r/theydidthemath 1d ago

[Request] Okay lets try this one

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1.3k Upvotes

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330

u/vwibrasivat 1d ago

This also works for limes, a stick of chalkboard chalk, or the length of a USB thumbdrive.

175

u/JumbledJay 1d ago

Or my... nevermind

198

u/humanino 1d ago

Cylinder

143

u/Background_Half_4568 1d ago

THE CYLINDER CANNOT BE DAMAGED

51

u/Hitotsudesu 1d ago

But can you remove it from a mini MnM container

49

u/TheMultiRounderGamer 1d ago

It is imperative that the cylinder remains unharmed.

13

u/GandalfTheSmol1 1d ago

Nah I would like to get rid of mine

7

u/r3d-v3n0m 1d ago

Is this a reference I'm missing, by chance?

10

u/Arthur_Burt_Morgan 1d ago

You havent heard of the cyllinder yet? Oh geez. Google cyllinder reddit.

0

u/HeyImGilly 20h ago

Oh yeah. Has to do with a male appendage roughly the size of a mini M&Ms container.

EDIT: Assumed to be male appendage, but nonzero chance it was mushy banana.

5

u/ThatGuyNikolas 9h ago

It's a cyllinder.

6

u/soyalguien335 1d ago

No, it is imperative that ...

6

u/JonasRahbek 3✓ 18h ago

Are you saying - that everything the size of a toothpick, is the size of a toothpick?🤯

156

u/chessgremlin 1d ago

No. The characteristic length for an atom is an angstrom ~10^-10 meters. A toothpick has a thickness of roughly a millimeter, or 10^-3 meters. The earth, at the equator, is roughly 10^7 meters "thick / wide". So a toothpick is around 10^7 times thicker than an atom while the earth is around 10^10 times thicker than a toothpick.

43

u/TheJWeed 1d ago

Alright, then what size object should we be looking for that does work for the meme?

96

u/1337k9 14h ago edited 14h ago

The atom in that pic has 3 reds, 4 blues and 3 electrons (assuming nothing's hidden behind the nucleus). It could be Lithium with a mass of 6.942 g/mol . A single atom of Lithium has a mass of 6.942 g divided by 6.02214076 × 1023 . Earth has a mass of 5.972 x 1027 grams. To calculate how much bigger the Earth is, use the formula

(5.972g x 1027 ) / (6.942g x 10-23 )

=5.972g / 6.942g x 1050

=5.972 / 6.942 x 1050

The Earth has 5.972 / 6.942 x 1050 times more mass than the atom. Something logarithmically between the two would have a mass of the root of this number. That is

2V[5.972/6.942] x 2V[1050 ]

=2V[5.972/6.942] x 1025

times the mass of the atom. If we're not just interested in ratios and want to know the real world mass, that's

6.941765353712grams / 6.02214076 × 10-23 x 2V[5.972/6.942] x 1025

=6.941765353712grams / 6.02214076 x 2V[5.972/6.942] x 102

≈1.069grams x 100

So 106.9 grams is the logarithmic middle between that Lithium atom and the Earth. An orange weighs about this much.

EDIT: Reddit's formatting

51

u/TheJWeed 14h ago

I’m glad you went with mass instead of length like everyone else. This answer wins in my book.

8

u/Dynespark 7h ago

What do you mean? An African orange or a European orange?

3

u/1337k9 3h ago

The final step doesn't matter. Any fruit at your closest grocery store that's around 106.9 grams works

2

u/Eucharism 12h ago

Legendary Intellect.

1

u/MovinOnUp2TheMoon 3h ago

There’s the Math They Did.

Thanks!

11

u/Shimetora 20h ago

With this calculation, something thats 1000 times smaller than earth, so ~10km across. Bit of an awkward size because that's much bigger than most parks/features, but much smaller than most cities. Maybe Manhattan Island?

17

u/syds 18h ago

you could've just say yo momma size

1

u/pickenmensch 17h ago

Huh? Your math doesn' check out. You have 17 orders of magnitude between an atom and the size of the earth (if we just take the prior comment at face value). So an object in between those orders should fall in between 10-1 m and 10-2 m. So maybe a toothpick fits the criterium if we take its length rather than its diameter.

1

u/math_is_best 17h ago

they answered to if we exchange the earth to something else

3

u/pickenmensch 17h ago

Oh, I see, I see. I guess that'd be another way to fix the scale. In that case an object of 104 m size does make sense.

1

u/al24042 17h ago

If we need to find something, the magnitude of the size of which is between 10-10m and 107m, wouldn't that object be between 10-2m and 10-1m? This would mean it's around 10 centimetres wide, say, an orange?

Also, something 1000 times smaller than earth is 104m which is not in between an atom and earth's sizes, no?

1

u/headsmanjaeger 23h ago

Idk a golf ball maybe

7

u/vwibrasivat 1d ago

Did you try length of toothpick?

1

u/chessgremlin 1d ago

Length of a toothpick ~10^-2 m, which is closer, but it seemed reasonable to use the radial dimension for consistent comparison

8

u/vwibrasivat 1d ago

Try the precise numbers.

lithium atom, 304 picometers

toothpick , 65 mm

earth , 12742018 meters

19

u/sdf15 23h ago

toothpick is 2.13x108 times as big as an atom earth is 1.96x108 times as big as a toothpick so actually that works really well

1

u/chessgremlin 22h ago

No, I think a question without reference to a specific element or type of toothpick is better solved as a fermi estimation.

4

u/Yung_Oldfag 21h ago

It's not labeled properly but if the image is depicting anything, it's lithium-7

1

u/Key-Tie2214 17h ago

What about length of the toothpick?

1

u/barney_trumpleton 15h ago

I remember reading somewhere that the mid-point (logarithmically) between the plank length and the observable universe is about the width of a human hair. Can you do that one?

2

u/Antisymmetriser 11h ago

Planck length: 10-35 m, log(Planck)=-35

Diameter of the universe (approx.): 54*1022 m, log (universe)=23.73

Width of a hair: ~100*10-6 m, log(hair)=-4

(23.73-(-35))/2 - 35=-5.63

So a hair is slightly over the midway between the scales, but close. If you take the minimum value for human hair I found, 17 microns, you get log(hair)=-4.77, a little closer bu not quite. If you take the size of a typical a cell nucleus, 6 microns, you get legg(nucleus)=-5.22, which is much closer

1

u/barney_trumpleton 10h ago

Incredible, thank you!

84

u/DrunkCommunist619 1d ago edited 13h ago

An atom is 0.0000001mm

A toothpick is 50mm

That's a 500,000,000 difference

A toothpick is 50mm

The earth is 12,756,000,000mm

That's a 225,120,000 difference

Edit: Sorry for the calculation error. I was off on the size of the world.

12

u/igotshadowbaned 23h ago edited 22h ago

An atom is 0.0000001mm

It's typically between a range of 0.0000001 - 0.0000005, so you should probably take a middle ground of 0.00000025.

Making the atom 200,000,000× smaller than a toothpick

The earth is 12,756,000,000,000mm

12,725km is equal to 12,725,000,000mm

You're off by a factor of 1000 in your calculations here

Making the toothpick 255,040,000× small than the earth.

Considering the magnitudes we're working with, 25% error is pretty decent. A comparison using a toothpick that is 56.40mm long would be accurate to 0.00009%

23

u/blaghed 1d ago

That's a cup of toothpicks, tho

4

u/DrunkCommunist619 1d ago

Regardless, that cup looks roughly as wide as it is tall. So the numbers should still work.

4

u/Sentient_Broccolini 1d ago

“a toothpick”

1

u/blaghed 1d ago

Hah, fair

6

u/sdf15 23h ago

the earth is 12.7 billion mm not trillion, making the difference 25x as large instead of 25000 times, which is bad but not that horrible

2

u/O_Martin 23h ago

You are a factor of 1000 off the diameter of the earth. It's 12,000km = 12,000,000m = 12,000,000,000mm

I've it's 12 billion mm, not 12 trillion

1

u/Hour_Ad5398 22h ago

The earth is 12,756,000,000,000mm

what the fuck? thats 12,756,000km which is absolutely not earth's size

1

u/mr_Cos2 20h ago

Really needed to be rude didn't you?

1

u/DrunkCommunist619 13h ago

Whoops my bad

8

u/igotshadowbaned 23h ago

An atoms diameter is typically between a range of 0.0000001 - 0.0000005mm, so you should probably take a middle ground of 0.00000025mm.

A toothpick is 65mm long

Making the atom 260,000,000× smaller than a toothpick

The earth is 12,725km in diameter which is equal to 12,725,000,000mm

Making the toothpick 195,769,000× smaller than the earth.

Considering the magnitudes we're working with, 33% error is pretty decent.

A comparison using a toothpick that is 56.40mm long would be accurate to 0.00009%

4

u/vwibrasivat 22h ago

Sandisk USB flash drive is 56.90mm

3

u/snacknight 20h ago

A mole is a wonderful number—easy to remember and surprisingly useful. ~6 × 10²³

A mole of carbon atoms weighs 12 grams, making the mole an excellent tool for scaling from the atomic level to the size of everyday human interactions.

Earth has a mass of roughly 10 moles of kilograms, making the mole equally handy for converting from human-scale objects to planetary-scale masses.

2

u/WearyGoal 19h ago

The toothpick is made of wood, which is about 50% carbon. It might contain other atoms that have a smaller radius (hydrogen, nitrogen, oxygen; yes, oxygen and nitrogen atoms are smaller than carbon despite being heavier!) and some atoms with a bigger radius, like phosphorus (in DNA) and sulfur (small amounts in proteins). For simplicity, let’s say all atomic radii average out to that of carbon, which is about 70pm. Let’s also assume that all atoms are perfect cubes. The reason I want to do that is because if we assume them to be perfect spheres, there will be some space between eight spheres stacked 2x2x2, which adds to calculation challenges. Assuming cubes eliminates that challenge.

Let’s begin by calculating the volume of one of these “cubes”, and also the volume of a toothpick, which according to google, is 2.0mm in diameter and 65-80mm in length. Let’s assume the smallest toothpick and say it has a 2.0mm diameter and 65mm height. Then, the volumes are:
Carbon: 140pm x 140pm x 140pm = 2,774,000 pm^3

Toothpick: πr^2h=π(1.0)(1.0)(65)=65π mm^3

If we divide the volume of the toothpick with the volume of the atom, we get 7.36 x 10^22 atoms.

Now, we do a similar calculation where we compare the ratio of the volume of the earth to the volume of a toothpick. According to google, the radius of the earth at the equator is 6378Km, and the polar radius is 6357Km. We take the average of the two and say the average radius is 6367.5Km. This might not be the most sound approach, but since we're doing a lot of approximations, I will just go for this too. Given that, we find the volume of the earth to be:

Volume of earth: (4/3)πr^3=(4/3)π(6367.5)^3=1.08x10^12Km^3

We also assume that the surface of the earth is completely flat; no mountains, valleys, etc. (fun fact: according to Neil deGrasse Tyson, if we shrink the earth to the size of a billiards ball, the mountains and valleys, even Mount Everest, will be so insignificant that the earth would be smoother than an actual billiards ball)

Now, if we divide the volume of a single toothpick from this volume, we get that the earth is the same volume as 5.3x10^27 toothpicks. So, really, if all these assumptions hold, 71,940 toothpicks (give or take about a 1000 for simplification errors) contain as many atoms as toothpicks could contain earth (if the building blocks of earth were toothpicks)

1

u/66bananasandagrape 2✓ 17h ago

The geometric mean of the earth’s diameter and a hydrogen atom’s diameter is about an inch.

WolframAlpha will just do this for you

1

u/AccelerandoRitard 17h ago

I think I see what they were going for. The geometric mean is particularly useful when comparing quantities that span several orders of magnitude, as it provides a central value that reflects the multiplicative nature of the differences. Unlike the arithmetic mean, which is more appropriate for additive differences, the geometric mean balances ratios, making it ideal for scenarios involving exponential growth, rates, or scales that differ vastly.

The geometric mean of two numbers a and b is ab divided by the square root of ab​. In this case, we are looking at:

  1. The diameter of a typical atom (approximately 10^-10 meters).
  2. The diameter of the Earth (approximately 1.3 x 10^7 meters).

Step 1: Multiply the two diameters (10^-10 m) x (1.3 x 10^7 m) = 1.3 x 10^-3 m^2

Step 2: Take the square root sqrt(1.3 x 10^-3) ≈ 3.6 x 10^-2 m = 3.6 cm

Therefore, a length of roughly 3–4 cm is the midpoint on a logarithmic scale between the size of a typical atom and the diameter of the Earth. So, not quite a toothpick, just under the diameter of a silver dollar. Maybe a large shooter marble. But it's close enough they could have just picked a different value for the diameter of an atom, which one could argue a few ways.

-3

u/Nowheel_Nodeal 23h ago

Well, a toothpick is about 3” long, which makes it 3” longer than an atom. The Earth is 7926 miles in diameter, which is more than 3”. Nowhere near the same size