Wow! Great suggestion. Whether to count coconut milk as milk is a decision I had not yet had to make.
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."
These look great, squarely in my breakfast wheelhouse, definitely eggy, but with - well, I guess what I now know as a hint of darkness. Thanks for sharing.
Going to share my advice to people going to Sri Lanka (other than what food to try and where) is to not focus on beaches (which are mostly pretty average) and go to the mountains and ancient cities which are unique and amazing.
Hoppers are not only Sri Lankan - also found in parts of South India and I think in some places in SE Asia.
I've tried their Kings X restaurant, and while their food is delicious, and the decor is also really nice, its not particularly authentic, and I felt its a bit overpriced (tbf its an extremely central location). I don't live in London so can't suggest a better alternative, but I'm sure there are much more authentic restaurants.
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.
It sounds like you're kind of a little up and to the right of the Ugandan Rolex[1]. This sounds like some hideous method of gangland killing, but is in fact a tomato omelette rolled up in a big fresh chapati. Nothing to stop you using any other variety of omelette and indeed I've had excellent results with cheese ones.
In Malaysia, a common breakfast is roti telur + teh tarik which is close to the dark breakfast region. It's like paratha, with an egg, and milk tea.
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.
Roti Telur is basically Egg Paratha. Indian migrants brought Paratha to Malaysia and Singapore, and it underwent some "localization" to suit the local palate, including being drenched in palm oil as it is cooked on the flat griddle. Sure fire way to clog your arteries, if eaten on a daily basis.
Haha. I'd suggest that what's missing in the um "latent space" here, is that the triangle should be a pentagon involving some form of bacon/sausage, and some form of potato.
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 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.
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!
Yeah meat is another dimension, as is potato. So we're up to 4 dimensional breakfast latent space. I hate to think what's in the dark breakfast black hole of that 4 dimensional latent space...
I feel like there's a lot of unexplored area in the carb-soaked-in-egg category that French Toast fits into. The major analogues being chiliqiles and matzoh brie. I recently did something like french toast bites where I cubed some sourdough bread, soaked it with egg and fried it up with small pieces of bacon mixed in. But what if you did that with a glazed donut? Or a waffle?
[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.
If you add baking powder and butter, that dark breakfast recipe is very close to crepes.
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
Interesting, we could add in an 'arm' from the pancake local group, heading out from American pancakes, via traditional English pancakes (approx 1 cup of whole milk, 2 eggs, 3/4 cup of flour) to Crepes.
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...
As an Englishman that has been transplanted to another country, I find myself making them more in Germany than I ever made them back home.
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.
Someone else may have said this but strictly speaking breakfast is something like a cone in a vector space, unless you want to explain to me how to eat negative eggs.
There’s a guy in overalls in a Brooklyn bistro who used to scramble them but that was very late-summer 2025 and you’ll get funny looks for asking about them.
I think I will attempt to just eat the negative eggs because at least I recognize, and can define what both "negative" and "eggs" mean. Can't say the same for literally half the words in the OP's graph.
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.
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.
(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):
ARFaceAnchor.BlendShapeLocation:
Identifiers for specific facial features, for use with coefficients describing the relative movements of those features.
Going against the spirit of TFA here, but I believe you can eat anything at any time of day, and my favorite breakfast tends to be regular food. Chili, soup, pasta, baked potatoes. Something warm, filling, and usually involving salt. I do sometimes have something more traditional, like oatmeal, but it's not as satisfying. I also sometimes have oatmeal as my final meal of the day if I'm hungry but already a bit tired, as it's more calm than something like spicy chili.
I really wonder why breakfast is the way it is, because since it's the first meal of the day, you'd think a big breakfast (e.g. dinner) would make sense as you need the calories during the (work)day. But maybe breakfast is relatively light because a lot of people have some digestive issues or sensitive stomachs in the mornings.
Circadian rhythm. In many people their liver releases a bunch of glucose right before waking, something diabetics term the "dawn effect" and need to control for. This helps get the metabolism going first thing in the morning (at least, in non-diabetics) and delays the need for caloric intake until later in the day. This effect is easily witnessed if you check your blood glucose before bed and when you rise in the morning: for example earlier in the week mine was around 5.4 mmol/L (normal) at bed time and 8.6 mmol/L (high for fasting blood glucose) before breakfast the next 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.
"IHOP Transgression" cracked me up. There's a restaurant in my town that does the same thing and I hate them for that reason. And like IHOP, what they call an omelette is actually a frittata.
The recipe at the end sounds a lot like the crepes I'd make in college. It was pre-WWW and I had no idea what I was doing but it seemed to work. The one thing I had going for me in college was a Costco membership. 25lb bags of flour, gallons of milk, and flats of eggs.. all cheap. I'd barter with roommates for crepe toppings (sour cream and jelly usually).
Eastern European pan-fried cottage cheese fritters (mix and fry 150g cottage cheese, 5 tbsp flour, 1 egg, 3 tbsp sugar, salt) are great. That's all I have to say.
The concept of a 'Dark Breakfast Abyss' in the Breakfast Simplex is hilarious, but it got me thinking—maybe the reason we lack foods in that specific ratio of milk, flour, and eggs isn't because they'd destroy the world, but because they simply don't cook well conceptually. Like, an overly-battered omelette just turns into a gummy mess, not a crepe. It's fascinating how our culinary traditions naturally naturally sort themselves into these distinct mathematical vertices over centuries of trial and error.
This whole thing is simply missing all milk based diary products like cheese, yogurt, white cheese, etc. When that is included then there is no gap or any mysterious quadrant.
While you are technically correct, since any triangle is a simplex, this is not relevant to this visualization.
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.
I sympathize with the author, I've had similar thoughts about snacks. We need more non-sweet snacks. Ideally something that tastes good, is not too salty, is healthy and satisfies your cravings.
Nuts (in the culinary sense) and cheese are good for that - a mini-cheeseboard with a couple of different bits of cheese (maybe 20-30g of each), a handful of cashews and walnuts, and maybe a dab of fig jam or membrillo on the side.
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.
South African food actually has this in abundance, biltong is like beef jerky but better in almost every conceivable way - not to mention all the other variants of it for example dry stix, droevors, etc.
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.
Is the "Dutch Baby" in the pancake group some alternative name for "flensjes" that I'm not familiar with? It's a thin dessert variation of Dutch pancakes that has relatively high egg and milk ratios compared to flour.
Dutch Baby or German Pancake is probably right in that abyss.
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.
Things made of flour? I think they were explicitly covered. Perhaps "Flour" should have been more generally "Grain-based" to cover oatmeal and rye bread. Maybe add potatoes for a "starches" category.
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.
French toast isn't plotted because the recipe doesn't customarily start with flour, but if you do plot it it ends up in the lower middle. If you have exceptionally eggy challah, then you might be able to push it into the abyss, but really exceptionally eggy, like 1:1 egg to flour by weight.
While a delightfully funny article it also touches on something I’ve thought about many times before: just how uniform basically 99% of all (American) restaurant breakfast menus are. It’s really quite extraordinary in some ways how these menus are nearly entirely interchangeable from on to another. They might have very small tweaks and of course the ingredient specifics aren’t exact (specific butter brands, slight proportion differences)but by and large you could predict with almost perfect accuracy every single item that a typical breakfast menu in the US might have.
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!
This is the kind of creative thinking that makes HN great. Using the framework of dark matter detection to explore unobserved breakfast possibilities is both hilarious and oddly rigorous. The breakfast phase space is clearly under-explored.
As a certified expert breakfast cook, I bristle at the idea that scrambled eggs includes any ingredient other than eggs or seasoning.
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?
Applying Tom Ngo's Embedded Constraint Graphics to Direct-Manipulation Breakfast Selection (Direct manipulation over simplicial complexes using barycentric interpolation: they're not just for breakfast any more.)
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:
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:
ECG defines example states at vertices. Compatible examples span simplices. The full state space is a simplicial complex. Interior points are barycentric blends.
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.
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
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.)
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.)
That's because such breakfasts are forbidden by the Geneva Convention articles, section 23, subsection 8, paragraph 4. It's not one of the more well known provisions, but it does exist, and nobody wants to break international law.
Please allow me to introuce you to the sri lankan egg hopper (https://www.lavenderandlovage.com/2016/05/sri-lankan-egg-hop...) which lives squarely in the aforementioned dark abyss.
Wow! Great suggestion. Whether to count coconut milk as milk is a decision I had not yet had to make.
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."
What dimension would you add to make a tetrahedron?
These look great, squarely in my breakfast wheelhouse, definitely eggy, but with - well, I guess what I now know as a hint of darkness. Thanks for sharing.
I've been meaning to go to Sri Lanka..
Going to share my advice to people going to Sri Lanka (other than what food to try and where) is to not focus on beaches (which are mostly pretty average) and go to the mountains and ancient cities which are unique and amazing.
Hoppers are not only Sri Lankan - also found in parts of South India and I think in some places in SE Asia.
FYI, You may not have to travel to Sri Lanka to try one.
You can get them e.g. in London UK: https://www.hopperslondon.com/
I've tried their Kings X restaurant, and while their food is delicious, and the decor is also really nice, its not particularly authentic, and I felt its a bit overpriced (tbf its an extremely central location). I don't live in London so can't suggest a better alternative, but I'm sure there are much more authentic restaurants.
[delayed]
sadly, they don't open for breakfast.
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.
And is a decidedly superior breakfast.
This article doesn't do it justice, but the Womelette at the short-lived Royal Canadian Pancake House in NYC lived in the dark abyss.
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.
Blending an omelette and a waffle should be totally doable; I've made waffle frittatas before and they turned out great.
It sounds like you're kind of a little up and to the right of the Ugandan Rolex[1]. This sounds like some hideous method of gangland killing, but is in fact a tomato omelette rolled up in a big fresh chapati. Nothing to stop you using any other variety of omelette and indeed I've had excellent results with cheese ones.
[1] https://www.theregister.com/2015/08/23/post_pub_nosh_ugandan...
This is fantastic, I'm dying to eat a Womelette.
In Malaysia, a common breakfast is roti telur + teh tarik which is close to the dark breakfast region. It's like paratha, with an egg, and milk tea.
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.
There is prior art in the grilled cheese and eggs area: Denny’s Moons Over My Hammy.
Or the even more legendary Breakfast Dagwood https://violasmithcooking.blogspot.com/2009/08/breakfast-dag...
For a continental vibe see the french "Croque Madame"
Egg paratha is a common Indian dish. The dough can be made with flour and water, but is traditionally made with flour and milk.
https://www.thedeliciouscrescent.com/omelette-stuffed-parath...
Roti Telur is basically Egg Paratha. Indian migrants brought Paratha to Malaysia and Singapore, and it underwent some "localization" to suit the local palate, including being drenched in palm oil as it is cooked on the flat griddle. Sure fire way to clog your arteries, if eaten on a daily basis.
We can certainly drink eggs, Rocky had a few in the morning before he went to punch hanging frozen meat.
It has been speculated that over half of the breakfast in the universe is dark breakfast.
There's a lot more dark breakfast in the winter time.
Can be confirmed by the fact anytime I eat a diner breakfast I feel more full than can be justified by the weight of the food alone.
By mass, sure - but have you considered delectability?
Haha. I'd suggest that what's missing in the um "latent space" here, is that the triangle should be a pentagon involving some form of bacon/sausage, and some form of potato.
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 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.
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!
Yeah meat is another dimension, as is potato. So we're up to 4 dimensional breakfast latent space. I hate to think what's in the dark breakfast black hole of that 4 dimensional latent space...
I feel like there's a lot of unexplored area in the carb-soaked-in-egg category that French Toast fits into. The major analogues being chiliqiles and matzoh brie. I recently did something like french toast bites where I cubed some sourdough bread, soaked it with egg and fried it up with small pieces of bacon mixed in. But what if you did that with a glazed donut? Or a waffle?
[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.
I'd add oil as another dimension - sunflower oil, olive oil, butter, lard, whatever.
If you add baking powder and butter, that dark breakfast recipe is very close to crepes.
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
Looks like very close to the dark breakfast recipe
Interesting, we could add in an 'arm' from the pancake local group, heading out from American pancakes, via traditional English pancakes (approx 1 cup of whole milk, 2 eggs, 3/4 cup of flour) to Crepes.
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...
As an Englishman that has been transplanted to another country, I find myself making them more in Germany than I ever made them back home.
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.
Someone else may have said this but strictly speaking breakfast is something like a cone in a vector space, unless you want to explain to me how to eat negative eggs.
> unless you want to explain to me how to eat negative eggs.
I'm sure there's a good joke in there involving trans-dimensional vegans, but I'm still on my first cup of coffee.
Are you telling me this dimensional vegan is trans.
Chicken maybe? It would represent an egg that did exist, but now doesn't.
Now I want some chicken & waffles
https://en.wikipedia.org/wiki/Chicken_and_waffles
Also, in the other direction in space time, it's an egg that could have been, but now won't.
Schrödinger’s egg?
More like Minkowski's egg in this case.
In the breakfast space, I'm afraid, chicken is orthogonal to eggs.
There’s a guy in overalls in a Brooklyn bistro who used to scramble them but that was very late-summer 2025 and you’ll get funny looks for asking about them.
Easy. Start eating two eggs a day. Then one day, eat three eggs taking the extra egg from tomorrows two spare eggs left in the carton.
Maybe the axes are actually logarithmic so 0 is 1 egg, 1 is e eggs, -1 is 1/e eggs, etc.
I think I will attempt to just eat the negative eggs because at least I recognize, and can define what both "negative" and "eggs" mean. Can't say the same for literally half the words in the OP's graph.
What is menstruation but negative egg entering the body?
These are a Barycentric Coordinate System, so the weights on the targets are normalized, must sum to one, and must be positive.
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...
Surprised not to see “breakfast pasta” aka Carbonara. Wheat in the noodles, egg and cheese sauce.
Going against the spirit of TFA here, but I believe you can eat anything at any time of day, and my favorite breakfast tends to be regular food. Chili, soup, pasta, baked potatoes. Something warm, filling, and usually involving salt. I do sometimes have something more traditional, like oatmeal, but it's not as satisfying. I also sometimes have oatmeal as my final meal of the day if I'm hungry but already a bit tired, as it's more calm than something like spicy chili.
I really wonder why breakfast is the way it is, because since it's the first meal of the day, you'd think a big breakfast (e.g. dinner) would make sense as you need the calories during the (work)day. But maybe breakfast is relatively light because a lot of people have some digestive issues or sensitive stomachs in the mornings.
Circadian rhythm. In many people their liver releases a bunch of glucose right before waking, something diabetics term the "dawn effect" and need to control for. This helps get the metabolism going first thing in the morning (at least, in non-diabetics) and delays the need for caloric intake until later in the day. This effect is easily witnessed if you check your blood glucose before bed and when you rise in the morning: for example earlier in the week mine was around 5.4 mmol/L (normal) at bed time and 8.6 mmol/L (high for fasting blood glucose) before breakfast the next 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.
It depends on whether you're making the breakfast, or someone else is making your breakfast.
Light breakfasts are popular if you have to make it yourself
Eating a lot of food is tiring and you have plenty of energy stored from the previous day(s) food.
That is not what we are talking about.
Also, amazing article, haven't laughed that hard all day!
"IHOP Transgression" cracked me up. There's a restaurant in my town that does the same thing and I hate them for that reason. And like IHOP, what they call an omelette is actually a frittata.
A soufflé would fit some of the void. It has some flour in the bechamel.. whipped egg whites etc.
The recipe at the end sounds a lot like the crepes I'd make in college. It was pre-WWW and I had no idea what I was doing but it seemed to work. The one thing I had going for me in college was a Costco membership. 25lb bags of flour, gallons of milk, and flats of eggs.. all cheap. I'd barter with roommates for crepe toppings (sour cream and jelly usually).
It would be remiss not to also mention The Cube Rule
https://cuberule.com/
Eastern European pan-fried cottage cheese fritters (mix and fry 150g cottage cheese, 5 tbsp flour, 1 egg, 3 tbsp sugar, salt) are great. That's all I have to say.
Salzburger Nockerln seems to fit in that area. https://www.austria.info/en-gb/recipes/salzburger-nockerl/
I suggest that the forbidden breakfast is tantamount to an eggs benedict, but with the hollandaise sauce replaced by a roux.
If it's forbidden, it is rather "eggs interdict".
The eggs had better be deviled, too.
https://en.wikipedia.org/wiki/Deviled_egg
How about a Sauce Mornay?
béchamel (roux + milk) + egg yolks + cheese
So eggs with white gravy? I think I’ve had that combo and it was pretty banging.
Even easier - egg sandwich using a basic milk bread.
Add crawfish and you're really talkin
The concept of a 'Dark Breakfast Abyss' in the Breakfast Simplex is hilarious, but it got me thinking—maybe the reason we lack foods in that specific ratio of milk, flour, and eggs isn't because they'd destroy the world, but because they simply don't cook well conceptually. Like, an overly-battered omelette just turns into a gummy mess, not a crepe. It's fascinating how our culinary traditions naturally naturally sort themselves into these distinct mathematical vertices over centuries of trial and error.
Love how at at the milk apex, there is cafe latte. Of course, it couldn't just be milk, perish the thought!
I am humbled by the genius of this article.
I feel like excluding French toast is a serious faux pas here!
Breakfast burritos are also at least as important as quiche (as in, neither are as tasty without addins - just like omelettes).
This whole thing is simply missing all milk based diary products like cheese, yogurt, white cheese, etc. When that is included then there is no gap or any mysterious quadrant.
What sort of projection is this that turns a 3-dimensional space into a triangle!
Fancy projection math is only for after coffee!
https://en.wikipedia.org/wiki/Simplex
Common for visualizing mixtures of three things!
While you are technically correct, since any triangle is a simplex, this is not relevant to this visualization.
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.
2-sphere, I believe you mean to say.
It's easy, if you normalize it so that it sums to one, it drops a dimension, and becomes an equilateral triangle.
I sympathize with the author, I've had similar thoughts about snacks. We need more non-sweet snacks. Ideally something that tastes good, is not too salty, is healthy and satisfies your cravings.
Nuts (in the culinary sense) and cheese are good for that - a mini-cheeseboard with a couple of different bits of cheese (maybe 20-30g of each), a handful of cashews and walnuts, and maybe a dab of fig jam or membrillo on the side.
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.
South African food actually has this in abundance, biltong is like beef jerky but better in almost every conceivable way - not to mention all the other variants of it for example dry stix, droevors, etc.
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
That's a hard one.
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.
This is amazing - and somehow channels both Douglas Adams and Randall Munroe at the same time...
Frankly I'm surprised Randall Munroe hasn't done this. (I assume he hasn't because nobody has linked to it.)
So basically omelette with matzo
> IHOP omelettes include pancake batter.
Wait what? I've never heard of such a thing.
Does that make them better in any way? Or strictly worse, but cheaper?
It’s a landmine for low carb gluten free people
Is the "Dutch Baby" in the pancake group some alternative name for "flensjes" that I'm not familiar with? It's a thin dessert variation of Dutch pancakes that has relatively high egg and milk ratios compared to flour.
Dutch Baby or German Pancake is probably right in that abyss.
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.
https://www.allrecipes.com/recipe/36900/german-pancakes-ii/
No egg banjo*? That sounds like it would sit close to, if not in, the abyss.
* https://en.wikipedia.org/wiki/Egg_sandwich#Fried_egg_sandwic...
Posts like this are why I read Hacker News.
My Egg McMuffin will never look the same!
It misses a lot of other breakfast options:
- croissants - muesli/porridge/oatmeal - cookies - toasts - bread & butter (nutella too)
Things made of flour? I think they were explicitly covered. Perhaps "Flour" should have been more generally "Grain-based" to cover oatmeal and rye bread. Maybe add potatoes for a "starches" category.
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.
pizza
Congratulations. You've reinvented the souffle.
Might choux hit that dark breakfast abyss? They aren't breakfast per se, but it might show that you can do things with those proportions.
Aggakake or oeuf au lait.
3 eggs, 2 cups milk, 1 cup flour. Makes a nice flan/pudding consistency. Eggy and delicious.
Aggakake is on the chart!
Breakfast is just generally milquetoast then?
Isn't it something like pancake with more eggs?
The real question is why this isn't already a dish.
What about eggs eggs and milk breakfast? (Omelette with cheese). Plenty of protein and little sugar
A Japanese Souffle Pancake might be in the Dark Breakfast realm.
Colombia's bandeja paisa.
What about french toast? I feel like there is a lot of egg in it, might place it near the bottom of the abyss.
French toast isn't plotted because the recipe doesn't customarily start with flour, but if you do plot it it ends up in the lower middle. If you have exceptionally eggy challah, then you might be able to push it into the abyss, but really exceptionally eggy, like 1:1 egg to flour by weight.
While a delightfully funny article it also touches on something I’ve thought about many times before: just how uniform basically 99% of all (American) restaurant breakfast menus are. It’s really quite extraordinary in some ways how these menus are nearly entirely interchangeable from on to another. They might have very small tweaks and of course the ingredient specifics aren’t exact (specific butter brands, slight proportion differences)but by and large you could predict with almost perfect accuracy every single item that a typical breakfast menu in the US might have.
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!
Egg scramble fluffed with milk plus slice of toast might qualify
This is the kind of creative thinking that makes HN great. Using the framework of dark matter detection to explore unobserved breakfast possibilities is both hilarious and oddly rigorous. The breakfast phase space is clearly under-explored.
The ones who walk away from omelettes.
As a certified expert breakfast cook, I bristle at the idea that scrambled eggs includes any ingredient other than eggs or seasoning.
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?
french toast was dismissed far too lightly, it's exactly what goes into the gap. also savoury bread pudding.
maybe Portugal’s Pastel de Nata falls in the dark zone?
It’s a baked custard (so plenty of eggs) in a pie.
not sure the proportions match.
I love the idea and the writing, but the execution seems off. Cake has a very well-defined spot and Weetabix doesn’t? More work needed
This reminded me of <https://xkcd.com/2893/>.
where's porridge?
Breakfast has way more dimensions.
I agree, that is an oversight.
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.
No mention of Eggs Florentine?
No jam and butter option when writing that long of a post on what to eat for breakfast is criminal.
Applying Tom Ngo's Embedded Constraint Graphics to Direct-Manipulation Breakfast Selection (Direct manipulation over simplicial complexes using barycentric interpolation: they're not just for breakfast any more.)
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...
I usually have egg on toast with plenty of butter. The combination sits squarely in the dark region I think.
I also get up early and it is often actually dark.
What about vegetable-forward breakfasts? Completely not on this chart.
That's because such breakfasts are forbidden by the Geneva Convention articles, section 23, subsection 8, paragraph 4. It's not one of the more well known provisions, but it does exist, and nobody wants to break international law.
> nobody wants to break international law
I was with you up to this point, but my suspension of disbelief has its limits.
Dietary fiber is clearly off charts for OP.