^th at ASC 16; came 3^rd at FSGP 17; came 4^th at ASC 18; won FSGP 19; and came 3^rd in the ASC 22 Cruiser class. Their team number (55) is the year that Western Electric began to sell licenses for silicon PV technology.

picture credit (click image to zoom)

66    Rutgers University 

Asymmetric challenger (Arctan) – they seem to have been given Stanford’s 2015 car.

picture credit (click image to zoom)

87    University of Virginia 

Monohull challenger (Rivanna 2) – I understand that they are racing their existing car.

Previously, UVA came 12^th at FSGP 22.

picture credit (click image to zoom)

96    Sunstang (Western University) 

Two-seat cruiser (new car) – they are building a new car for this event. The car uses Sunpower Maxeon Gen II monocrystalline silicon cells.

Previously, Sunstang came 14^th at FSGP 15 and came equal 15^th at FSGP 17.

picture credit (click image to zoom)

99    North Carolina State University (SolarPack) 

Two-seat cruiser (SPX) – this team will race their modified commercial ICE vehicle.

Previously, NCSU came 4^th in the FSGP 21 Cruiser class and came 6^th in the FSGP 22 Cruiser class.

 
Left: credit / Right: credit (click images to zoom)

540    Virginia Tech 

Challenger (new team) – I am not sure of the status of this team. Their team number (540) is the telephone area code for parts of northwestern Virginia.

placeholder photo: Dennis Cantrell / US Navy

608    Badgerloop (Wisconsin) 

Challenger (new team) – I am not sure of the status of this team. Their team number (608) is the telephone area code for southwestern Wisconsin.

placeholder photo: Dennis Cantrell / US Navy

614    Buckeye (Ohio State) 

Symmetric challenger (Farasi) – they seem to have been given Sunseeker’s Farasi. Their team number (614) is the telephone area code for parts of Columbus, Ohio.

picture credit (click image to zoom)

777    Tigres Del Sol (University of the Pacific) 

Challenger (new team) – I am not sure of the status of this team.

placeholder photo: Dennis Cantrell / US Navy

786    Western Michigan (Sunseeker) 

Asymmetric challenger (Aethon) – I understand that they are racing their existing car.

Previously, W Mich came 7^th at ASC 14; came 14^th at FSGP 17; came 9^th at ASC 18; came 5^th at FSGP 19; came 8^th at FSGP 21; and came 8^th at FSGP 22. Their team number (786) is the sequence of digits for S-U-N on old phones (but last year they raced as 30).

photo: WMU team (click image to zoom)

828    Appalachian State University (Sunergy) 

Two-seat cruiser (ROSE) – this team has a fantastic, well-tested car.

Previously, AppState came 6^th at ASC 16; came 2^nd at FSGP 17; came equal 2^nd in the ASC 18 Cruiser class; won the ASC 21 Cruiser class; and came 2^nd in the ASC 22 Cruiser class. Their team number (828) is the telephone area code for western North Carolina.

 
Left: credit / Right: credit (click images to zoom)

850    West Florida 

Monohull challenger (new team) – I am not sure of the status of this team. Their team number (850) is the telephone area code for parts of the Florida Panhandle.

placeholder photo: Dennis Cantrell / US Navy

This page last updated 19:27 on 17 March 2023 AEDT.

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Pi Day once more!

In honour of Pi Day (March 14), the chart shows six ways of randomly selecting a point in a unit disc. Four of the methods are bad, for various reasons.

A. Midpoint of random p, q on circumference

p = (cos(𝜃₁), sin(𝜃₁)) is a point on the circumference

q = (cos(𝜃₂), sin(𝜃₂)) is another point on the circumference

x = ½ cos(𝜃₁) + ½ cos(𝜃₂) and

y = ½ sin(𝜃₁) + ½ sin(𝜃₂), for random 𝜃₁ and 𝜃₂, define their midpoint.

B. Random polar coordinates

x = r cos(𝜃)

and y = r sin(𝜃), for random angle 𝜃 and radius r ≤ 1. This gives choices biased towards the centre.

C. Random y, then restricted x

Random y, followed by random x in the range −√(1−y²) to √(1−y²). This gives choices biased towards the top and bottom.

D. Random point on chord in A

Similar to A, but x = a cos(𝜃₁) + (1−a) cos(𝜃₂)

and y = a sin(𝜃₁) + (1−a) sin(𝜃₂), for random 𝜃₁ and 𝜃₂ on the circumference of the circle and random a between 0 and 1. This gives choices biased towards the periphery.

E. Random polar with sqrt(r)

Similar to B, but x = √r cos(𝜃)

and y = √r sin(𝜃), for random angle 𝜃 and radius r. The square root operation makes the selection uniform across the disc.

F. Random x, y within disc

Random x and y, repeating the choice until x² + y² ≤ 1. This is uniform, and the selection condition restricts the final choice to the disc.

Oh, and here are some Pi Day activities.

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The train crash in East Palestine, Ohio

A great deal has been written about the train derailment in East Palestine, Ohio. The preliminary NTSB report is one of the few solidly factual responses. One important question has been: what happens when vinyl chloride burns? Theoretically, in the presence of enough oxygen, you get this:

More realistically, based on the above reaction and this paper, you also get carbon monoxide, black soot (carbon), and traces of phosgene:

Why was it burned? Because vinyl chloride, if it gets too warm, can spontaneously polymerise into PVC plastic. That reaction is heat-producing and can lead to an explosion. Since the fire following the derailment had heated the rail cars containing vinyl chloride to dangerous levels, authorities believed an explosion was imminent. A controlled burn was probably a rational decision at the time.

Hydrogen chloride (HCl) in the smoke cloud was probably the immediate threat resulting from the controlled burn (and the likely cause of dead birds), although the HCl would soon have been safely diluted by rain. Unburned vinyl chloride in the subsoil is probably the longer-term threat, and (I understand) the focus of cleanup efforts.

Why did the train derail? This (somewhat fuzzy) map and chart is my best guess at a timeline. As the train travelled east, hotbox detectors (HBDs) noted increasing wheel bearing temperatures on car 23. Some media reports suggest flames of burning axle grease were seen in Columbiana, Ohio. The HBD at East Palestine noted a temperature of 253°F above ambient, higher than the railway company’s critical threshold. The crew immediately began to further slow the already slowing train, at which point the faulty wheel bearing on car 23 failed catastrophically, triggering the derailment and fire:

The map at database.defectdetector.net suggests that there used to be hotbox detector near Columbiana, Ohio (at MP 60.8), and this would presumably have caught the fault in time to avoid a derailment. One wonders what became of that HBD. It’s a pity that we need to wait 18 months for the NTSB’s final report.

Meanwhile, the EPA has put all their air, soil, and water testing results online.

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Do gun laws save lives?

Do gun laws save lives? The chart above shows homicide rates for U.S. states (data from here) together with an A to F ranking of state gun laws from the Giffords organisation. As with my post from 2017, there is actually no statistically significant correlation (this is particular noticeable among the F’s, which include both the seven states with the highest murder rate and the two states with the lowest). In other words, the answer seems to be no.

Rather, it seems that guns don’t kill people, people kill people. The murder rate in the U.S. is driven by social factors which differ from state to state – factors which make New Hampshire and Maine pretty safe, but which produce a murder rate ranging from 14 per 100,000 to 20.5 per 100,000 in Missouri, Alabama, Louisiana, and Mississippi. For comparison, New Hampshire has a murder rate similar to that of Australia, but Louisiana and Mississippi, if they were countries, would rank among the most murderous 20 countries in the world.

There is some evidence that keeping guns out of the hands of criminals would reduce the murder rate in the U.S., but this is extremely difficult to do. The U.S. has a lengthy, porous southern border, across which there is a free flow of people, guns, and illegal drugs.

In addition, a concept from catastrophe theory is useful here. In the cusp pictured below, it is possible to “drop” from the top of the fold to the bottom, but a long roundabout journey would be required to get back up. Similarly, it is very easy to introduce guns into a society, but very difficult to remove them. Although such removal has been done elsewhere, laws forbidding gun ownership are likely to be ignored by precisely those violent criminals that one would not wish to carry them. And, of course, there is the 2^nd Amendment.

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The Austin airport incident

Above is a chart of altitude data for the recent near-miss of two aircraft at Austin-Bergstrom International Airport in Texas, involving a Southwest 737 and FedEx 767, which the NTSB is investigating. Data for the chart is from Flightradar24 (at 25 foot resolution, and not totally accurate because it is calculated from air pressure). See that link also for the story, or this tweet and this other tweet.

In the chart, the emergency go-around by the FedEx aircraft is obvious (it began about 5 seconds before minimum separation, when horizontal separation was about 650 feet). The temporary rise of Southwest to 25 feet appears to be an artifact. I estimate minimum separation as 187 feet. FedEx was about 0.58 nautical miles (1.08 km) away, on course to land, when Southwest started rolling.

The landing approach was CAT III ILS due to heavy fog. It seems to me that the clear-thinking FedEx pilots saved lives that day. A recording of the radio communication is here. Transcript is as follows (accurate to-the-second timestamps are not available):

• FDX1432: Austin tower, FedEx 1432 heavy passing 5.4 for that CAT III ILS 18L.
• Tower: FedEx 1432 heavy, Austin Tower, 18L RVR [Runway Visual Range] touchdown 1400, midpoint 600, rollout 1800, 18L cleared to land.
• FDX1432: Cleared to land 18L, FedEx 1432 heavy.
• SWA708: Tower, Southwest 708, we’re short of 18L and we’re ready.
• Tower: Southwest 708, Austin Tower, runway 18L RVR 1200, midpoint 600, rollout 1600, fly heading 170, runway 18L, cleared for takeoff, traffic 3 mile final is a heavy 767.
• SWA708: Okay, 170 cleared for takeoff, 18L, copy the traffic, Southwest 708.
• FDX1432: Tower, confirm FedEx 1432 heavy is cleared to land on 18L [hearing the message to SWA708 and seeing a potential problem].
• Tower: FedEx 1432 heavy that is affirmative, 18L you are cleared to land, traffic departing prior to your arrival is a 737.
• FDX1432: Roger.
• Tower: Southwest 708 confirm on the roll.
• SWA708: Rolling now.
• FDX1432: Southwest abort [seeing SWA708 on the runway in front of him at the limit of visibility in the fog].
• FDX1432: FedEx is on the go [go-around].
• Tower: Southwest 708, roger [apparently believing “abort” came from SWA708], you can turn right when able.
• SWA708: Negative.
• Tower: FedEx 1432, climb and maintain 3000 [feet], when able you can turn left heading 080.
• FDX1432: Left turn to 080, 3000, FedEx 1432 heavy.
• Tower: Southwest 708, you can turn left heading 170.

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Australian slang

The Bridgestone World Solar Challenge will be held next October. Here is a taste of Australian slang for any Americans participating:

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Timeline of mathematical notation

Following up on my earlier timelines about zero and about Hindu-Arabic numerals, here is a timeline for some other mathematical notation, starting with the square root symbol (click to zoom).

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0123456789 in Europe: an infographic

Following up on my earlier post about 0 and 1 in Greek mathematics and my timeline of zero in Europe, here is a timeline for the use of Hindu-Arabic numerals in Europe up to René Descartes (click to zoom).

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0 and 1 in Greek mathematics

Following up on an earlier post about zero in Greek mathematics and this timeline of zero, I want to say something more about the role of 0 (zero) and 1 (one) in ancient Greek thought. Unfortunately, some of the discussion on Greek mathematics out there is a bit like this:

0 and 1 as quantities

The ancient Greeks could obviously count, and they had bankers, so they understood credits and debts, and the idea of your bank account being empty. However, they had not reached the brilliant insight of Brahmagupta, around 628 AD, that you could multiply a debt (−) and a debt (−) to get a credit (+).

The ancient Greeks had three words for “one” (εἷς = heis, μία = mia, ἑν = hen), depending on gender. So, in the opening line of Plato’s Timaeus, Socrates counts: “One, two, three; but where, my dear Timaeus, is the fourth of those who were yesterday my guests … ? (εἷς, δύο, τρεῖς: ὁ δὲ δὴ τέταρτος ἡμῖν, ὦ φίλε Τίμαιε, ποῦ τῶν χθὲς μὲν δαιτυμόνων … ; )”

The Greeks had two words for “nothing” or “zero” (μηδέν = mēden and οὐδέν = ouden). So, in the Christian New Testament, in John 21:11, some fisherman count fish and get 153, but in Luke 5:5, Simon Peter says “Master, we toiled all night and took nothing (οὐδὲν)!”

0 and 1 in calculations

In ordinary (non-positional) Greek numerals, the Greeks used α = 1, ι = 10, and ρ = 100. There was no special symbol for zero. Greek mathematicians, such as Archimedes, wrote numbers out in words when stating a theorem.

Greek astronomers, who performed more complex calculations, used the Babylonian base-60 system. Sexagesimal “digits” from 1 to 59 were written in ordinary Greek numerals, with variations of ō for zero. The overbar was necessary to distinguish ō from the letter ο, which denoted the number 70 (since an overbar was a standard way of indicating abbreviations, it is likely that the symbol ō was an abbreviation for οὐδὲν).

Initially (around 100 AD) the overbar was quite fancy, and it became shorter and simpler over time, eventually disappearing altogether. Here it is in a French edition of Ptolemy’s Almagest of c. 150 AD:

In Greek-influenced Latin astronomical calculations, such as those used by Christians to calculate the date of Easter, “NULLA” or “N” was used for zero as a value. Such calculations date from the third century AD. Here (from Gallica) is part of a beautiful late example from around 700 AD (the calendar of St. Willibrord):

Outside of astronomy, zero does not seem to get mentioned much, although Aristotle, in his Physics (Book 4, Part 8) points out, as if it is a well-known fact, that “there is no ratio of zero (nothing) to a number (οὐδὲ τὸ μηδὲν πρὸς ἀριθμόν),” i.e. that you cannot divide by zero. Here Aristotle may have been ahead of Brahmagupta, who thought that 0/0 = 0.

0 and 1 as formal numbers?

We now turn to the formal theory of numbers, in the Elements of Euclid and other works. This is mathematics in a surprisingly modern style, with formal proofs and (more or less) formal definitions. In book VII of the Elements (Definitions 1 & 2), Euclid defines the technical terms μονάς = monas (unit) and ἀριθμὸς = arithmos (number):

1. A monas (unit) is that by virtue of which each of the things that exist is called one (μονάς ἐστιν, καθ᾽ ἣν ἕκαστον τῶν ὄντων ἓν λέγεται).
2. An arithmos (number) is a multitude composed of units (ἀριθμὸς δὲ τὸ ἐκ μονάδων συγκείμενον πλῆθος).

So 1 is the monas (unit), and the technical definition of arithmos excludes 0 and 1, just as today the technical definition of natural number is taken by some mathematicians to exclude 0. However, in informal Greek language, 1 was still a number, and Greek mathematicians were not at all consistent about excluding 1. It remained a number for the purpose of doing arithmetic. Around 100 AD, for example, Nicomachus of Gerasa (in his Introduction to Arithmetic, Book 1, VIII, 9–12) discusses the powers of 2 (1, 2, 4, 8, 16, 32, 64, 128, 256, 512 = α, β, δ, η, ιϛ, λβ, ξδ, ρκη, σνϛ, φιβ) and notes that “it is the property of all these terms when they are added together successively to be equal to the next in the series, lacking a monas (συμβέβηκε δὲ πᾱ́σαις ταῖς ἐκθέσεσι συντεθειμέναις σωρηδὸν ἴσαις εἶναι τῷ μετ’ αὐτὰς παρὰ μονάδα).” In the same work (Book 1, XIX, 9), he provides a multiplication table for the numbers 1 through 10:

The issue here is that Euclid was aware of the fundamental theorem of arithmetic, i.e. that every positive integer can be decomposed into a bag (multiset) of prime factors, in no particular order, e.g. 60 = 2×2×3×5 = 2×2×5×3 = 2×5×2×3 = 5×2×2×3 = 5×2×3×2 = 2×5×3×2 = 2×3×5×2 = 2×3×2×5 = 3×2×2×5 = 3×2×5×2 = 3×5×2×2 = 5×3×2×2.

Euclid proves most of this theorem in propositions 30, 31 and 32 of his Book VII and proposition 14 of his Book IX. The number 0 is obviously excluded from consideration here, and the number 1 is special because it represents the empty bag (even today we recognise that 1 is a special case, because it is not a prime number, and it is not composed of prime factors either – although, as late as a century ago, there were mathematicians who called 1 prime, which causes all kinds of problems):

• If two numbers (arithmoi) by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers (ἐὰν δύο ἀριθμοὶ πολλαπλασιάσαντες ἀλλήλους ποιῶσί τινα, τὸν δὲ γενόμενον ἐξ αὐτῶν μετρῇ τις πρῶτος ἀριθμός, καὶ ἕνα τῶν ἐξ ἀρχῆς μετρήσει) – i.e. if a prime p divides ab, then it divides a or b or both
• Any composite number is measured by some prime number (ἅπας σύνθετος ἀριθμὸς ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται) – i.e. it has a prime factor
• Any number (arithmos) either is prime or is measured by some prime number (ἅπας ἀριθμὸς ἤτοι πρῶτός ἐστιν ἢ ὑπὸ πρώτου τινὸς ἀριθμοῦ μετρεῖται) – this would not be true for 1
• If a number be the least that is measured by prime numbers, it will not be measured by any other prime number except those originally measuring it (ἐὰν ἐλάχιστος ἀριθμὸς ὑπὸ πρώτων ἀριθμῶν μετρῆται, ὑπ᾽ οὐδενὸς ἄλλου πρώτου ἀριθμοῦ μετρηθήσεται παρὲξ τῶν ἐξ ἀρχῆς μετρούντων) – this is a partial expression of the uniqueness of prime factorisation

The special property of 1, the monas or unit, was sometimes expressed (e.g. by Nicomachus of Gerasa) by saying that it is the “beginning of arithmoi … but not itself an arithmos.” As we have already seen, nobody was consistent about this, and there was, of course, no problem in doing arithmetic with 1. Everybody agreed that 1 + 2 + 3 + 4 = 10. In modern mathematics, we would avoid problems by saying that natural numbers are produced using the successor function S, and distinguish that function from the number S(0) = 1.

The words monas and arithmos occur in other Greek writers, not always in the Euclidean technical sense. For example, in a discussion of causes and properties in the Phaedo (105c), Plato tells us that “if you ask what causes an arithmos to be odd, I shall not say oddness, but the monas (οὐδ᾽ ᾧ ἂν ἀριθμῷ τί ἐγγένηται περιττὸς ἔσται, οὐκ ἐρῶ ᾧ ἂν περιττότης, ἀλλ᾽ ᾧ ἂν μονάς).” Aristotle, in his Metaphysics, spends some time on the philosophical question of what the monas really is.

In general, the ancient Greeks seem to have had quite a sophisticated understanding of 0 and 1, though hampered by poor vocabulary and a lack of good symbols. Outside of applied mathematics and astronomy, they mostly worked with what we would call the multiplicative group of the positive rational numbers. What they were missing was any awareness of negative numbers as mathematical (not just financial) concepts. That had to wait until Brahmagupta, and when it came, 0 suddenly became a whole lot more interesting, because it eventually became possible to define more advanced mathematical concepts like fields.

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