# Fractangular Frolics

Here’s an interesting shape that looks like a distorted and dissected capital S:

A distorted and dissected capital S

If you look at it more closely, you can see that it’s a fractal, a shape that contains itself over and over on smaller and smaller scales. First of all, it can be divided completely into three copies of itself (each corresponding to a line of the fractangle seed, as shown below):

The shape contains three smaller versions of itself

The blue sub-fractal is slightly larger than the other two (1.154700538379251…x larger, to be more exact, or √(4/3)x to be exactly exact). And because each sub-fractal can be divided into three sub-sub-fractals, the shape contains smaller and smaller copies of itself:

Five more sub-fractals

But how do you create the shape? You start by selecting three lines from this divided equilateral triangle:

A divided equilateral triangle

These are the three lines you need to create the shape:

Fractangle seed (the three lines correspond to the three sub-fractals seen above)

Now replace each line with a half-sized set of the same three lines:

Fractangle stage #2

And do that again:

Fractangle stage #3

And again:

Fractangle stage #4

And carry on doing it as you create what I call a fractangle, i.e. a fractal derived from a triangle:

Fractangle stage #5

Fractangle stage #6

Fractangle stage #7

Fractangle stage #8

Fractangle stage #9

Fractangle stage #10

Fractangle stage #11

Here’s an animation of the process:

Creating the fractangle (animated)

And here are more fractangles created in a similar way from three lines of the divided equilateral triangle:

Fractangle #2

Fractangle #2 (anim)

(open in new window if distorted)

Fractangle #2 (seed)

Fractangle #3

Fractangle #3 (anim)

Fractangle #3 (seed)

Fractangle #4

Fractangle #4 (anim)

Fractangle #4 (seed)

You can also use a right triangle to create fractangles:

Divided right triangle for fractangles

Here are some fractangles created from three lines chosen of the divided right triangle:

Fractangle #5

Fractangle #5 (anim)

Fractangle #5 (seed)

Fractangle #6

Fractangle #6 (anim)

Fractangle #6 (seed)

Fractangle #7

Fractangle #7 (anim)

Fractangle #7 (seed)

Fractangle #8

Fractangle #8 (anim)

Fractangle #8 (seed)

# Wineshine

«Il vino è la luce del sole tenuta insieme dall’acqua.» — Galileo (1564-1642)

“Wine is sunlight held together by water.” — Galileo

Fractal leaves of Heuchera “Red Lightning

Fractal river network in Shaanxi province, China

Post-Performative Post-Scriptum

Because “Heuchera” comes from the name of the German botanist J.H. Heucher (1677–1747), it should strictly speaking be pronounced something like “HOI-keh-ruh”. But people often say “HYOO-keh-ruh” or variations thereon.

# Toxic Turntable #20

Currently listening…

• Ultravirago, Force Majeure (1979)
• Vuo Taş, Uorjao (1963)
• Plotting with Heini, Flaxen (2004)
• Eye Sway Tiger, Zoo of Deleuze (1984)
• Locked Zodiacal Nailbar, Zodiac III (2009)
• Arkham Daylight, It’s A Given (1981)
• Crastic, Cool Your Jets (2014)
• VII Blades, Oceanic Panic (1981)
• Imperil, La Japonesa (1997)
• Quicksilver Gothlings, Paolo il Lupo (1994)
• Philip Molyneaux Orchestra, File under Fog (1959)
• Ocean of Ice, My Peony (2007)
• Hiq Nsujuir, G’Ykuq Iw EP (2001)
• Quail in Morse, Peel Session 2 (1989)
• Mví Yíjó, Ajax + Ulysses (1990)
• Whipt Quiff, Under the Sward (2018)
• Joseph Bastermoe, God Be With (1984)
• Domenico Scarlatti, Harpsichord Sonatas (1986)
• Msakimoh, Pvalroh (1996)

Previously pre-posted:

Toxic Turntable #1#2#3#4#5#6#7#8#9#10#11#12#13#14#15#16#17#18#19

# Hour Re-Re-Re-Powered

Here’s a set of three lines:

Three lines

Now try replacing each line with a half-sized copy of the original three lines:

Three half-sized copies of the original three lines

What shape results if you keep on doing that — replacing each line with three half-sized new lines — over and over again? I’m not sure that any human is yet capable of visualizing it, but you can see the shape being created below:

Morphogenesis #3

Morphogenesis #4

Morphogenesis #5

Morphogenesis #6

Morphogenesis #7

Morphogenesis #8

Morphogenesis #9

Morphogenesis #10

Morphogenesis #11 — the Hourglass Fractal

Morphogenesis of the Hourglass Fractal (animated)

The shape that results is what I call the hourglass fractal. Here’s a second and similar method of creating it:

Hourglass fractal, method #2 stage #1

Hourglass fractal #2

Hourglass fractal #3

Hourglass fractal #4

Hourglass fractal #5

Hourglass fractal #6

Hourglass fractal #7

Hourglass fractal #8

Hourglass fractal #9

Hourglass fractal #10

Hourglass fractal #11

Hourglass fractal (animated)

And below are both methods in one animated gif, where you can see how method #1 produces an hourglass fractal twice as large as the hourglass fractal produced by method #2:

Two routes to the hourglass fractal (animated)

Elsewhere other-engageable:

# Maximal Moz

Morrissey in Conversation: The Essential Interviews, ed. Paul A. Woods (Plexus 2016)

It’s very Mozzean that one of the most Mozzean things in this book is marginal. That is, it’s not in the interviews or anything Moz himself says: it’s in the mini-bios of the “Contributors” section at the end of the book. For example, Dave McCullough interviewed Moz for the long-defunct Sounds in 1983. And I thought it was a joke when McCullough’s mini-bio ended with “His current whereabouts are unknown.”

But it happened again for Shaun Philips, who interviewed Moz, again for Sounds, in 1988: “His
current whereabouts are unknown.” And again for Elissa Van Poznak, who interviewed Moz for The Face in 1984: “Her current whereabouts are unknown.” And that sentence is the last in the book, apart from the acknowledgements. What happened to these three journalists? They had lives and careers, friends and family. Their writing was once regularly read by many thousands or even millions of people. And then read again in this book. But “Their current whereabouts are unknown.” They’ve dropped out of sight, even maybe out of life, and the editor of the book, Paul A. Woods, hasn’t been able to find out what happened to them. Not even in this ultra-connected internet age.

That’s very Mozzean. You could even wonder whether they’ve succumbed to a belated form of the Curse of Moz, or the career-failure that strikes bands after Morrissey praises them or takes them on tour as support. Or you could wonder whether, like Morrissey himself for so long, they were struggling with depression and an urge-to-self-annihilation even as they achieved professional success. You’d certainly expect the first publication of this book in 2007 to have flushed them out. But it didn’t. Nor did the second publication in 2011. But perhaps the third publication did in 2016.

I don’t know and I’d rather not know. I like the Mozzeanism of three missing journalists. And I liked this book too. A lot. Obviously a lot of other people did too, or it wouldn’t have been printed three times. But I suspect it won’t be re-printed again. Why not? Coz of Moz on Muz. Guardian-readers were not pleased by Morrissey’s comments on Muslims and Muslim immigration after the Manchester bombing in 2017 or by his support for Brexit and the “far-right” For Britain party. You can get T-shirts now that say “Shut Up, Morrissey!” and there have been a string of anathemas and excommunications issued at Moz from woke bastions like the Quietus (where bad English goes to die). Guardian-readers feel deeply betrayed by Morrissey, who once said all the right things about economics, animal rights, vegetarianism, and the evilness of the Conservative and Republican parties – as you can read here.

But you’ll also read here about disturbing early signs – or sounds – that Moz wasn’t prepared to buzz with the hive-mind on everything. After he began his solo career in 1988 he released songs with titles like “The National Front Disco” and “Bengali in Platforms”, the latter of which opined “Life is hard enough when you belong here.” But there was enough ambiguity and authorial distance in the songs for him to deny plausibly that he was being racist or sympathizing with racism. And he still had a whole heap of good-will from the Smiths, so he survived the first campaign to cancel him and came back as strong as ever.

Well, the good-will has disappeared now. Moz has burned all his bridges to the Guardian and I don’t think there’s any chance of this book being re-re-re-printed. Indeed, I bet a lot of former fans have thrown out their copies or even ritually burned them. It’s their loss, because Morrissey is one of the wittiest, most interesting, and most intelligent interviewees who ever lived. As the back cover says of an earlier edition of Morrissey: In Conversation:

It’s proof, lest we forget, that in terms of great copy, Morrissey has rarely been anything other than interview gold. – Q magazine

But that quote itself needs trimming of its Guardianist fat: “It’s proof, lest we forget, that Morrissey has rarely been anything other than interview gold.” Moz himself is rarely guilty of saying more than he needs to. He’s both articulate and acute. It’s hard to believe that he came from a big working-class Irish family in Manchester and spent years on the dole after being shunted into a bad school by failing his eleven-plus. If he’d passed that selective exam he would have gone to a better school and most probably on to university. But I think university would have been bad for him. He probably wouldn’t have had a career in music and he certainly wouldn’t have become the Morrissey that millions of people either love or loathe.

But he would have become someone who habitually said “in terms of” and “prior to”. Alas, he does sometimes say “in terms of” in later interviews here, but it’s a minor blemish and I read everything in the book. Except – speak of “in terms of” and the windbag appears – Will Self’s “The King of Bedsit Angst Grows Up” from 1995. As usual with Self, I began losing the will to live half-a-paragraph in and gave up. If it had been a proper interview rather than Self blotivating on themes Mozzean, I might have persevered. But it wasn’t, so I didn’t.

Most of the other pieces were proper interviews, but either way I always persevered. You can read how Moz’s ideas and allegiances changed. And you can also see how Moz himself changed, because there are some good photos too. I bet some of the interviewers now regret their association with Morrissey and their appearance in this book, but that adds to its appeal for me. Moz has bitten the hands that typed about him and they’ll never forgive him for it. But they were warned:

Only inwardly. (“The Importance of Being Morrissey”, Jennifer Nine for Melody Maker, August 1997)

And here’s more from the man himself:

What else could you do [besides perform]?

Nothing. I’m entirely talentless… it was all a great big accident – I just came out of the wrong lift. (“Mr Smith: All Mouth and Trousers”, Dylan Jones for i-D magazine, October 1987)

Well, they wear heavy overcoats and stare at broken lightbulbs. That’s the way it’s always been for me! (“Wilde Child”, Paul Morley for Blitz, April 1988)

“I often pass a mirror,” he confides, loving the attention he’s getting, “and I glance into it slightly, and I don’t really recognize myself at all. You can look into a mirror and wonder – where have I seen that person before? And then you remember. It was at a neighbour’s funeral, and it was the corpse.” (“Wilde Child”)

What was it like playing live again when you appeared in Wolverhampton in December [1988]?

It was nice. I did enjoy it. It was nice to be fondled.

Was it good to be back on stage again?

No, it was just nice to be fondled. (“Playboy of the Western World”, Eleanor Levy, Q magazine, January 1989)

My perfect audience are skinheads in nail varnish. And I’m not trying to be funny, that really is the perfect audience for me. But I am incapable of racism, and the people who say I am racist are basically just the people who can’t stand the sight of my physical frame. I don’t think we should flatter them with our attention. (“Morrissey Comes Out (For a Drink)”, Stuart Maconie for New Musical Express, May 1991)

I would rather eat my own testicles than reform the Smiths – and that’s saying something for a vegetarian. (“The Last Temptation of Morrissey”, Paul Morley for Uncut, May 2006)

My best friend is myself. I look after myself very, very well. I can rely on myself never to let myself down. I’m the last person I want to see at night and the first in the morning. I am endlessly fascinating – at eight o’clock at night, at midnight, I’m fascinated. It’s a lifelong relationship and divorce will never come into it. That’s why, as I say, I feel privileged. And that is an honest reply. (“The man with the thorn in his side”, Lynn Barber for The Observer, September 2002)

Favourite shop?

Rymans, the stationers. To me it’s like a sweetshop. I go in there for hours, smelling the envelopes. As I grew up I used to love stationery and pens and booklets and binders. I can get incredibly erotic about blotting paper. So for me, going into Rymans is the most extreme sexual experience one could ever have. (“Morrissey Answers Twenty Questions”, Smash Hits Collection, 1985)

# Slanted and Enchanted

(click for larger)

Post-Performative Post-Scriptum

I assume I got the name “Slanted and Enchanted” from a subconscious memory of an album of the same name by the American band Pavement, though I might just have come up with it independently.

# Kim Pickings

As a keyly committed core component of the anti-racist community, I’ve always been a passionate admirer of Kimberlé Crenshaw, the Black legal genius who conceived the corely committed key concept of intersectionality, the pro-feminist, anti-racist ideo-matrix whereby multiply impactive factors of oppression around race, gender and class are recognized to overlap in terms of toxic impact on corely vulnerable communities of color, gender, and class…

So, imagine my excitement when I saw that the Guardian was engaging core issues around Ms Crenshaw in a keynote article itself passionately penned by a Journalist of Color:

Kimberlé Crenshaw: the woman who revolutionised feminism – and landed at the heart of the culture wars, by Aamna Mohdin

From police brutality to sexual harassment, the lawyer fights to ensure black women’s experiences are not ignored. So why are her ideas being denounced? — The Guardian, 12xi20

“Why indeed?” I interrogated to myself as I began to read. But imagine my horror when I came across this passage in terms of the core article:

Crenshaw’s early academic work, meanwhile, was also an important building block in the development of critical race theory, which revolutionised the understanding of race in the US’s legal system and is taught in law schools across the country. — Kimberlé Crenshaw

What is it coming to when the Guardian uses everyday English to engage issues around the keyly vital work of a Black legal genius? Huh? The Guardian should of course have put it like this:

Crenshaw’s early academic work, meanwhile, was also a core building block in terms of the development of critical race theory, which revolutionised the understanding of race in the US’s legal system and is taught in law schools across the country.

And “core foundational keystone in terms of the gestational development…” would have been even better

Elsewhere other-engageable:

Ex-term-in-nate! — incendiarily interrogating issues around “in terms of” dot dot dot

# Tri Again (Again (Again))

Like the moon, mathematics is a harsh mistress. In mathematics, as on the moon, the slightest misstep can lead to disaster — as I’ve discovered again and again. My latest discovery came when I was looking at a shape called the L-tromino, created from three squares set in an L-shape. It’s a rep-tile, because it can be tiled with four smaller copies of itself, like this:

Rep-4 L-tromino

And if it can be tiled with four copies of itself, it can also be tiled with sixteen copies of itself, like this:

Rep-16 L-tromino

My misstep came when I was trying to do to a rep-16 L-tromino what I’d already done to a rep-4 L-tromino. And what had I already done? I’d created a beautiful shape called the hourglass fractal by dividing-and-discarding sub-copies of a rep-4 L-tromino. That is, I divided the L-tromino into four sub-copies, discarded one of the sub-copies, then repeated the process with the sub-sub-copies of the sub-copies, then the sub-sub-sub-copies of the sub-sub-copies, and so on:

Creating an hourglass fractal #1

Creating an hourglass fractal #2

Creating an hourglass fractal #3

Creating an hourglass fractal #4

Creating an hourglass fractal #5

Creating an hourglass fractal #6

Creating an hourglass fractal #7

Creating an hourglass fractal #8

Creating an hourglass fractal #9

Creating an hourglass fractal #10

Creating an hourglass fractal (animated)

The hourglass fractal

Next I wanted to create an hourglass fractal from a rep-16 L-tromino, so I reasoned like this:

• If one sub-copy of four is discarded from a rep-4 L-tromino to create the hourglass fractal, that means you need 3/4 of the rep-4 L-tromino. Therefore you’ll need 3/4 * 16 = 12/16 of a rep-16 L-tromino to create an hourglass fractal.

So I set up the rep-16 L-tromino with twelve sub-copies in the right pattern and began dividing-and-discarding:

A failed attempt at an hourglass fractal #1

A failed attempt at an hourglass fractal #2

A failed attempt at an hourglass fractal #3

A failed attempt at an hourglass fractal #4

A failed attempt at an hourglass fractal #5

A failed attempt at an hourglass fractal (animated)

Whoops! What I’d failed to take into account is that the rep-16 L-tromino is actually the second stage of the rep-4 triomino, i.e. that 4 * 4 = 16. It follows, therefore, that 3/4 of the rep-4 L-tromino will actually be 9/16 = 3/4 * 3/4 of the rep-16 L-tromino. So I tried again, setting up a rep-16 L-tromino with nine sub-copies, then dividing-and-discarding:

A third attempt at an hourglass fractal #1

A third attempt at an hourglass fractal #2

A third attempt at an hourglass fractal #3

A third attempt at an hourglass fractal #4

A third attempt at an hourglass fractal #5

A third attempt at an hourglass fractal #6

A third attempt at an hourglass fractal (animated)

Previously (and passionately) pre-posted:

# Count Amounts

One of my favourite integer sequences is what I call the digit-line. You create it by taking this very familiar integer sequence:

• 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20…

And turning it into this one:

• 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0… (A033307 in the Online Encyclopedia of Integer Sequences)

You simply chop all numbers into single digits. What could be simpler? Well, creating the digit-line couldn’t be simpler, but it is in fact a very complex object. There are hidden depths in its patterns, as even a brief look will uncover. For example, you can try counting the digits as they appear one-by-one in the line and seeing whether the digit-counts compare. Do the 1s of the digit-line always outnumber the 0s, as you might expect? Yes, they do (unless you start the digit-line 0, 1, 2, 3…). But do the 2s always outnumber the 0s? No: at position 2, there’s a 2, and at position 11 there’s a 0. So that’s one 2 and one 0. Does it happen again? Yes, it happens again at the 222nd digit of the digit-line, as below:

1, 2count=1, 3, 4, 5, 6, 7, 8, 9, 1, 0count=1, 1, 1, 1, 22, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 23, 02, 24, 1, 25, 26, 27, 3, 28, 4, 29, 5, 210, 6, 211, 7, 212, 8, 213, 9, 3, 03, 3, 1, 3, 214, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 04, 4, 1, 4, 215, 4, 3, 4, 4, 4, 5, 4, 6, 4, 7, 4, 8, 4, 9, 5, 05, 5, 1, 5, 216, 5, 3, 5,4, 5, 5, 5, 6, 5, 7, 5, 8, 5, 9, 6, 06, 6, 1, 6, 217, 6, 3, 6, 4, 6, 5, 6, 6, 6, 7, 6, 8, 6, 9, 7, 07, 7, 1, 7, 218, 7, 3, 7, 4, 7, 5, 7, 6, 7, 7, 7, 8, 7, 9, 8, 08, 8, 1, 8, 219, 8, 3, 8, 4, 8, 5, 8, 6, 8, 7, 8, 8, 8, 9, 9, 09, 9, 1, 9, 220, 9, 3, 9, 4, 9, 5, 9, 6, 9, 7, 9, 8, 9, 9, 1, 010, 011, 1, 012, 1, 1, 013, 221, 1, 014, 3, 1, 015, 4, 1, 016, 5, 1, 017, 6, 1, 018, 7, 1, 019, 8, 1, 020, 9, 1, 1, 021

So count(2) = count(0) = 1 at digit 11 of the digit-line in the 0 of what was originally 10. And count(2) = count(0) = 21 @ digit 222 in the 0 of what was originally 110. Is a pattern starting to emerge? Yes, it is. Here are the first few points at which the count(2) = count(0) in the digit-line of base 10:

1 @ 11 in 10
21 @ 222 in 110
321 @ 3333 in 1110
4321 @ 44444 in 11110
54321 @ 555555 in 111110
654321 @ 6666666 in 1111110
7654321 @ 77777777 in 11111110
87654321 @ 888888888 in 111111110
987654321 @ 9999999999 in 1111111110
10987654321 @ 111111111110 in 11111111110
120987654321 @ 1222222222221 in 111111111110
[...]

The count(2) = count(0) = 321 at position 3333 in the digit-line, and 4321 at position 44444, and 54321 at position 555555, and so on. I don’t understand why these patterns occur, but you can predict the count-and-position of 2s and 0s easily until position 9999999999, after which things become more complicated. Related patterns for 2 and 0 occur in all other bases except binary (which doesn’t have a 2 digit). Here’s base 6:

1 @ 11 in 10 (1 @ 7 in 6)
21 @ 222 in 110 (13 @ 86 in 42)
321 @ 3333 in 1110 (121 @ 777 in 258)
4321 @ 44444 in 11110 (985 @ 6220 in 1554)
54321 @ 555555 in 111110 (7465 @ 46655 in 9330)
1054321 @ 11111110 in 1111110 (54121 @ 335922 in 55986)
12054321 @ 122222221 in 11111110 (380713 @ 2351461 in 335922)
132054321 @ 1333333332 in 111111110 (2620201 @ 16124312 in 2015538)
1432054321 @ 14444444443 in 1111111110 (17736745 @ 108839115 in 12093234)
15432054321 @ 155555555554 in 11111111110 (118513705 @ 725594110 in 72559410)
205432054321 @ 2111111111105 in 111111111110 (783641641 @ 4788921137 in 435356466)
2205432054321 @ 22222222222220 in 1111111111110 (5137206313 @ 31345665636 in 2612138802)

And what about comparing other pairs of digits? In fact, the count of all digits except 0 matches infinitely often. To write the numbers 1..9 takes one of each digit (except 0). To write the numbers 1 to 99 takes twenty of each digit (except 0). Here’s the proof:

11, 21, 31, 41, 51, 61, 71, 81, 91, 12, 01, 13, 14, 15, 22, 16, 32, 17, 42, 18, 52, 19, 62, 110, 72, 111, 82, 112, 92, 23, 02, 24, 113, 25, 26, 27, 33, 28, 43, 29, 53, 210, 63, 211, 73, 212, 83, 213, 93, 34, 03, 35, 114, 36, 214, 37, 38, 39, 44, 310, 54, 311, 64, 312, 74, 313, 84, 314, 94, 45, 04, 46, 115, 47, 215, 48, 315, 49, 410, 411, 55, 412, 65, 413, 75, 414, 85, 415, 95, 56, 05, 57, 116, 58, 216, 59, 316, 510, 416, 511, 512, 513, 66, 514, 76, 515, 86, 516, 96, 67, 06, 68, 117, 69, 217, 610, 317, 6
11
, 417, 612, 517, 613, 614, 615, 77, 616, 87, 617, 97, 78, 07, 79, 118, 710, 218, 711, 318, 712, 418, 713, 518, 714, 618, 715, 716, 717, 88, 718, 98, 89, 08, 810, 119, 811, 219, 812, 319, 813, 419, 814, 519, 815, 619, 816, 719, 817, 818, 819, 99, 910, 09, 911, 120, 912, 220, 913, 320, 914, 420, 915, 520, 916, 620, 917, 720, 918, 820, 919, 920

And what about writing 1..999, 1..9999, and so on? If you think about it, for every pair of non-zero digits, d1 and d2, all numbers containing one digit can be matched with a number containing the other. 100 → 200, 111 → 222, 314 → 324, 561189571 → 562289572, and so on. So in counting 1..999, 1..9999, 1..99999, you use the same number of non-zero digits. And once again a pattern emerges:

count(0) = 0; count(1) = 1; count(2) = 1; count(3) = 1; count(4) = 1; count(5) = 1; count(6) = 1; count(7) = 1; count(8) = 1; count(9) = 1 (writing 1..9)
count(0) = 9; count(1) = 20; count(2) = 20; count(3) = 20; count(4) = 20; count(5) = 20; count(6) = 20; count(7) = 20; count(8) = 20; count(9) = 20 (writing 1..99)
0: 189; 1: 300; 2: 300; 3: 300; 4: 300; 5: 300; 6: 300; 7: 300; 8: 300; 9: 300 (writing 1..999)
0: 2889; 1: 4000; 2: 4000; 3: 4000; 4: 4000; 5: 4000; 6: 4000; 7: 4000; 8: 4000; 9: 4000 (writing 1..9999)
0: 38889; 1: 50000; 2: 50000; 3: 50000; 4: 50000; 5: 50000; 6: 50000; 7: 50000; 8: 50000; 9: 50000 (writing 1..99999)
0: 488889; 1: 600000; 2: 600000; 3: 600000; 4: 600000; 5: 600000; 6: 600000; 7: 600000; 8: 600000; 9: 600000 (writing 1..999999)
0: 5888889; 1: 7000000; 2: 7000000; 3: 7000000; 4: 7000000; 5: 7000000; 6: 7000000; 7: 7000000; 8: 7000000; 9: 7000000 (writing 1..9999999)
[...]

And here’s base 6 again:

0: 0; 1: 1; 2: 1; 3: 1; 4: 1; 5: 1 (writing 1..5)
0: 5; 1: 20; 2: 20; 3: 20; 4: 20; 5: 20 (writing 1..55 in base 6)
0: 145; 1: 300; 2: 300; 3: 300; 4: 300; 5: 300 (writing 1..555)
0: 2445; 1: 4000; 2: 4000; 3: 4000; 4: 4000; 5: 4000 (writing 1..5555)
0: 34445; 1: 50000; 2: 50000; 3: 50000; 4: 50000; 5: 50000 (writing 1..55555)
0: 444445; 1: 1000000; 2: 1000000; 3: 1000000; 4: 1000000; 5: 1000000 (writing 1..555555)
0: 5444445; 1: 11000000; 2: 11000000; 3: 11000000; 4: 11000000; 5: 11000000 (writing 1..5555555)
0: 104444445; 1: 120000000; 2: 120000000; 3: 120000000; 4: 120000000; 5: 120000000 (writing 1..55555555)
0: 1144444445; 1: 1300000000; 2: 1300000000; 3: 1300000000; 4: 1300000000; 5: 1300000000 (writing 1..555555555)