The Hunt for Dark Breakfast(moultano.wordpress.com) |
The Hunt for Dark Breakfast(moultano.wordpress.com) |
There's also no mention of the fourth leg of the breakfast triad: maple syrup.
The Breakfast Simplex is a space of recipes parameterized by {egg, milk, flour} ratios, normalized onto a simplex. Add butter or sugar and the dimension increases. Add prep method and you create adjacent regions. A breakfast buffet is a larger, possibly disconnected simplicial complex spanning multiple ingredient families.
That structure is exactly what Tom Ngo formalized and patented in Embedded Constraint Graphics in 1996 at Interval Research Corporation. I wrote about it when the patent expired in 2016:
https://news.ycombinator.com/item?id=12572696
US Patent #5933150: System for image manipulation and animation using embedded constraint graphics
https://patents.google.com/patent/US5933150
When I asked Tom about applying ECG to other applications after the patent expired, he wrote:
>I am, of course, partial to the idea that gluing high-dimensional simplices at their edges and faces is an extremely general way to represent blending manifolds, in the same way that gluing polygons together has done us so much good in the 3D modeling space. I also think the >2 decades of progress since ECG have put us in a better position to do something really cool based on direct manipulation.
Golan Levin, Malcolm Slaney, and Tom Ngo used the ECG graphical editor to build the vector face cartoons for Mouther, simply by dragging eyes, mouths, and features directly on the drawing:
https://web.archive.org/web/20180717222910/http://www.flong....
ECG defines example states at vertices. Compatible examples span simplices. The full state space is a simplicial complex. Interior points are barycentric blends.
https://en.wikipedia.org/wiki/Simplicial_complex
https://en.wikipedia.org/wiki/Barycentric_coordinate_system
When you drag something in screen space, the system maps that motion into the n-dimensional interpolation space and solves for blend weights via the Moore–Penrose pseudoinverse of the Jacobian matrix, the same linear algebra used in inverse kinematics.
https://en.wikipedia.org/wiki/Moore%E2%80%93Penrose_inverse
https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinan...
https://en.wikipedia.org/wiki/Inverse_kinematics
You don’t indirectly adjust abstract sliders. You directly manipulate concrete outcomes. The solver recovers coordinates.
The same formulation applies to interpolating vector drawings, mesh blending, facial animation, pose spaces, and other example-based interfaces where states are meaningful and compatibility matters.
The same geometric intuition appears in large language models. Tokens and concepts are represented as high-dimensional vectors, and model activations are computed through weighted linear combinations in embedding space. Interpolating between embeddings corresponds to moving through that vector space via weighted blends, just in many more dimensions. ECG makes the simplices explicit and topologically structured, while LLM representations are implicit and learned. In both cases, behavior emerges from interpolation in high-dimensional spaces.
Apple’s ARKit already exposes facial expression as a set of named blend shape coefficients via ARFaceAnchor — values like mouthSmileLeft, jawOpen, and eyeBlinkRight driving a 3D face mesh in real time. Bring Mouther into 3D and you can drag the mouth corners upward to interpolate toward smiling targets, mapping that motion through the same barycentric machinery into blend weights instead of hard-coded sliders. This would make a great Blender plug-in for directly manipulating facial animation, to use with FaceIt!
Faceit : Facial Expressions And Performance Capture
https://superhivemarket.com/products/faceit
Breakfast is a concrete instance. Pancake, crepe, and omelette define a simplex over ingredient ratios. Drag toward eggs and the egg weight increases. Drag toward milk and you move along that axis. Cross the egg-milk edge shared by the crepe simplex {flour, egg, milk} and the custard simplex {egg, milk, sugar}, and you move from thin batters into sweet custards without leaving the manifold. The Dark Breakfast region is simply an unoccupied part of a valid simplex -- suggesting adjacent, unexplored Dark Custard subspaces rather than forbidden states.
Simplicial complexes are useful UI primitives. They provide local linear interpolation inside zones and explicit global topology across zones. They scale to higher dimensions, while maintaining a user friendly 2D direct manipulation user interface. They encode constraints structurally instead of procedurally.
A pie menu can be viewed as a radial parameterization of a simplex. A direct-manipulation pie menu over ingredient space lets you drag in the direction of the crusts and fillings you want, with barycentric weights accumulating as you move.
The Design and Implementation of Pie Menus (Dr. Dobb’s Journal, Dec. 1991, cover story, user interface issue.)
https://donhopkins.medium.com/the-design-and-implementation-...
An Empirical Comparison of Pie vs. Linear Menus (Jack Callahan, Don Hopkins, Mark Weiser and Ben Shneiderman. Presented at ACM CHI’88 Conference, Washington DC, 1988.)
https://donhopkins.medium.com/an-empirical-comparison-of-pie...
Breakfast has way more dimensions.
Porridge is an extremely common breakfast here in Sweden. There is a huge number of different variations of it as well.
And then there is müsli, with and without yogurt / sour milk.
> Confusion continues because: People who live in the Hollands are called Hollanders, but all citizens of the Netherlands are called Dutch as is their language.
> But in Dutch they say: "Nederlands sprekende Nederlanders in Nederland" which sounds like they'd rather we call them Netherlanders speaking Netherlandish.
> Meanwhile, next door in Germany, they're "Deutsche sprechen Deutsch in Deutschland". Which sounds like they'd rather be called Dutch.
Very eggy, with some flour/milk. It's essentially a souffle, puffs up to like 6" high in the oven. Tasty with maple syrup, powdered sugar/lemon, or just butter.
6 Eggs, 1 C flour, 1C milk.
Tasty, nutrient-dense, surprisingly filling. Great as a mid-afternoon snack, or add some fruit and a bit of bread or some oatcakes to make it into a decent lunch.
Not to say we don't also have other sweet snacks, but compared to other places I've been nowhere quite has the same level of "savory" snacks
In terms of beverages alcohol-free beer gained popularity in recent years because it's not sweet, has half the calories of juice/soda yet actually carries some taste.
If I were to imagine something fitting this description, it would have to be something you have to chew for extended periods of time.
How about doughnut with just the right amount of custard filling? Or perhaps a brioche with the right amount of French vanilla ice cream/gelato? A classic egg and cheese sandwich in your desired proportions?
WHERE ARE THE BISCUITS???
It’s a baked custard (so plenty of eggs) in a pie.
not sure the proportions match.
* https://en.wikipedia.org/wiki/Egg_sandwich#Fried_egg_sandwic...
I was with you up to this point, but my suspension of disbelief has its limits.
I also get up early and it is often actually dark.
- croissants - muesli/porridge/oatmeal - cookies - toasts - bread & butter (nutella too)
What it's missing is fish, fruit, preserved meats (bacon, ham, sausage, black pudding?). It's very culturally biased to typical North American breakfast choices.
Wait what? I've never heard of such a thing.
Does that make them better in any way? Or strictly worse, but cheaper?
Edit: looked it up and apparently they still use 3 eggs but the batter makes it super fluffy (like 2x) so the omelette looks enormous.
My thinking at the moment is that I probably would not. It seems like further research would reveal a whole new region in the upper left, clustering with Dan Bing, of "asian milkless crepes."
You can get them e.g. in London UK: https://www.hopperslondon.com/
I wonder if this is why I've missed them? I've lived within a few hundred metres of their Soho place for the best part of the last decade.
I've been meaning to go to Sri Lanka..
Hoppers are not only Sri Lankan - also found in parts of South India and I think in some places in SE Asia.
They are delicious, similar to noodles but I eat them cold with a dry chutney and a great breakfast.
https://www.eater.com/2015/1/26/7860903/amanda-cohen-royal-c...
It wasn't just an omelette on top of a waffle (and both of them the size of a medium pizza). As you strayed from the edges toward the center it became difficult to see where the waffle ended and the omelette began.
Such a shame they went out of business.
This cracked me up, because I had a fantastic dream the other night where I had a tour through a donut factory. But the best thing I had (in the dream) was something I'd never tried before, never seen, and which I intend to make at the earliest opportunity. It was slightly salty french fries, buttered and coated in sugar and cinnamon, like cinnamon toast. Bang on. Makes a lot of sense too, if you think about it. Definitely would fit in the "dark breakfast" polygon.
[edit] the potato and bacon theory also comes from what ends up deliciously mixed on your plate at the end, which along with syrup and ketchup is also an integral part of any egg/flour/milk breakfast.
It is difficult to put milk into food. Why not just drink it? Alternatively, can we drink eggs and flour?
Cheese is another variation for milk. What about grilled cheese and eggs? Or some variation on Mac and Cheese?
You can also consider other dimensions like vegetables and spices. According to this plane, shakshuka is pure egg. Add spices to milk and you have chai. Add eggs to chai and you have cursed eggnog.
https://www.thedeliciouscrescent.com/omelette-stuffed-parath...
My crepe recipe - cook on medium heat pan:
Blend on low: 4 eggs- 3/4 cup whole milk, 1/2 stick of melted butter, and 1/4cup to 1/2 cup plain flower, 1 heaped tbsp of baking powder, 1/2 tsp salt, vanilla optional and to taste
I guess the only difficulty there is we English don't eat those for breakfast, and really only make them on one day of the year. Which I missed this year!
Dammit, we're going to be having a belated pancake day here soon...
But that's because my wife requests it.
It would never occur to me to up the egg ratio so high to reach into that void though. My wife manages to mess up the proportions every time though, so maybe we'll unwittingly explore that region one day.
A normal person will need caloric replenishment during the day, especially later in the day before (typically) an 8 to 10 hour fast during the night.
So, small proteine-heavy breakfast, large balanced dinner in the evening with a smaller balanced mid-day meal in the middle.
Light breakfasts are popular if you have to make it yourself
Also, amazing article, haven't laughed that hard all day!
- 1/4 c. milk
- 1/2 c. flour
- 4 eggs
- 1/3 c. sugar
- some salt
- cinnamon
- cloves
- nutmeg
- poppyseeds
Did the first two cakes without baking powder, turned into something between a crepe and a tortilla. Did the last two cakes with baking powder and they were just a very squishy pancake.
Breakfast burritos are also at least as important as quiche (as in, neither are as tasty without addins - just like omelettes).
And don’t get me started on breakfast burritos, top 10 food that’s just ridiculous if you’re ordering it after 3pm?
béchamel (roux + milk) + egg yolks + cheese
Also, hollandaise is pretty integral to eggs benedict, I've had lots of variations but the traditional with poached egg, canadian bacon, english muffin, and hollandaise is really by far the best.
One important property of pudding is that unlike pancakes, the space of pudding isn't "chaotic and fractal" because whether you add just enough flour to make it somewhat sticky or so much that you can cut it with a knife, it's still pudding. This means that if we take flour-heavy pudding and somehow add shitload of eggs, we should be able to venture into the Dark Breakfast area. I have a pudding recipe that calls for egg yolks, but I feel like this isn't a good lead. Still it proves further how flexible "pudding" is.
Going back to pancakes, there's one funny variation. "Naleśniki" with cottage cheese and cream (or yoghurt). Cottage cheese and cream are basically milk with extra steps, which means that the entire dish becomes mainly milk. This means that the Pancake Local Group actually stretches much closer to the "milk" vertex than your diagram suggests.
If I were to prepare something that passes as food and sits in the Dark Breakfast Abyss, I'd try scrambled eggs on the thinnest bread I could reasonably make. Something like "scrambled eggs taco".
And I feel like that’s really a missed opportunity? Like even recognizing that of all meals breakfast is the most “comforting” and the one most likely for customers to want familiar, at the same time there are so many unexplored variants using the same ingredients (as this article shows!) that almost no restaurants will ever experiment with or offer. Lunch and dinner menus have massive variation in comparison!
Fancy projection math is only for after coffee!
Common for visualizing mixtures of three things!
For this visualization: get positive quantities in 3D space, normalize to 1, now you have dot on a triangle on 1-sphere in a positive octant. Project triangle into 2D space a this is your visualization.
Also, while I know that omelette is technically the whipping of large amounts of air into what is otherwise scrambled eggs, it feels wrong to me that "omelette" is categorized as "pure egg singularity". Is an omelette worth the time and effort over scrambled eggs if it does not include bits of vegetables, meat, and/or cheese folded inside like a taco?
My Egg McMuffin will never look the same!
- There's nothing Canadian about a pancake house. We love pancakes but they aren't really ingrained with our identity. Maple syrup on the other hand, is EXTREMELY important to a lot of Canadians. Serving table syrup instead of real maple syrup is an affront. I found a Reddit thread[1] where a user espouses "tons of free syrup" you were given at RCPH. That's NOT a good thing if you ask me!
- In Canada (and I assume other British Commonwealth countries) you aren't legally allowed to have "Royal" in the name of your business without Royal consent from the Governor General of Canada[2]
Just a bit of Canadiana sparked by your comment I thought I'd share. I always get a kick of the small but conspicuous cultural differences between Canada and USA. They give me that Ingluorious Basterds "number 3" moment.
[1] https://www.reddit.com/r/newyorkcity/comments/1ajujhi/who_re...
[2] https://www.canada.ca/en/canadian-heritage/services/royal-sy...
[1] https://www.theregister.com/2015/08/23/post_pub_nosh_ugandan...
[*] of course, here I mean proper omelettes, which are an egg shell around ingredients, not scrambled eggs with ingredients mixed in.
Your ingredient mixing distinction doesn't reflect what I've encountered. That seems to have more to do with the nature of a given ingredient or alternatively with presentation or other concerns specific to a given recipe.
Yeah if you wanted "cooked separately and also circular" you'd need to make a two omelette sandwich. I've yet to encounter that at a restaurant.
The closest existing food I know to this are churros, which can be truly excellent when made well. In places like Barcelona, they dip them in chocolate sauce.
I support your experiments in potato-based churro analogs!
[edit] just also why this post touched my heart - I think form is as important as ingredients whenever you're dealing with relatively few ingredients. I have a breakfast I particularly love making that's just hash browns, egg and cheese. But the trick is, you griddle the hash browns, then flip them and smash them on griddled cheese, then crack an egg on top while the cheese fries and flip the whole thing again. The result is a crispy potato pancake where one side is fried cheese and the other is embedded fried egg. The same 3 ingredients, but it can be held in hand and it's got the perfect balance in each bite.
yes, a much better vector space. Thank you, Noduerme, you are one of the faithful.
https://www.okonomikitchen.com/daigaku-imo-japanese-candied-...
I'm sure there's a good joke in there involving trans-dimensional vegans, but I'm still on my first cup of coffee.
When you divide by x by sum(x), the resulting vector p has p >= 0 and sum(p) = 1. That new sum constraint is how you get the simplex.
https://en.wikipedia.org/wiki/Barycentric_coordinate_system
Barycentric coordinates are the local coordinate system inside a simplex. A simplicial complex is what you get when you glue multiple simplices together along shared k-faces for k = 0 … n -- vertices (0-faces), edges (1-faces), triangles (2-faces), tetrahedra (3-faces), and higher-dimensional faces -- to form a larger state space.
https://en.wikipedia.org/wiki/Simplicial_complex
It's not possible to have negative eggs, but you can apply the same machinery to many other things, like facial animation mesh blend shapes (Apple ARKit, Blender Blend Shapes and the FaceIt plugin, Unity SkinnedMeshRenderer, etc), where weights are often allowed to be overdriven >1 or even underdriven <0 for exaggerated or monstrous effects.
Nouveau Art Pipeline Demo: Blend Shapes:
https://youtu.be/phM8Wnzs_-g?t=104
(None of Epstein's spiritually close friends and shameless guru confidants were harmed or embarrassed in this demo, alas.)
Eric Hedman - "Doppel" Character Modelling with Blendshapes for Animation (ARKit):
https://www.youtube.com/watch?v=6L7jtgRD5rs
(Eric "Irk" Hedman designed and created the character animations and objects in The Sims 1, and as you can see is extremely skilled and delightful to work with! Hire him if you need professional high quality creative artwork and animation, and can pay him in bananas: https://erichedman.artstation.com/projects/8wJDgw )
Faceit: Facial Expressions And Performance Capture (Blender):
https://superhivemarket.com/products/faceit
Unity SkinnedMeshRenderer:
https://docs.unity3d.com/6000.3/Documentation/ScriptReferenc...
Apple ARKit Tracking and visualizing faces:
https://developer.apple.com/documentation/ARKit/tracking-and...
ARFaceAnchor.BlendShapeLocation: Identifiers for specific facial features, for use with coefficients describing the relative movements of those features.
https://developer.apple.com/documentation/arkit/arfaceanchor...
FWIW, for Thailand (where I have been), I'd have a similar recommendation.
Thailand has some great beaches, but as someone from great-beaches-land, I would recommend the mountains and ancient cities.
That describes all of the restaurants in the Kings Cross newly built area. They are all different cuisines, but they all fit.
It's a little simpler than the given recipe too. Just a regular paratha, crack an egg directly onto it, then cook on a griddle.
I'm also overseas these days, in Australia with my English partner, and have rarely thought of the English pancake in some time. Even when I'm making breakfast pancakes they're usually American style and I'm supposed to be watching my weight and cholesterol these days so that's become a lot less frequent too.
https://www.youtube.com/watch?v=hB42iztkzVQ
i've done the french toast pizza and it wasn't bad. not sure if it was worth the effort. maybe there's an ideal type of pizza or combo of toppings that makes this spectacular. either way it's worth trying once just to say you did.
Egg nog is listed on the triangle as away from flour, but it is extremely high in carbs. When I was a kid, I loved egg nog and a couple of years ago I decided to purchase some. I liked it so much I drank the whole half gallon in a day. That night I had horrible painful bloating and looked at the ingredients label to find "sugar", "cane sugar". "corn syrup", and "high fructose corn syrup".