Why the 2026 World Cup Ball Has Deeper Seams

 In 2010 the South African World Cup had a terrible football, the Jabulani​

​It was too smooth resulting in serious problems. The football had a lot less seam which the technology at the time was not ready for

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Jabulani: eight thermally bonded panels, unusually short/shallow seams

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The 2010 ball had shallow thin seams

When you kick a football really fast a layer of air develops around it that reduces the drag. This is why golf balls have dimples. An amount of surface roughness can actually reduce drag.

At a certain point the air buffer leaves the football and it gets full drag. This is the critical speed.

Trionda: four panels, deeper seams and textured icons.

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​The Jabulani​ had a really high critical speed which means as soon as it got to 24 meters per second it started to slow down quickly. The Trionda's, this year's ball, seams are deeper allowing it to 'catch' air and avoid the critical speed of drag for longer. Also the surface of the ball itself is slightly bumpy in a way the ​Jabulani​'s were not.

Goalkeepers did not like this as they said it made the ball seem like a beach ball that did not have a predictable movement. But the ​number of goals at the World Cup were really low. So the goalkeeper discomfort at the weird ball paths were probably made up for by the ball going slower in a way that reduced goals.

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This book on the mathematics of the football and this article lead to this post 

I got my hands on one of the new balls. It has a groove down each of the panels. And finger print like dimples in the surface. 

There are 11064 Cars in Luxembourg

I was in Luxembourg recently and saw 4 cars when I was there. Their registration plates were 1396, 1871, 2383 and 8852

This made me wonder if I could estimate the total number of cars in Luxembourg. 

This is the German Tank problem where the British in WW2 found some serial numbers on German tanks in north Africa and wanted to use those to guess how many tanks had been produced. If you are sure the serial number on tanks goes up by one each time and they started with number 1. Then if you have seen the highest number 100 in looking at 3 tanks of after looking at 50 you can have a better estimate to the highest number of tank made.  Also if you see a tank with number 1000 you know at least 1000 have been made. 

A formula to estimate the total number based on seeing k serial (registration) numbers where m is the highest number seen is the MVUE (Minimum Variance Unbiased Estimator) N ≈ m + (m/k − 1)

Which gives an estimate of 11064 total cars in Luxembourg.

Euclid, Fermat, and the 65537 Reasons We Have Never Drawn Their Last Shape

 1. Straight Edge and Compass Euclidean Constructions

From the time of ancient Greece, geometers have been fascinated by what can be built using only two tools: a compass and a straightedge. These "Euclidean constructions" formed the foundation of classical geometry. You can copy a line segment, bisect an angle, draw perpendiculars Euclid's Elements are full of constructions and theorems that are built from these two tools.

But some polygons cannot be drawn this way. A regular 7-gon is impossible. Why? Because the set of constructible polygons is tied deeply to number theory, to Fermat primes.

2. Fermat Primes and Drawing Them

A Fermat prime is a prime number of the form

Only five such primes are known: 3, 5, 17, 257, 65537

Carl Friedrich Gauss at age 19 by proved that a regular polygon with sides is constructible if and only if is a product of a power of 2 and distinct Fermat primes. This means a regular 65537-gon is theoretically constructible.

One mathematician actually attempted to do it. Johann Gustav Hermes spent over a decade and 200 pages , detailing how to construct a 65537-gon using Euclidean methods in Ueber die Teilung des Kreises in 65537 gleiche Teile  in 1894. 

3. The 65537-gon

If you tried to actually draw a 65537-gon on a circle the size of the Earth (with of 6,371 km), each side would be approximately 611 meters long. Which means on a flat lake one side of the 65537-gon would not follow the earth's curvature by 3cm. A 65537-gon would be 3cm per 611 meter side off being a perfect circle. 

No one has ever made a by hand 65537-gon because it would just be too involved. though Hermes did spend ten years working out how it could be done in principle.

A 65537-gon (not a circle)

4. A Machine to Draw It for Us?

Could we build a machine that uses only a compass and a straightedge, like Euclid himself? Plotters could draw the 65537-gon by calculating angles and coordinates, but Euclid has famously tough lawyers and anyone making his shape not using his methods is likely to be mired in law suits

Such a Euclid Geometry machine would have to:

• Set distances using a mechanical compass

• Draw arcs

• Align a straightedge through marked points

• Draw straight lines with a pen

• Combine these steps tens of thousands of times

It would follow the ancient Greek rules of construction.

This would be rock hard to make and not practically useful. It would involve, aligning a  ruler through two points. Resetting a compass to exactly the distance between two previous marks, which is mechanically fiddly. Detecting intersections and points on paper. All of this is simple for Ancient Greek humans but not for machines.

And yet, it would be beautiful. It would make visible not just the result of geometric thinking, but the process. A Euclid-bot wouldn’t just draw geometry it would demonstrate it.

There are cool online geometry tools but something about plotters and physical drawing things still appeals. Maybe it would be possible to make a ruler and compass wielding machine that could finally draw the 65537 sides of the last Polygon.

Calculus Made Easy and Memorable

I made a website to help people learn calculus. I have taken a 1910 book Calculus Made Easy by Silvanus P. Thompson that has already been digitized and put on the web in a nice format at https://calculusmadeeasy.org/. This was created by nadvornix and other volunteers. This edition has added some memorisation reminders to the text.

The book is famous for being accessible to teenagers. Some of the text has dated in the century since. Martin Gardner created a new edition in the 1980s with some of the text deolded and some extra background chapters. Making similar updates might improve this book some more.

Orbit is a tool that tests your knowledge of text you just read. And then retests you on a regular basis. This vastly increases the amount you remember for very little extra time or effort. Some of the evidence that timed repetition aids recall is given in this essay 

Calculus made Easy and Memorable is on the web at http://calculusmadememorable.org/ and the code is here. 

Why put a new more memorable calculus book online? ‘1 million students take a college-level Calculus 1 course in the United States, at an average cost of $2,500. And then 40% of them fail.’ That is 1 billion loss a year in one country from failing one course. Anything that helps reduce that rate could be a boon.

Pi Digits High Low Game

Suppose you take a long number and for each digit if it is bigger then the previous one increase a counter by one. If it is less then the previous number reduce the counter by one. You keep a running total and graph that total. Noting when it passes 0. This total number will go up and down and can get to zero many times. If you play this game with random numbers in a million digits on average the number of times you will have crossed 0 is 1594.4 and the standard deviation of the number of times crossed 0 is 1207.3. Though as the number of times zero is crossed cannot be less then 0 this is a bit odd.

       
import random

numcrossed=[]
j=0
while j < 200:
	i = 0
	last=0
	total=0
	x=[]
	y=[]
	crossed=0
	while i < 1000000:
		ran= random.randint(0, 10)
		if ran==last:
			total=total
		elif ran>last:
			total=total+1
		else:
			total=total-1
		if total==0:
			x.append(i)
			y.append(total)
			crossed=crossed+1        
		i=i+1        
		last=ran
	numcrossed.append(crossed)
	j=j+1
       
 

If instead of random numbers the digits of pi are used. This is what the path of total counts looks like

       
file = open("pi1000000.txt", "r") 
#3.14159265358979323846264338327950 pi2.txt
x = []
y = []
text=file.read() 

pi = list(text)
total =0
i = 0
crossed=0
i=0

while i < len(pi):
	if pi[i]>pi[i-1]:
		#print(pi[i])
		total=total+1
	if pi[i]
 

Pi has a 0 total 657 times. Which is more than 51 out of 200 random million long sequences did in my tests. None of this means anything. Going up or down based on digits in a base ten number but i like these pattern sort of sequences.

E crosses 0 1725 times

sqrt2 crosses 0 1300 times

and with 2 million digits

The python code for visualisation is 
       
import numpy as np
import matplotlib.pyplot as plt

plt.scatter(x, y, alpha=0.5, color='green')
plt.title('Sqrt 2 High Low Game')
plt.show()
       
 

The Great Stagnation, Football Ball Edition

'Since 2002, panel numbers have roughly halved every four years: 32 in 2002, 14 in 2006 and eight in 2010. Thus, by the 2022 World Cup, players should be kicking a single-panel ball around the pitch.' claimed Ken Bray here. I will call this Brays law 'The number of panels on the World Cup football ball will half every tournament'. This is not quite as epoch defining as Moore's law but still cool I think.

This year according to projections the soccer ball in the world cup should have at most 4 panels. Instead the ball has 6. 50% more panels then you would expect if progress continued at the rate Bray predicted. The Great Stagnation is the belief that things are not improving as quickly as they used to and is used to explain why we still have homeless people but not flying cars.

The number of panels each world cup ball has is found on each balls individual Wikipedia page.

2014 the Adidas Brazuca: 'The ball has been made of six polyurethane panels'

2010 the Adidas Jabulani: 'The ball was constructed consisting of eight (down from 14 in the 2006 World Cup) thermally bonded, three-dimensional panels'

2006 the Adidas Teamgeist:'The Teamgeist ball differs from previous balls in having just 14 curved panels rather than the 32 that have been standard since 1970. Like the 32 panel Roteiro which preceded it'

Fewer panels mean the ball is smoother and should fly more true. The ball flying true involves the interaction of several variables other than the panel number though. The 2010 ball was notorious for wobbly flight for example. For this reason just reducing the number of panels at the expense of the quality of the ball is a bad idea and may explain why Bray's law has failed. Still not being able to make a ball work with fewer panels indicates a technological innovation slow down to me. The aerodynamics of soccer balls and why there is a race to fewer panels is described in Brays article 'A fly walks round a football'.

Becoming President with 22% of the Votes

Political pundits talk about the possibility of winning the Presidential election with less votes than your opponent. Assuming only two candidates what is the lowest percentage of votes you could get and still win the election? In the case where everyone votes this becomes an interesting question. In America some states have a higher ratio of people to electoral college votes than others. If the ones that are preferentially treated banded together a small percentage of people could decide the election.

In my last post I created a program to work out in an overly complicated way what was the least amount of land a president could be elected from in the US. In this post I want to figure out the best states to win to get you 270+ electoral college seats using the smallest number of voters.

I got the estimated population in July 2011 from the census website here. Using this data in the glpk program below and got the result shown in this map. If everyone voted and the people who get the most power in votes all voted one way then the states on the winning side would have 135936335 voters to get 270 electoral college seats. The total population is 311591917 so 43.67% of the population

The winner would win 40 districts of the 51 states+dc and lose Virginia, Georgia, New Jersey, Michigan, Ohio, Illinois, Pennsylvania, Florida, New York, Texas, California.*

Now if the candidate in these states just squeaked a win by one vote and got zero votes in all the other states that means in a two party election you could win an election, where everyone voted, with under 22% of the vote. If you think of this happening in a senate election where each state has 2 senators (and DC none) then you could control 80% of the senate with under 22% of the vote.

There are all sorts of other questions similar to this. What is the smallest block where all the states touch? What is the shortest distance between all her state capitals a winner could have? I think some analysis that included Senate and Congress seats could be interesting. If you have any ideas please comment. * Thanks to Hakan Kjellerstrand who pointed out here I had read the solution file wrong and North Carolina was not present.

/* sets */
set STATES;
set NEED;

/* parameters */
param VotesTable {i in STATES, j in NEED};
param Pop {i in STATES};
param Need {j in NEED};


/* decision variables: x1: alabama, x2: , x3: , x4:  x51: Wyoming*/
      var x {i in STATES} binary >= 0;

/* objective function */
      minimize z: sum{i in STATES} Pop[i]*x[i];

/* Constraints */
s.t. const{j in NEED} : sum{i in STATES} VotesTable[i,j]*x[i] >= Need[j];


/* data section */
data;

set STATES :=  Alaska Delaware "District of Columbia" Montana "North Dakota" "South Dakota" Vermont Wyoming Hawaii Idaho Maine "New Hampshire" "Rhode Island" Nebraska Nevada "New Mexico" Utah "West Virginia" Arkansas Kansas Mississippi Connecticut Iowa Oklahoma Oregon Kentucky "South Carolina" Alabama Colorado Louisiana Arizona Maryland Minnesota Wisconsin Indiana Missouri Tennessee Washington Massachusetts Virginia Georgia "New Jersey" "North Carolina" Michigan Ohio Illinois Pennsylvania Florida "New York" Texas California;
set NEED := Votes;

param VotesTable: Votes:=
 Alabama 9
 Alaska 3
 Arizona 11
 Arkansas 6
 California 55
 Colorado 9
 Connecticut 7
 Delaware 3
 "District of Columbia" 3
 Florida 29
 Georgia 16
 Hawaii 4
 Idaho 4
 Illinois 20
 Indiana 11
 Iowa 6
 Kansas 6
 Kentucky 8
 Louisiana 8
 Maine 4
 Maryland 10
 Massachusetts 11
 Michigan 16
 Minnesota 10
 Mississippi 6
 Missouri 10
 Montana 3
 Nebraska 5
 Nevada 6
 "New Hampshire" 4
 "New Jersey" 14
 "New Mexico" 5
 "New York" 29
 "North Carolina" 15
 "North Dakota" 3
 Ohio 18
 Oklahoma 7
 Oregon 7
 Pennsylvania 20
 "Rhode Island" 4
 "South Carolina" 9
 "South Dakota" 3
 Tennessee 11
 Texas 38
 Utah 6
 Vermont 3
 Virginia 13
 Washington 12
 "West Virginia" 5
 Wisconsin 10
 Wyoming 3;

param Pop:=
Alabama 4802740
Alaska 722718
Arizona 6482505
Arkansas 2937979
California 37691912
Colorado 5116796
Connecticut 3580709
Delaware 907135
"District of Columbia" 617996
Florida 19057542
Georgia 9815210
Hawaii 1374810
Idaho 1584985
Illinois 12869257
Indiana 6516922
Iowa 3062309
Kansas 2871238
Kentucky 4369356
Louisiana 4574836
Maine 1328188
Maryland 5828289
Massachusetts 6587536
Michigan 9876187
Minnesota 5344861
Mississippi 2978512
Missouri 6010688
Montana 998199
Nebraska 1842641
Nevada 2723322
"New Hampshire" 1318194
"New Jersey" 8821155
"New Mexico" 2082224
"New York" 19465197
"North Carolina" 9656401
"North Dakota" 683932
Ohio 11544951
Oklahoma 3791508
Oregon 3871859
Pennsylvania 12742886
"Rhode Island" 1051302
"South Carolina" 4679230
"South Dakota" 824082
Tennessee 6403353
Texas 25674681
Utah 2817222
Vermont 626431
Virginia 8096604
Washington 6830038
"West Virginia" 1855364
Wisconsin 5711767
Wyoming 568158;

param Need:=
Votes        270;

end;