games with numbers

I was watching videos about Collatz and these thoughts occurred to me.

There are more interesting graphs around a julia set like set up.

 

n 1 2 3 4 5 6 7 8 9 10
n^2 1 4 9 16 25 36 49 64 81 100
3(n)+1 4 7 10 13 16 19 22 25 28 31
(3(n+1))/2 2 3.5 5 6.5 8 9.5 11 12.5 14 15.5
subtract 3 5 7 9 11 13 15 17 19 21
tot/n 4 3.5 3.33333333333333 3.25 3.2 3.16666666666667 3.14285714285714 3.125 3.11111111111111 3.1
n/tot 0.5 0.571428571428571 0.6 0.615384615384615 0.625 0.631578947368421 0.636363636363636 0.64 0.642857142857143 0.645161290322581
n 11 12 13 14 15 16 17 18 19 20
n^2 121 144 169 196 225 256 289 324 361 400
3(n)+1 34 37 40 43 46 49 52 55 58 61
(3(n+1))/2 17 18.5 20 21.5 23 24.5 26 27.5 29 30.5
subtract 23 25 27 29 31 33 35 37 39 41
tot/n 3.09090909090909 3.08333333333333 3.07692307692308 3.07142857142857 3.06666666666667 3.0625 3.05882352941176 3.05555555555556 3.05263157894737 3.05
n/tot 0.647058823529412 0.648648648648649 0.65 0.651162790697674 0.652173913043478 0.653061224489796 0.653846153846154 0.654545454545455 0.655172413793103 0.655737704918033
n 21 22 23 24 25 26 27 28 29 30
n^2 441 484 529 576 625 676 729 784 841 900
3(n)+1 64 67 70 73 76 79 82 85 88 91
(3(n+1))/2 32 33.5 35 36.5 38 39.5 41 42.5 44 45.5
subtract 43 45 47 49 51 53 55 57 59 61
tot/n 3.04761904761905 3.04545454545455 3.04347826086957 3.04166666666667 3.04 3.03846153846154 3.03703703703704 3.03571428571429 3.03448275862069 3.03333333333333
n/tot 0.65625 0.656716417910448 0.657142857142857 0.657534246575342 0.657894736842105 0.658227848101266 0.658536585365854 0.658823529411765 0.659090909090909 0.659340659340659
n 31 32 33 34 35 36 37 38 39 40
n^2 961 1024 1089 1156 1225 1296 1369 1444 1521 1600
3(n)+1 94 97 100 103 106 109 112 115 118 121
(3(n+1))/2 47 48.5 50 51.5 53 54.5 56 57.5 59 60.5
subtract 63 65 67 69 71 73 75 77 79 81
tot/n 3.03225806451613 3.03125 3.03030303030303 3.02941176470588 3.02857142857143 3.02777777777778 3.02702702702703 3.02631578947368 3.02564102564103 3.025
n/tot 0.659574468085106 0.65979381443299 0.66 0.660194174757282 0.660377358490566 0.660550458715596 0.660714285714286 0.660869565217391 0.661016949152542 0.661157024793388
n 41 42 43 44 45 46 47 48 49 50
n^2 1681 1764 1849 1936 2025 2116 2209 2304 2401 2500
3(n)+1 124 127 130 133 136 139 142 145 148 151
(3(n+1))/2 62 63.5 65 66.5 68 69.5 71 72.5 74 75.5
subtract 83 85 87 89 91 93 95 97 99 101
tot/n 3.02439024390244 3.02380952380952 3.02325581395349 3.02272727272727 3.02222222222222 3.02173913043478 3.02127659574468 3.02083333333333 3.02040816326531 3.02
n/tot 0.661290322580645 0.661417322834646 0.661538461538462 0.661654135338346 0.661764705882353 0.661870503597122 0.661971830985916 0.662068965517241 0.662162162162162 0.662251655629139
n 51 52 53 54 55 56 57 58 59 60
n^2 2601 2704 2809 2916 3025 3136 3249 3364 3481 3600
3(n)+1 154 157 160 163 166 169 172 175 178 181
(3(n+1))/2 77 78.5 80 81.5 83 84.5 86 87.5 89 90.5
subtract 103 105 107 109 111 113 115 117 119 121
tot/n 3.01960784313725 3.01923076923077 3.0188679245283 3.01851851851852 3.01818181818182 3.01785714285714 3.01754385964912 3.01724137931034 3.01694915254237 3.01666666666667
n/tot 0.662337662337662 0.662420382165605 0.6625 0.662576687116564 0.662650602409639 0.662721893491124 0.662790697674419 0.662857142857143 0.662921348314607 0.662983425414365
n 61 22 23 24 25 26 27 28 29 30
n^2 3721 484 529 576 625 676 729 784 841 900
3(n)+1 184 67 70 73 76 79 82 85 88 91
(3(n+1))/2 92 33.5 35 36.5 38 39.5 41 42.5 44 45.5
subtract 123 45 47 49 51 53 55 57 59 61
tot/n 3.01639344262295 3.04545454545455 3.04347826086957 3.04166666666667 3.04 3.03846153846154 3.03703703703704 3.03571428571429 3.03448275862069 3.03333333333333
n/tot 0.663043478260869 0.656716417910448 0.657142857142857 0.657534246575342 0.657894736842105 0.658227848101266 0.658536585365854 0.658823529411765 0.659090909090909 0.659340659340659
n 31 32 33 34 35 36 37 38 39 40
n^2 961 1024 1089 1156 1225 1296 1369 1444 1521 1600
3(n)+1 94 97 100 103 106 109 112 115 118 121
(3(n+1))/2 47 48.5 50 51.5 53 54.5 56 57.5 59 60.5
subtract 63 65 67 69 71 73 75 77 79 81
tot/n 3.03225806451613 3.03125 3.03030303030303 3.02941176470588 3.02857142857143 3.02777777777778 3.02702702702703 3.02631578947368 3.02564102564103 3.025
n/tot 0.659574468085106 0.65979381443299 0.66 0.660194174757282 0.660377358490566 0.660550458715596 0.660714285714286 0.660869565217391 0.661016949152542 0.661157024793388

Three photos from a little journey

During the summer we went for a little break to the Ardmore in Waterford.

These are three photos that remind me of those warmer days.

 

pump

This little beauty is a hand bellows for a firew. It is in a tiny rural pub on the outskirts of Ardmore and it still works over a hundred years since it was made.

half-penny

This cast of a half penny adorns the wall of a small dining area to the rear of The Spire Cafe, in Lismore. As I remember predecimal coins this was a bit of a surprise.

However this cafe gave me even more of a surprise, when I went inside I found the meter below on a shelf and they let me take it down and place it on a table to take a photo.

In a former life I used to use analogue meters on a regular basis.

My father had one from this manufacturer when I was a very young. That was when we had lead in solder.

meter

Sometimes you should look at the back of a thing.

backmeter

Kings Of The Wyld

Kings Of The Wyld

I finished it. If you like fantasy all action books, you could to a lot worse than this. 5/5

My usual reading in fiction are more Space Opera, this is outright fantasy.  It doesn’t suffer with an over complicated story. I enjoyed it from beginning to end.

It follows some good plot writing toolchains and this is a good thing.

 

If you need to spend some hours reading fiction, go on, treat yourself it’s great.

 

#kingsofthewyld #fiction #fantasy

 

My Favourite Mathematics Book

It is a hard a question to pick one book  to represent my favourite book. This challenge was set by person who started to organise a group of people to read mathematics books in a book club of sorts. What follows is what I wrote for them on that subject.

I’m going to exclude standard text books as they are used to practice the ideas we are playing with.
I’m going to exclude history of mathematics books, while I read them there are parts of history where I get excited, usually by thinking about how the person being discussed got to the point where they did whatever thing it was that is documented. For instance, Ian Stewart’s very nice ‘Taming the infinite the story of mathematics’, starts with the wonderful line, ‘Mathematics did not spring into being fully formed. It grew from the cumulative efforts of many people, from many cultures, who spoke many languages.’ Knowing this means that I can’t know how or why that happened to be, I can, I suppose, guess but I’ll not be certain without evidence, and then at this remove that evidence is not viewed as it was at the time, we can’t know what went on in the Babylonians minds. Frank Sweetz wrote a book titled Capitalism and Arithmetic, the new math of the 15th Century. We also know that people approximately counted seasons and years from the Mayans to the Celts through the legacy of stones in various trigonometric designs. So I will abandon these books for their distance of the journey.
This thinking has me reflecting on three of my favorite books, in the series about people, in no particular order:
Siobhan Roberts excellent book on John Conway, Genius At Play. Levels of Infinity by Hermann Weyl a collection of his essays including one on Emmy Noether, which is reason alone enough to read the book.
Lastly in a category all on it’s own Lymm Gladwell’s excellent Mathematics and Art, which if you want to characterise it you might say it is a coffee table book, it is also a nice history of mathematics but not in the style of the general history books which tend to be very much aimed at course work. It is beautifully illustrated and just a joy to dip into now and again.
This however leaves whole subsections of mathematics alone, it causes me some grief not to mention the Princeton Companion to Mathematics, edited by Timothy Gowers, or Mathematics From The Birth of Numbers, by Jan Gullberg both of which are tomes and well worth the investment. In this short survey of my shelves I’ve left out a few brilliant things, anything by the popularising group Keith Devlin, Ian Stewart, Marcus Du Sautoy.
Now comes the reason I can can’t name a single book, there is a series that takes a bit of work at times, be that as it may, you can skip the hard parts now and again and still get significant gain from working though this little set of books. It is called The Best Writing in Mathematics and it has been an annual release since 2010 edited by Pitici they are a joy in and of themselves and cover a broad swathe of the subject.
With all that in mind I think you might see why I have problems picking a single book.

John Horton Conway on Surreal Numbers

This is a great lecture (in my opinion) that John Conway gave in Toronto in 2016 on how he came to discover Surreal Numbers. With that in mind be aware that he doesn’t do proofs early on, he states assumptions which he deals with later.

I’ve not written anything in a couple of weeks here because I started to write about infinity and I’m still unsure what I want to say about it. There is so much you could say about it.

Popular Television & Physics Comedy

There is a mathematician named Douglas R Hofstader, another one named Sheldon Katz, both rather interesting in their own way.

The former is the author of several books. Godel Esher Bach is most likely the most well known of them.

The latter Sheldon H. Katz is an American mathematician, specializing in algebraic geometry and its applications to string theory. In 1973 Katz won first prize in the U.S.A. Mathematical Olympiad (Wikipedia). This is him talking about something I understand some of the words but not in that order, it is way over my head. Sheldon Katz – Elliptically fibered Calabi-Yau threefolds: mirror symmetry and Jacobi forms

Books in a Series

James S.A. Corey Leviathan Wakes and Caliban’s War the first two books in The Expanse have left me impressed. I’ve not picked up books in a long time that were that easy to read.

The second of these from four pm Saturday on and off until about 10PM Sunday both read over the last day or so.  The words flowed from the page outlining a series of mini adventures. It was so enjoyable.