Poll

No votes (0%)
1 vote (14.28%)
4 votes (57.14%)
1 vote (14.28%)
No votes (0%)
3 votes (42.85%)
1 vote (14.28%)
No votes (0%)
1 vote (14.28%)
3 votes (42.85%)

7 members have voted

Wizard
Administrator
Wizard
  • Threads: 1528
  • Posts: 27473
Joined: Oct 14, 2009
Thanked by
smoothgrh
May 25th, 2025 at 7:50:48 PM permalink
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?
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Johnzimbo
Johnzimbo
  • Threads: 4
  • Posts: 1208
Joined: Sep 29, 2010
May 25th, 2025 at 8:09:41 PM permalink
Sorry, I like the shredded wheat mini pillows.

That's all I got
Dieter
Administrator
Dieter
  • Threads: 17
  • Posts: 6701
Joined: Jul 23, 2014
May 25th, 2025 at 8:31:01 PM permalink
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.
May the cards fall in your favor.
AutomaticMonkey
AutomaticMonkey
  • Threads: 10
  • Posts: 603
Joined: Sep 30, 2024
May 25th, 2025 at 8:46:52 PM permalink
A flake! A Sugar Frosted Flake, they're grrrrr-eat!

An infinitesimally thin flake will yield an infinite ratio of glaze/cereal.
DJTeddyBear
DJTeddyBear
  • Threads: 215
  • Posts: 11113
Joined: Nov 2, 2009
Thanked by
Wizardcamapl
May 25th, 2025 at 8:56:45 PM permalink
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. 😵‍💫
I invented a few casino games. Info: http://www.DaveMillerGaming.com/ ————————————————————————————————————— Superstitions are silly, irrational, childish rituals, born out of fear of the unknown. But how much does it cost to knock on wood? 😁
EvenBob
EvenBob
  • Threads: 443
  • Posts: 30555
Joined: Jul 18, 2010
May 25th, 2025 at 10:40:45 PM permalink
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.
"It's not called gambling if the math is on your side."
Wizard
Administrator
Wizard
  • Threads: 1528
  • Posts: 27473
Joined: Oct 14, 2009
May 26th, 2025 at 1:55:07 AM permalink
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
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
Wizard
Administrator
Wizard
  • Threads: 1528
  • Posts: 27473
Joined: Oct 14, 2009
May 26th, 2025 at 2:03:50 AM permalink
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
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
camapl
camapl
  • Threads: 8
  • Posts: 616
Joined: Jun 22, 2010
May 26th, 2025 at 2:44:51 AM permalink
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…
It’s a dog eat dog world. …Or maybe it’s the other way around!
odiousgambit
odiousgambit
  • Threads: 329
  • Posts: 9946
Joined: Nov 9, 2009
May 26th, 2025 at 3:58:11 AM permalink
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!!

the next time Dame Fortune toys with your heart, your soul and your wallet, raise your glass and praise her thus: “Thanks for nothing, you cold-hearted, evil, damnable, nefarious, low-life, malicious monster from Hell!”   She is, after all, stone deaf. ... Arnold Snyder
rxwine
rxwine
  • Threads: 227
  • Posts: 13115
Joined: Feb 28, 2010
Thanked by
camapl
May 26th, 2025 at 3:59:31 AM permalink
Does the glaze get on the inside and the outside of the "hole"?

(weirdest sentence I've made all week)
Sanitized for Your Protection
billryan
billryan
  • Threads: 281
  • Posts: 18654
Joined: Nov 2, 2009
May 26th, 2025 at 8:14:55 AM permalink
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.
The older I get, the better I recall things that never happened
DRich
DRich
  • Threads: 91
  • Posts: 13661
Joined: Jul 6, 2012
Thanked by
camapl
May 26th, 2025 at 8:30:12 AM permalink
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%?
You can't know everything, but you can know anything.
ThatDonGuy
ThatDonGuy 
  • Threads: 126
  • Posts: 7131
Joined: Jun 22, 2011
May 26th, 2025 at 9:54:06 AM permalink
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
billryan
billryan
  • Threads: 281
  • Posts: 18654
Joined: Nov 2, 2009
May 26th, 2025 at 9:55:33 AM permalink
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.
The older I get, the better I recall things that never happened
gordonm888
Administrator
gordonm888
  • Threads: 63
  • Posts: 5509
Joined: Feb 18, 2015
Thanked by
camapl
May 26th, 2025 at 11:07:48 AM permalink
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.
So many better men, a few of them friends, are dead. And a thousand thousand slimy things live on, and so do I.
rxwine
rxwine
  • Threads: 227
  • Posts: 13115
Joined: Feb 28, 2010
May 26th, 2025 at 1:28:35 PM permalink
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)
Sanitized for Your Protection
rxwine
rxwine
  • Threads: 227
  • Posts: 13115
Joined: Feb 28, 2010
May 26th, 2025 at 2:45:05 PM permalink
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.
Sanitized for Your Protection
AutomaticMonkey
AutomaticMonkey
  • Threads: 10
  • Posts: 603
Joined: Sep 30, 2024
May 26th, 2025 at 3:40:48 PM permalink
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.
billryan
billryan
  • Threads: 281
  • Posts: 18654
Joined: Nov 2, 2009
May 26th, 2025 at 4:55:47 PM permalink
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.
The older I get, the better I recall things that never happened
Wizard
Administrator
Wizard
  • Threads: 1528
  • Posts: 27473
Joined: Oct 14, 2009
May 26th, 2025 at 5:08:07 PM permalink
Quote: DRich

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



No, I'm not. It was an optimistically low estimate.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
TomG
TomG
  • Threads: 16
  • Posts: 2546
Joined: Sep 26, 2010
May 26th, 2025 at 6:52:02 PM permalink
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.
Joeman
Joeman
  • Threads: 36
  • Posts: 2465
Joined: Feb 21, 2014
May 27th, 2025 at 4:52:01 AM permalink
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
are all the same flavor, regardless of the individual loop's color!
"Dealer has 'rock'... Pay 'paper!'"
Dieter
Administrator
Dieter
  • Threads: 17
  • Posts: 6701
Joined: Jul 23, 2014
May 27th, 2025 at 5:01:06 AM permalink
(snip!)
Quote: Joeman


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

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.

That's still what, 60 million broadly age appropriate you could talk to? I simply don't have that kind of time.
May the cards fall in your favor.
camapl
camapl
  • Threads: 8
  • Posts: 616
Joined: Jun 22, 2010
May 27th, 2025 at 10:04:45 AM permalink
Quote: Dieter

(snip!)

Quote: Joeman


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

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.

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?
It’s a dog eat dog world. …Or maybe it’s the other way around!
  • Jump to: