Will Apple get on the bus?
January 11, 2026 at 9:51 PM by Dr. Drang
Do you remember the Acid Tests? Not Ken Kesey’s Acid Tests; even I’m not old enough to remember those; I only read about them in Tom Wolfe’s book. No, I’m talking about the tests that checked browsers on how well they complied with web standards. Back in the mid-aughts, Apple talked a lot about how well Safari and WebKit did on these tests, in contrast to Internet Explorer. I was reminded of this while listening to the most recent episode of Connected.
In the show, Myke Hurley’s risky pick was that by the end of 2026, Apple will not have shipped Apple Intelligence features equivalent to those shown at 2024’s WWDC. You can use Overcast’s convenient web interface to listen to his pick. The part of his pick that reminded me of the Acid Tests was this, where he explains why he thinks they’ll be delayed furthur:
I think they will not commit to App Intents as the way that [actions across apps] works, and that instead they will move to MCP… They need to go into an open standard that everybody else could potentially use, that there is more incentivization to use, and use that as a way to make this work.
You may think that shifting from a home-grown system to an open standard is not Apple’s way, and it certainly hasn’t been over the past 15 years or so—quite the opposite, in fact. But in the early days of OS X, when Apple was struggling to re-establish the Mac, one of the groups it wanted to appeal to was the growing number of web developers who wanted to work in a standard Unix environment, but one with a friendly face. Remember when System Preferences had a graphical UI for controlling your local Apache webserver? Remember when /usr/bin was filled with interpreters before you installed Xcode? Those were part of the same mindset that touted Safari’s Acid Test score.
This doesn’t mean that Myke is right and that Apple will embrace a standard it didn’t help create. It just means that Apple used to see the value in using standards to catch up when it’s behind. The question is whether it will remember that after so many years of success.
A Zeckendorf table in Python
January 10, 2026 at 5:15 PM by Dr. Drang
After writing this morning’s post, I went down to Channahon for a walk along the I&M Canal towpath. On the drive there and back, I thought about redoing the project in Python instead of Mathematica. It seemed like a fairly easy problem, and it was.
Recall that the goal was to reproduce this table from a recent Numberphile video:
Instead of using someone else’s code for creating the Zeckendorf representation of an integer (examples of which are relatively easy to find), I decided to write my own using the greedy algorithm outlined by Tony Padilla in the video. Here’s the script:
python:
1: #!/usr/bin/env python3
2:
3: from collections import defaultdict
4:
5: def zeck(n):
6: 'Return the Zeckendorf list of Fibonacci numbers for n.'
7:
8: # Greedily subtract Fibonacci numbers from n until zero.
9: z = []
10: left = n
11: for f in reversed(fibs):
12: if (m := left - f) >= 0:
13: left = m
14: z.append(f)
15: return z
16:
17: # Fibonacci numbers less than 100, excluding the initial 1.
18: fibs = [1, 1]
19: while (n := fibs[-2] + fibs[-1]) < 100:
20: fibs.append(n)
21: del fibs[0]
22:
23: # Build the table.
24: ztable = defaultdict(list)
25: for i in range(1, 101):
26: for z in zeck(i):
27: ztable[z].append(i)
28:
29: # Print it.
30: for f in fibs:
31: print(', '.join(str(n) for n in ztable[f]))
32: print()
The greedy algorithm is in the zeck function on Lines 5–15. It assumes the existence of a list of Fibonacci numbers saved in the global variable fibs. It goes through fibs in reverse order. If the current Fibonacci number can be subtracted from the number without going below zero, it is, and it’s also appended to the Zeckendorf representation list. This process is repeated, subtracting—if possible—each Fibonacci number in turn from the remaining difference until we get to the end of the reversed fibs list.
Unlike the ZeckendorfRepresentation function in the Wolfram Language, zeck doesn’t return a list of ones and zeros whose positions are associated with Fibonacci numbers; it returns the Fibonacci numbers themselves. So zeck(50) returns [34, 13, 3].
You probably see some things in zeck that could be made more efficient. Me too. But in a small problem like this, I didn’t think those efficiencies were worth the effort.
Also note that zeck is not a general-purpose function. I wrote it specifically for this script, and it’s built on certain assumptions. The assumptions have to do with how it’s called and how the global fibs list is constructed:
- The argument passed to
zeckis a number small enough that its Zeckendorf representation consists entirely of numbers from thefibslist. - The
fibslist is in increasing order. - The
fibslist does not contain the initial one of the Fibonacci sequence.
Lines 17–21 create the fibs list to meet these conditions. After the deletion on Line 21, it’s
[1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
My favorite part of this bit of code is the walrus operator (:=) in Line 19. Apparently, there are people who don’t like the walrus operator. Don’t listen to them.
Lines 23–27 then build the table. In this case, I used a defaultdict called ztable whose keys are the Fibonacci numbers and whose values are the lists of numbers that have that key in their Zeckendorf representation.
Finally, Lines 29–32 print out the values of ztable. Here they are, formatted to fit better in this space:
1, 4, 6, 9, 12, 14, 17, 19, 22, 25, 27, 30, 33, 35,
38, 40, 43, 46, 48, 51, 53, 56, 59, 61, 64, 67, 69,
72, 74, 77, 80, 82, 85, 88, 90, 93, 95, 98
2, 7, 10, 15, 20, 23, 28, 31, 36, 41, 44, 49, 54,
57, 62, 65, 70, 75, 78, 83, 86, 91, 96, 99
3, 4, 11, 12, 16, 17, 24, 25, 32, 33, 37, 38, 45,
46, 50, 51, 58, 59, 66, 67, 71, 72, 79, 80, 87, 88,
92, 93, 100
5, 6, 7, 18, 19, 20, 26, 27, 28, 39, 40, 41, 52, 53,
54, 60, 61, 62, 73, 74, 75, 81, 82, 83, 94, 95, 96
8, 9, 10, 11, 12, 29, 30, 31, 32, 33, 42, 43, 44,
45, 46, 63, 64, 65, 66, 67, 84, 85, 86, 87, 88, 97,
98, 99, 100
13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51,
52, 53, 54, 68, 69, 70, 71, 72, 73, 74, 75
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54
55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67,
68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88
89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
This matches both the video screenshot above and the list given in this morning’s post.
A Zeckendorf table
January 10, 2026 at 9:29 AM by Dr. Drang
As a general rule, I don’t get the fascination mathematicians have with Fibonacci numbers, but I did enjoy this recent Numberphile video. It’s about the Zeckendorf representation of numbers, which I’d never heard of, and it inspired me to build a short Mathematica notebook that reproduces one of the things in the video.
Zeckendorf’s theorem says that any positive integer can be uniquely represented as the sum of non-consecutive Fibonacci numbers. For example,
which are the ninth, seventh, and fourth Fibonacci numbers. In the video, Tony Padilla uses this fact in a crappy magic trick to divine the number Brady guesses. The trick involves this table of numbers:
The trick goes like this: Brady chooses a number from 1 to 100. He then checks the rows in this table that contain his number. Tony then scans the table—not as quickly as he should to make it seem magical—and tells Brady the number he chose.
The trick is in the construction of the table. Each row of the table starts with a Fibonacci number and contains every number (from 1 through 100) that includes that Fibonacci number in its Zeckendorf representation. To determine the number guessed, the “magician” adds up the first number in each checked row.
I thought it would be fun to build the table. I opened Mathematica and started exploring its Fibonacci-related functions. It didn’t take long to see that the Wolfram Function Repository (a sort of external library for the Wolfram Language) has a ZeckendorfRepresentation function. When given a number, it returns a list of ones and zeros corresponding to the Zeckendorf representation of that number. For example,
ResourceFunction["ZeckendorfRepresentation"][50]
returns
{1, 0, 1, 0, 0, 1, 0, 0}
The Fibonacci numbers included in the Zeckendorf representation correspond to the ones, and those that are skipped correspond to the zeros. The most significant Fibonacci number (i.e., the largest) comes first in the list. You may have noticed that although 34 is the ninth Fibonacci number, the list above is only eight digits long. Recall that the Fibonacci sequence starts with two ones:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, …
The first one is redundant in a Zeckendorf representation, so it’s ignored.
To make the table, I started with the list of Fibonacci numbers less than 100, not including the initial one. The Fibonacci function returns the nth Fibonacci number, so I built the list this way:
fibs = Select[Map[Fibonacci, Range[2, 15]], # < 100 &]
The call to Map created a list of the 2nd through 15th Fibonacci numbers. I couldn’t remember how many Fibonacci numbers are less than 100, but I figured the 15th had to be above 100. I then used Select and a pure function to filter the higher numbers out of the list.1 This returned
{1, 2, 3, 5, 8, 13, 21, 34, 55, 89}
Next I created a list of the Zeckendorf representations of every integer from 1 through 100. Because the Zeckendorf representations are themselves lists of varying length, I decided it would be easier to work with reversed Zeckendorf representations, where the least significant digit comes first.
rzeck = Map[Reverse,
Map[ResourceFunction["ZeckendorfRepresentation"],
Range[1, 100]
]
];
The rzeck list is kind of long, but here are the first ten and last two entries:
{{1},
{0, 1},
{0, 0, 1},
{1, 0, 1},
{0, 0, 0, 1},
{1, 0, 0, 1},
{0, 1, 0, 1},
{0, 0, 0, 0, 1},
{1, 0, 0, 0, 1},
{0, 1, 0, 0, 1},
[etc]
{0, 1, 0, 0, 1, 0, 0, 0, 0, 1},
{0, 0, 1, 0, 1, 0, 0, 0, 0, 1}}
The reversed Zeckendorf representation of 50 is
{0, 0, 1, 0, 0, 1, 0, 1}
so to get the Fibonacci numbers in that representation, we do this:
fibs[[Flatten[Position[rzeck[[50]], 1]]]]
The Position function returns a nested list of where the ones are,
{{3}, {6}, {8}}
which is why it had to be run through Flatten.
This idea is how I built a list of the Fibonacci terms from the Zeckendorf representations. I created a function like the code above and mapped it to every integer from 1 through 100.
fibTerms[n_] := fibs[[Flatten[Position[rzeck[[n]], 1]]]]
f = Map[fibTerms, Range[1, 100]]
So now I have a list of 100 lists. Each sublist contains the Fibonacci numbers of the Zeckendorf representation for the corresponding index number of the outer list. Like this:
1: {1} 51: {1, 3, 13, 34}
2: {2} 52: {5, 13, 34}
3: {3} 53: {1, 5, 13, 34}
4: {1, 3} 54: {2, 5, 13, 34}
5: {5} 55: {55}
6: {1, 5} 56: {1, 55}
7: {2, 5} 57: {2, 55}
8: {8} 58: {3, 55}
9: {1, 8} 59: {1, 3, 55}
10: {2, 8} 60: {5, 55}
11: {3, 8} 61: {1, 5, 55}
12: {1, 3, 8} 62: {2, 5, 55}
13: {13} 63: {8, 55}
14: {1, 13} 64: {1, 8, 55}
15: {2, 13} 65: {2, 8, 55}
16: {3, 13} 66: {3, 8, 55}
17: {1, 3, 13} 67: {1, 3, 8, 55}
18: {5, 13} 68: {13, 55}
19: {1, 5, 13} 69: {1, 13, 55}
20: {2, 5, 13} 70: {2, 13, 55}
21: {21} 71: {3, 13, 55}
22: {1, 21} 72: {1, 3, 13, 55}
23: {2, 21} 73: {5, 13, 55}
24: {3, 21} 74: {1, 5, 13, 55}
25: {1, 3, 21} 75: {2, 5, 13, 55}
26: {5, 21} 76: {21, 55}
27: {1, 5, 21} 77: {1, 21, 55}
28: {2, 5, 21} 78: {2, 21, 55}
29: {8, 21} 79: {3, 21, 55}
30: {1, 8, 21} 80: {1, 3, 21, 55}
31: {2, 8, 21} 81: {5, 21, 55}
32: {3, 8, 21} 82: {1, 5, 21, 55}
33: {1, 3, 8, 21} 83: {2, 5, 21, 55}
34: {34} 84: {8, 21, 55}
35: {1, 34} 85: {1, 8, 21, 55}
36: {2, 34} 86: {2, 8, 21, 55}
37: {3, 34} 87: {3, 8, 21, 55}
38: {1, 3, 34} 88: {1, 3, 8, 21, 55}
39: {5, 34} 89: {89}
40: {1, 5, 34} 90: {1, 89}
41: {2, 5, 34} 91: {2, 89}
42: {8, 34} 92: {3, 89}
43: {1, 8, 34} 93: {1, 3, 89}
44: {2, 8, 34} 94: {5, 89}
45: {3, 8, 34} 95: {1, 5, 89}
46: {1, 3, 8, 34} 96: {2, 5, 89}
47: {13, 34} 97: {8, 89}
48: {1, 13, 34} 98: {1, 8, 89}
49: {2, 13, 34} 99: {2, 8, 89}
50: {3, 13, 34} 100: {3, 8, 89}
To get Tony’s table, I have to do a sort of inversion of the list f. This consists of going through every Fibonacci number in fibs and selecting the indices of f in which that Fibonacci number appears. Here’s the code:
Table[Select[f, MemberQ[i] -> Index], {i, fibs}]
MemberQ is a Boolean function, returning True if the item is in the list. I’m using the operator form of it. The Wolfram Language has lots of test functions that end with Q, which I think of as meaning “Question” or “Query.” It’s a convention taken from Lisp, where predicate functions tend to end with the letter p.
Here’s the resulting table, formatted to make it easier to read:
1, 4, 6, 9, 12, 14, 17, 19, 22, 25, 27, 30, 33, 35,
38, 40, 43, 46, 48, 51, 53, 56, 59, 61, 64, 67, 69,
72, 74, 77, 80, 82, 85, 88, 90, 93, 95, 98
2, 7, 10, 15, 20, 23, 28, 31, 36, 41, 44, 49, 54,
57, 62, 65, 70, 75, 78, 83, 86, 91, 96, 99
3, 4, 11, 12, 16, 17, 24, 25, 32, 33, 37, 38, 45,
46, 50, 51, 58, 59, 66, 67, 71, 72, 79, 80, 87, 88,
92, 93, 100
5, 6, 7, 18, 19, 20, 26, 27, 28, 39, 40, 41, 52, 53,
54, 60, 61, 62, 73, 74, 75, 81, 82, 83, 94, 95, 96
8, 9, 10, 11, 12, 29, 30, 31, 32, 33, 42, 43, 44,
45, 46, 63, 64, 65, 66, 67, 84, 85, 86, 87, 88, 97,
98, 99, 100
13, 14, 15, 16, 17, 18, 19, 20, 47, 48, 49, 50, 51,
52, 53, 54, 68, 69, 70, 71, 72, 73, 74, 75
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33,
76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88
34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54
55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67,
68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83, 84, 85, 86, 87, 88
89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100
If you compare it to the image of the table above, you’ll see that they match.
Here’s the complete notebook:
-
For the record, 89 is the 11th Fibonacci number. ↩
Choosing a driving route in CarPlay
January 6, 2026 at 9:05 PM by Dr. Drang
If you pay any attention at all to the Apple-centric web, you saw an explosion of links yesterday to this article by Nikita Prokopov, which rightly eviscerates the proliferation of menu icons in macOS 26. I may write up my thoughts on menu icons in a future post (short version: they don’t belong), but today I have something else on my mind. Like Prokopov’s post, it addresses the question, “Why the fuck has Apple forgotten the UI principles it knew forty years ago?”
If you’re using CarPlay and ask for directions, you’ll see a screen that looks like this:
That the default route’s Go button is gray while the alternates are green is a stupidity addressed by Sage Olson and Joe Rosensteel1, so I won’t bother.
What I will address is that whichever route you choose, you have to tap its Go button. Even though the full description of each route looks like a button, the only part that’s tappable is the part that looks like a button inside another button.
Is this just as stupid as having a dull color as the default and a bright color as the alternate? Yes. And Apple has known that descriptions should be click/tap targets since the very beginning of the Mac. Here, courtesy of Infinite Mac, is MacWrite 1.0 running on a simulation of an original Macintosh.
I’ve brought up the Find window. It’s currently set to do a Whole Word search, but I can switch to a Partial Word search by clicking on the Partial Word radio button or anywhere on that button’s label. Even with a label that doesn’t look like a button, Apple knew it would be helpful to select the radio button if I clicked on its label. After all, what else could I possibly mean if I clicked on the word Partial?
Apple understood this in 1984. But now, in 2026, if I’m driving and ask Siri for directions to Starved Rock State Park, CarPlay doesn’t understand what I mean if I tap in the middle of either of those large gray buttons-that-aren’t-buttons.
Why, you may ask, would I even consider tapping outside the Go button? Because Apple has trained me for forty years to expect that I can tap on its label.
-
Even Joe couldn’t find the Mastodon post where he complains about it, but we both remember that he did. ↩