Optimal size for breakfast cereal

I've already spent hours on math behind this post, which will mean it will probably get very little traction. However, I'm used to it. Probably explains why 97% of women find me incredibly boring.

Let me introduce some pictures I took myself at the local supermarket.


Front of box


Back of box.


Math equations on back of box.

So, Kellogg's has introduced spherical shaped versions of Frosted Flakes, Apple Jacks, and Krave (which I never heard of before).

The back of the boxes claim that the spherical shape is "genius" because it delivers more glaze. I assume they mean the ratio of surface area to volume is greater comparing the sphere to a torus (donut shape).

Before going further, let's review the formulas for volume and surface area for both shapes.

Sphere:
Volume = (4/3)𝝅r³
Surface area = 4𝝅r²

Torus:
Volume = 2𝝅²r²*R
Surface area = 4𝝅²r¹*R
where r = radius of the circle that is rotated 360 degrees around the center creating the torus, R = distance of torus from center

In the case of the torus, r represents the radius of the red circle as shown in the following image.



On the back of the box, the A clearly refers to surface area.

However, their formula has a 2 instead of a 4. Why? Perhaps they think they glaze only half of torus shaped cereals like Apple Jacks or Fruit Loops. However, they don't. It's obvious from looking at classic Fruit Loops or Apple Jacks they glaze the whole thing.

Let's do some math to determine the optimal shape of cereal to maximize glaze.

If we have a torus with r=1 and R=1, we get:
Volume = 19.7392088
Surface area = 39.4784176
Ratio surface area to volume = 2

Next, let's carefully choose a radius of a sphere to equalize the volume.

Sphere with r=1.67653921
Volume = 19.7392088
Surface area = 35.32134996
Ratio surface area to volume = 1.78940044

With 2>1.78940044, I have shown that a torus delivers more glaze than a sphere.

However, in all fairness, if torus were only glazed half way, the ratio of glaze to volume would be 1, in favor of a sphere.

However, again, they don't just glaze half their torus shape. I submit for your consideration that Kellogg's didn't do their math correctly.

In fact, I submit that the Isoperimetric Inequality says the sphere actives the LOWEST ratio of surface area to volume.

I still have more to say on this topic, but that is enough for one post. I'll close with some relevant links.

Isoperimetric Inequality

Torus

Internet spots big mistake on Kellogg's cereal box

Stellated Octahedron (I'll get to this one)

The question for the poll is what shape of breakfast cereal achieves the most glaze?

Sorry, I like the shredded wheat mini pillows.

That's all I got

Quote: Johnzimbo

Sorry, I like the shredded wheat mini pillows.

That's all I got
link to original post

Aren't they only frosted on one side?

Honeycomb was another, I thought. One side seemed shinier.

Quisp, on the other hand, seemed to be predominantly sugar glaze or glaze-equivalent, bound together with the bare minimum amount of grain to maintain structural integrity.

A flake! A Sugar Frosted Flake, they're grrrrr-eat!

An infinitesimally thin flake will yield an infinite ratio of glaze/cereal.

I haven’t eaten frosted flakes lately, but I kinda think that a flake, even if it’s only coated on one side, would offer the greatest surface area to volume. But I wonder how the thickness of the flake would come into play.

I voted flake. But I’m kind of curious to know how a fractal would be different from a flake. At least for this exercise, anyway.

But regarding the original question, isn’t it relatively well known that a sphere is the best shape for maximizing the volume to surface ratio? Therefore, it would seem to be the worst candidate for an area to volume ratio. Clearly, even without the math, it would seem that the torus is better than the sphere.

Mike, if I was you, I’d write a very strongly worded letter to Kelloggs. 😡
At another one to whatever federal organization is involved with truth in advertising. That is, assuming that Trump hasn’t dismantled it. 😵‍💫

Oh my dear God, Mike turns 60 this year and he's already going around the Bend. He's figuring out how to use math to eat cereal.

Thank you for the comments so far.

Here is an image of Apple Jacks close up, which I think shows they are frosted on both sides.


Image source: https://commons.wikimedia.org/wiki/File:2020-05-03_09_34_31_A_sample_of_Kellogg%27s_Apple_Jacks_cereal_in_the_Franklin_Farm_section_of_Oak_Hill,_Fairfax_County,_Virginia.jpg

You may notice that fractal was one of the choices in the poll for optimal cereal size to maximize ratio of surface area to volume.

In particular, I use a tetrahedron as a starting point. The first step would take us to a stellated octahedron.



The following table shows the volume and surface area for the first 20 fractal iterations.

Level	Volume	Surface area	Ratio SA to volume
1	0.117851	1.732051	14.696938
2	0.176777	2.598076	14.696938
3	0.220971	3.897114	17.636326
4	0.254116	5.845671	23.003904
5	0.278976	8.768507	31.431076
6	0.297620	13.152761	44.193114
7	0.311603	19.729141	63.314899
8	0.322091	29.593712	91.879989
9	0.329957	44.390568	134.534586
10	0.335856	66.585852	198.257286
11	0.340280	99.878777	293.519244
12	0.343598	149.818166	436.026868
13	0.346087	224.727249	649.337067
14	0.347954	337.090874	968.780691
15	0.349354	505.636311	1447.347971
16	0.350404	758.454466	2164.516810
17	0.351191	1137.681700	3239.495213
18	0.351782	1706.522550	4851.084900
19	0.352225	2559.783824	7267.476620
20	0.352557	3839.675736	10890.942993
21	0.352806	5759.513605	16324.877603
22	0.352993	8639.270407	24474.353438
23	0.353133	12958.905611	36696.960326
24	0.353238	19438.358416	55029.060808
25	0.353317	29157.537624	82525.173666

Quote: DJTeddyBear

I haven’t eaten frosted flakes lately, but I kinda think that a flake, even if it’s only coated on one side, would offer the greatest surface area to volume. But I wonder how the thickness of the flake would come into play.

I voted flake. But I’m kind of curious to know how a fractal would be different from a flake. At least for this exercise, anyway.

But regarding the original question, isn’t it relatively well known that a sphere is the best shape for maximizing the volume to surface ratio? Therefore, it would seem to be the worst candidate for an area to volume ratio. Clearly, even without the math, it would seem that the torus is better than the sphere.

Mike, if I was you, I’d write a very strongly worded letter to Kelloggs. 😡
At another one to whatever federal organization is involved with truth in advertising. That is, assuming that Trump hasn’t dismantled it. 😵‍💫
link to original post

It is true that bubbles are spherical, as their surfaces contain the greatest volume using the least surface area. (Or as DJTB said, “…maximizing volume to surface ratio…”)

@Wiz, Good catch! Give ‘em hell!

ETA: I voted flake, but I haven’t done the math on any shapes outside of math classes taken decades ago.

I also checked “I’m a bigot”, as I’m bigoted against bigots. Talk about a recipe for self loathing…

Quote: Wizard

In fact, I submit that the Isoperimetric Inequality says the sphere actives the LOWEST ratio of surface area to volume.
link to original post

I would say that is self evident, however, somebody had to do the math to prove it and I thank you.

the audacity for them to post a formula and claim it 'proves' what they say is outrageous

I say it smacks of girl math!!

Does the glaze get on the inside and the outside of the "hole"?

(weirdest sentence I've made all week)

Kellogg's has a bunch of middle-aged men talking about their cereals and seems to have inspired a debate about math and science.
Perhaps they know what they are doing, which is simply selling cereal.

Quote: Wizard

I've already spent hours on math behind this post, which will mean it will probably get very little traction. However, I'm used to it. Probably explains why 97% of women find me incredibly boring.

Are you sure, only 97%?

Quote: Wizard

So, Kellogg's has introduced spherical shaped versions of Frosted Flakes, Apple Jacks, and Krave (which I never heard of before).

I think Krave is chocolate "pillows" filled with some sort of vanilla creme. Spherical ones would be like "creme-filled Cocoa Puffs." It is one of its newer cereals.

Quote: Wizard

However, their formula has a 2 instead of a 4. Why? Perhaps they think they glaze only half of torus shaped cereals like Apple Jacks or Fruit Loops. However, they don't. It's obvious from looking at classic Fruit Loops or Apple Jacks they glaze the whole thing.

I have a feeling they meant to say (2𝝅)²Rr instead of 2(𝝅²)Rr

Quote: ThatDonGuy

Quote: Wizard

So, Kellogg's has introduced spherical shaped versions of Frosted Flakes, Apple Jacks, and Krave (which I never heard of before).

I think Krave is chocolate "pillows" filled with some sort of vanilla creme. Spherical ones would be like "creme-filled Cocoa Puffs." It is one of its newer cereals.

Quote: Wizard

However, their formula has a 2 instead of a 4. Why? Perhaps they think they glaze only half of torus shaped cereals like Apple Jacks or Fruit Loops. However, they don't. It's obvious from looking at classic Fruit Loops or Apple Jacks they glaze the whole thing.

I have a feeling they meant to say (2𝝅)²Rr instead of 2(𝝅²)Rr
link to original post

If they correct the math, watch for the early box versions to soar in value.

I have some misgivings about this analysis and the comparable analyses found on many sites on the Internet.

Specifically I wonder whether "surface area/volume of shape" is the correct figure of merit. Instead, I imagine that the correct figure of merit is "surface area/volume that the shape occupies."

Given that you have a cereal bowl that will be filled with liquid milk after you have put the glazed cereal shapes in the bowl. The cereal shapes are a "delivery system for sugary glaze." You want to maximize the surface area of all the shaped cereal in a bowl of fixed volume.

A torus may have more surface area than a comparable sphere, but the torus would (I think) occupy more volume in the cereal bowl. Because the hole in the torus is inefficient. Thus the ratio of 'surface area to volume occupied' of the torus seems like it might be lower for a torus than for a sphere.

Rigorously, the packing density of the shapes might be a consideration as well. I really haven't considered packing density.

Quote: gordonm888

Rigorously, the packing density of the shapes might be a consideration as well. I really haven't considered packing density.
link to original post

(My Ai generator kung foo is weak)

Quote: rxwine

Quote: gordonm888

Rigorously, the packing density of the shapes might be a consideration as well. I really haven't considered packing density.
link to original post

(My Ai generator kung foo is weak)
link to original post

I think I need to trademark that name. Then my commercials would feature people trying to say "Wute Cubes" 3 times fast.

It starts to get moot when talking about a real cereal and coating instead of an ideal one. An ideal coating would be a superfluid of zero particle size, the ideal cereal substrate could take any shape or thickness and remain intact, infinite rigidity.

Real sugar is a large molecule and can produce some strong forces with other surfaces which is why it is so sticky. So it won't flow into an extreme fractal. Another funny thing that happens is substrates and coatings stress each other. This is a problem in optical design because if you coat a thin element on one side, the resulting stress from the dissimilar materials being in contact will bow the substrate so that it's not optically flat anymore, and the usual remedy is to coat the other side too, with a coating that will produce an equal stress at all temperatures the finished product will experience. If you were to coat only one side of a corn flake with sugar, it would probably crack the flakes over time and you could end up with a box of Crumbly Mess cereal.

So for the best way to do it with a real-world product, I'll go with what I knew as a little kid- that Sugar Frosted Flakes are the best because they have the most sugar, and even after the scoundrels changed the name to mere "Frosted Flakes" we all still knew.

I remember eating boxes of some sort of wheat chex gruel because they had the coolest premiums-each box had a toy battleship, submarine, or aircraft carrier.. Sugar Smacks had unique Civil War soldiers, but anything starting with sugar was a no-go.
The general rule was that sugar had to be the fourth ingredient or later, but it wasn't set in stone.
We were a Special K family, but switched to Total when that came out, as I recall.

Quote: DRich

Are you sure, only 97%?
link to original post

No, I'm not. It was an optimistically low estimate.

If the purpose is to maximize frosting to flake (or loop) ratio, then just make the entire cereal the frosting. The best shape is a zero-dimensional point, surrounded by sugar.

If there is any value to the starch part, first decide what ratio is best, then decide which shape best achieve that. Since today is Memorial Day, where we honor those who gave their lives in defense of corporate profits, I suggest we test a bunch of them and see which ones sell the best.

I have no idea if spheres are more cost-effective to produce than tori, but this smacks of marketing trying to convince the consumer that a new version of their product (which may just happen to be more economical to produce) is better than the old version. Kind of like triple-zero roulette ("More ways to win!")

I voted for flakes because, in addition to the reasons mentioned above, the irregularity of the flake's surface provides more area for the glaze to accumulate.

On the subject of cereal disinformation, was anyone else as devastated as I when they found out that

(snip!)

Quote: Joeman

On the subject of cereal disinformation, was anyone else as devastated as I when they found out that

Froot Loops...

are all the same flavor, regardless of the individual loop's color!

link to original post

I was only mildly surprised. My recollection of Froot Loops is that they are only glazed on one side.

Quote: Dieter

(snip!)

Quote: Joeman

On the subject of cereal disinformation, was anyone else as devastated as I when they found out that

Froot Loops...

are all the same flavor, regardless of the individual loop's color!

link to original post

I was only mildly surprised. My recollection of Froot Loops is that they are only glazed on one side.

As for 3% of women...

That's still what, 60 million broadly age appropriate you could talk to? I simply don't have that kind of time.

link to original post

Is that where the practice of calling women “broads” came from?