For someone rooted in graphic design and illustration, I typically hate running across visuals on the internet. Aside from being numbed by ads, the fact of the matter is that a large percentage of the graphical presentation on the web is just bandwidth-stealing window dressing with little impact on the surrounding content. Part of my plan with this blog was to avoid graphics almost entirely, and yet over the past month or so, I have littered this space with a handful of SVGs. I think, for the most part, they have added meaningful visual aids to the surrounding content, but I still don’t want to make too much of a habit of it.

I’m far more comfortable with SVGs (or, vector graphics in general) because I find it easier to have them settle onto the page naturally without becoming jarring. I could obviously restrict the palette of a raster image to the palette of my site, and render a high resolution PNG with manageable file size, but scaling will still come into play, type may be mismatched… aside from being accessibility issues, these things have subtle effects on visual flow. I’m thankful that SVG has been adopted as well as it has, and that it’s relatively simple to write or manipulate by hand. Following is the process I go through to make my graphics as seamless as possible.

Generally speaking, the first step is going to be to get my graphic into Illustrator. Inside Illustrator, I have a palette corresponding to my site’s colors. Making CSS classes for primary, secondary, tertiary colors is in my to-do list, but I need to ensure nothing will break with a class defining both color and fill. Groups and layers (mostly) carry over when Illustrator renders out an SVG, so I make a point of going through the layer tree to organize content. Appearances applied to groups cascade down in the output process, so (as far as SVG output is concerned) there’s no point in, say, applying a fill to a group – each individual item will get that fill in the end anyway. I use Gentium for all of the type, as that is ideally how it will be rendered in the end, though it’s worth quickly checking how it all looks in Times New Roman as well.

Once I get things colored and grouped as I need them, I crop the artboard to the artwork boundaries. This directly affects the SVG viewbox, and unless I need extra whitespace for the sake of visually centering a graphic, I can rely instead on padding or the like for spacing.

Once in the SVG Save dialog, I ensure that ‘Type’ is set to ‘SVG’. I don’t want anything converted to an outline, because I want the type to visually fall back with the rest of my page. I never actually save an SVG file from Illustrator, I just go to ‘SVG Code…’ from the Save dialog, and copypaste it elsewhere for further massaging. This involves:

Illustrator seemingly outputs SVG with the intent being structural accuracy if the file is read back in for editing, which is often counterproductive for web use, which would prioritize small filesize without a sacrifice in selection ordering or visual accuracy. To be fair, I just installed 2018 and haven’t tested its SVG waters yet, so we’ll see how Adobe managed to mess that up handle that.

Finally, it’s worth mentioning SVGO (and the web-based SVGOMG). Very customizable optimization, definitely more useful once one starts dealing with more intricate, complicated SVGs. I’m happy to optimize mine down by hand, and stop there – but I’m keeping them to a handful of kilobytes anyway.

Aztec diamonds: Testing the reversed Aztec numbers

My initial test of x=ceil(sqrt(2y)/2) to determine the size of a square given a value in A046092 corresponding to how many line segments would remain if the grid formed internally at every unit of height and width of the square was broken at every intersection was tested and shown to be valid for the first 1000 numbers in the sequence. I did this haphazardly in Excel, just to make sure it wasn’t going to fail quickly.

I have since scripted a test in python, which has thus far shown my method viable up to a 50907428697×50907428697 square. The script is called by passing it an argument containing the initial integer for the square’s size (default 2), and it loops infinitely until it either fails or is halted with SIGINT. Upon halting, it returns something like Last success: Square 39033633; azNum 3047249088424644; urNum 39033633.0, where Square is the height/width of the square being tested, azNum is the number of line segments (or squares in the equivalent Aztec diamond), and urNum is the calculation which (hopefully) is equal to Square. Revealing the last success in this way tells me where to start next time. The code:

import signal
import sys
from math import ceil,sqrt
def sigintHdl(signal,f):
    print "Last success: Square %s; azNum %s; urNum %s" % (tNum-1,azNum,urNum)
tNum = 2 if len(sys.argv)<2 or int(sys.argv[1])<2 else int(sys.argv[1])
while result==0:
print "FAILURE: Square %s; azNum %s; urNum %s" % (tNum-1,azNum,urNum)

There’s another way of looking at this whole thing. If we consider an isosceles right triangle with hypotenuse h, we know the length of either of the legs is equal to sqrt(h^2/2). Interestingly enough, if we work with a hypotenuse of one unit larger (which should never exist as a halved Aztec diamond), h^2/2 is equal to our A046092 value +0.5.

Adding one unit to our central line yields seven units; squaring seven and then halving the result yields 24.5. 7

Ultimately, the problem seems to be one of dealing with a not-quite-proper triangle. It’s easy to imagine additional nodes that make the triangle more… triangular. Doing so leads to more funny math, but it all sort of, kind of makes sense. I guarantee there’s an off-the-shelf solution out there, and it’s likely quite straightforward and, in hindsight, obvious. But this sort of math isn’t necessarily my forté, so I’ll just fidget around until I come up with something conclusive. At this point, it’s all for fun – I have far more known-valid values than I could ever imagine needing. My little python test snippet will easily be reused for other things as well, so I’ll call that a win.

Golfing in Eukleides

Eukleides is decidedly not a golfing language, but when a geometry-related question came up on PPCG, I had to give it a shot. Eukleides can be quite rigid; for starters it is very strongly typed. Functions are declared by what they intend to return, so while set would be the shortest way to declare a function, it can’t really be exploited (unless returning a set is, in fact, desired). Speaking of sets, one thing that is potentially golfable is ‘casting’ a point to a set. Given a point, p, attempting a setwise operation on p will fail because a point does not automatically cast to a set (strict typing). p=set(p) will overwrite point p with a single-item set containing the point that was p. If, however, it is okay to have two copies of the point in the set, p=p.p is three bytes shorter.

If user input is required, the command number("prompt") reads a number from STDIN. The string for the prompt is required, though it can be empty (""). Thus, if more than four such inputs (or empty strings for other purposes) are required, it saves bytes to assign a variable with a one-letter name to an empty string.

Whitespace is generally unavoidable, but I did come to realize that boolean operators do not need to be preceded by whitespace if preceded by a number. So, if a==7or a==6 is perfectly valid. aor a, 7ora, 7 or6 are all invalid, however. This may be an interpreter bug, but for the time being it is an exploitable byte.

Finally, loci. Loci are akin to for-loops that automagically make sets. Unfortunately, they don’t seem to care much for referencing themselves mid-loop, which meant that I couldn’t exploit how short of a construction they are compared to manually creating a set in a for-loop.

This was a fun exercise, and just goes to show that if you poke around at any language enough, you’ll find various quirks that may prove useful given some ridiculous situation or another.

Tetris has been implemented in Conway's Game of Life (external)

Over four years ago, a challenge was posed on the Programming Puzzles & Code Golf Stack Exchange: Build a working game of Tetris in Conway’s Game of Life (GoL). Yesterday, an incredibly dedicated team of seven posted a working solution that is just fantastic to read about. I am flabbergasted by the work that went into this, and what these folks achieved. Starting with the concept of a metapixel – essentially a giant block of GoL tiles that can be programmed to behave like a single GoL cell, but with any ruleset – they developed wiring, logic gates, an ALU, a RISC architecture, an assembly language, a higher-level language, and finally (for now) a working game of Tetris. Not the first example of advanced computation in GoL (Wireworld is well known), but almost certainly the largest in scope and likely the best-documented as well. I linked to the Stack Exchange page, and if you have any knowledge of GoL, logic gates, or low-level computing, the multiple write-ups are incredibly clear and fascinating.


Despite having failed out of geometry in my younger days, it has become my favorite sort of recreational math. I think, back in elhi geometry, too much emphasis was placed on potential practical applications instead of just distilling it down to the reality that this was what math was before there was any sort of formal mathematical system. Geometry is hands-on, it’s playful, and as such I have come to really enjoy playing with it. As much as I enjoy doing constructions with a straightedge and compass, I occasionally poke around to see what tools exist on the computer as well. Recently, I stumbled across a very neat thing: Eukleides.

I’m drawn to Eukleides because it is a language for geometry, and not a mouse-flinging WYSIWYG virtual compass. This seems contradictory given my gushing about geometry being hands-on, and don’t get me wrong – I love a hands-on GUI circle-canvas too1. But sometimes (often times) my brain is in code-mode and it’s easier to express what I’m trying to do in words than to fiddle around with a mouse. And a task like ‘intersecting a couple of circles’ is far more conducive to writing out than, say, laying down an SVG illustration from scratch.

a b c d e bi

There you have one of the first constructions learned by anyone studying geometry – bisecting an angle with three arcs (or, full-blown circles in this case). Angle ∠abc is bisected by segment bbi. Here’s the code:

% Percent signs are comments
a=point(7,0); b=point(0,0) % Semicolons and newlines separate commands
a b c triangle 5,-40°
d=intersection(g,a.b); e=intersection(g,b.c)
d=d[0]; e=e[0] % Intersections return sets (arrays), extract the points
h=circle(d,3); i=circle(e,3)
  a 0°; b 180°; c 40° % Label the points
  d -40° gray; e 90° gray
  bi 20°
  a,b,bi; bi,b,c 1.5 % Make the angle markers
  g lightgray; h gray
  i gray

Note that the code originally used shades of grey, I shifted these around to match my site’s colors when I converted the resulting EPS file to SVG. The code is pretty straightforward: define some points, make an angle of them, draw a circle, intersect segments ab and bc, make some more circles, intersect where the circles meet, and boom – a bisected angle. The language reminds me a bit of GraphViz/DOT – purpose-built for naturally expressing how things will be drawn.

We can actually prove that the construction works without even generating an image file. Removing the draw and label sections, and replacing them with some print (to stdout) statements calling angle measurement functions:

a=point(7,0); b=point(0,0)
a b c triangle 5,-40°
d=intersection(g,a.b); e=intersection(g,b.c)
h=circle(d,3); i=circle(e,3)
%%%%%%%% New content starts here:
print angle(a,b,c)
print angle(a,b,c)/2
print angle(a,b,bi)
print angle(bi,b,c)

We get ∠abc, ∠abc/2, ∠abbi, and ∠bcbi. The last three of these should be equal, and:


…they are! We can also do fun things like dividing the circumference of a circle by its diameter:

print (perimeter(circle(point(0,0),4))/8)

…to get 3.14159. Very cool. There are a few improvements I would like to see. Notably, while you can label points etc. with their names, I can’t find a way to add arbitrary labels, or even labels based on functions. So you can’t (seemingly) label a line segment with its length, or an angle with its measure. Also, the interpreter wants ISO 8859-1 encoding, and it uses the degree symbol2 (°) to represent angle measures. This gets all flippy-floppy when moving to UTF-8, and in fact if I forget to convert a file to ISO 8859-1, I’ll get syntax errors if I use degree symbols. Finally, it only outputs to EPS; native SVG would be incredibly welcome.

Eukleides is a lot of fun to play with, and it’s worth mentioning that it has looping and conditionals and the like, so it is programmable and not just computational or presentational. Presumably some pretty interesting opportunities are thus opened up.

Nth-order Fibonacci sequence in dc

After working on the nearest Fibonacci number problem in dc, I got to thinking about how one might implement other generalized Fibonacci sequences in the language. Nth-order sequences seemed like a fun little challenge, and since I was simply doing it for my own pleasure I went ahead and gave it prompts and user input:

% dc -e '[_nacci? ]n?1-sg[terms? ]n?2-snlgsi[0lid1-si1<Z]dsZx1[dls+ssli:flid1+silg>G]sG[li;flid1-si0<R]sRlgsz[0si0sslGxlgsilRxlslzd1+szln>M]dsMxf'
_nacci? 4
terms? 20

…gives us the first 20 terms of the tetranacci sequence. This interested me because unlike a simple Fibonacci iteration that can be handled on the stack with only rudimentary stack manipulation (dsf+lfr), higher order ‘naccis need summation of more terms. For a defined number, I could simply use registers, but dc does support arrays, so setting the order at runtime is plausible. There’s a bit going on here, so I’m going to start by rattling off all of the registers I use:

The order (so, a g-nacci sequence)
Number of terms to run through
General counter
Array used to temporarily hold the last sequence as it’s being summed
Generalized Fibonacci generating macro
Macro to retrieve previous sequence from f
Zero-seed macro
Counter for iterations (compares to n)
Main macro

Now to break down what’s going on. [_nacci? ]n?1-sg[terms? ]n?2-sn just prompts the user and inputs g and n. We reduce each of these in the process to appease the loops. After doing this, we need to seed the sequence with g-1 zeroes, and one one. lgsi sets counter i to g, and then the macro Z, does nothing but put a zero on the stack and loop: [0lid1-si1<Z]1. dsZx stores the macro and executes it; then 1 pushes the necessary one onto the stack such that we can begin.

[dls+ss]sS is our macro, S, which is a simple summer for register s. It duplicates the top of stack, recalls s, adds the two together, and then writes that back to s. The stack is returned to its original state.

Our next macro, G, has a bit more going on: [dls+ssli:flid1+silg>G]sG. It starts with a simple summer for register s, dls+ss. This duplicates the stack, recalls s, adds them and then overwrites s with the new sum. The stack returns to its original state. The next thing we need to do is move the top of the stack into our holding array, f. We’ll use our counter i as an index, so we load i and then do the array operation, li:f. Every time we do these two things, our summation (the next number in the sequence) nears its next value, and our stack shrinks. The rest of the macro, lid1+sig>G just handles incrementing i and comparing it to our order, g, determining whether or not to continue the loop.

Macro R, [li;flid1-si0<R]sR repopulates the stack from array f. Before calling R, i is set to g, and we use that as our array index to pull from, thus treating the array as a LIFO2. li;f does this, and then the rest of the macro is (surprise, surprise) counter/loop handling.

Before we run macro M, which is our main macro essentially, we set counter z to our order number g, which accounts for the fact that we already have our first few terms in assorted zeroes and a one. M, [0si0sslGxlgsilRxlslzd1+szln>M]dsMx, starts out by resetting counter i and the sum register s to zero: 0si0ss. lGx runs macro G, lgsi sets counter i to our order g, and then lRx runs macro R. ls puts our sum (the new Fibonacci value) at the top of the stack, and then the rest of the thing is counter/loop handling. dsMx saves the macro as M and also sets it running, while our last command, f prints the entire stack.

Finding the greatest Yahtzee score

I’ve been meaning to implement a way to incorporate style or script requirements into my posts using Hugo frontmatter. I’m not there yet, and before I get there, I need a test post that requires one or the other. I thought a little toy that lets one play a turn (three rolls) of Yahtzee, and then returns the highest possible score of the roll would be a fun and simple demonstration. Aside from small straights wigging me out a little (and I still have a nagging feeling this can be optimized), it was indeed simple1 to come up with an optimal score search. Fortunately, for a single-turn score, we don’t need to worry about a few scoring rules: bonus (joker) Yahtzees, the upper row bonus, nor chance. We could implement chance easily, but it really doesn’t make sense for single-turn scoring.

dc Syntax for Vim

This is an old post from an old blog; assets may be missing, links may be broken, and my opinions may differ considerably by this point…

I use dc as my primary calculator for day-to-day needs. I use other calculators as well, but I try to largely stick to dc for two reasons - I was raised on postfix (HP 41CX, to be exact) and I'm pretty much guaranteed to find dc on any *nix machine I happen to come across. Recently, however, I've been expanding my horizons, experimenting with dc as a programming environment, something safe and comfortable to use as a mental exercise. All of that is another post for another day, however - right now I want to discuss writing a dc syntax definition for vim.