Hi Steemians, today another math story. I had this idea for a while but I didn't have a nice way to frame the mathematical concept. While eating some cheese I suddenly knew what to write. Hope you guys enjoy this cheesy post!

Sandwich man

Bob is your average office worker. Five days a week from 8 to 5 he is a nobody making a nobody's pay. If you would see him you would think he is just an ordinary guy. But come Saturday morning and everything changes as his heart begins to pump with the fiery passion of a primordial god ready to create. On Saturday Bob becomes the `Sandwich man'.

Bob has been running a sandwich market stand for the last 5 years which is appropriately called ` the Sandwich man'. He truly is all about sandwiches. In his mind the smell of fresh bread conjures images of a field of grain on a warm autumn's day. In his mind the spreading of butter is like petting an adorable puppy. And in his mind the sight of the rough bread texture instantly projects perfect sandwich fillings. But most importantly, he loves sharing his tasty creations with the world.

Photo by Daria Svetsova

His sandwiches are by far the most popular market attraction. Bob also sells loaves of bread. But they do not seem to be blessed with the same idol-status. Since Bob's sandwiches are gods compared to the loaves the latter have to be sacrificed and ascend to the god-like status of sandwich. However, since this procedure requires the ritual of sandwich filling, of which Bob never has enough with him, the loaves remain on the sacrificial cutting board.

An old man with a keen eye spots the bread surplus. "Sir, what will you be doing with those breads?" the old man asks Bob. When Bob turns around and sees the old man he is bit shocked. Although the old man wears a thick tattered coat his figure is still too skinny. Bob cannot imagine what the old man would look like without the coat. He also notices that there is a strange cheese smell coming from his clothes. Bob feels a bit sorry for the old man so he replies, "I always sell fresh bread. So just take all the left-overs."

"Don't mind if I do." says the old man.

"I also have some cheese left. Would you like some?" adds Bob.

"Don't worry about me good sir" replies the old man and he rushes off with the breads.

"Well that is bit strange" says Bob to himself. Wonder if will ever see him again.

A week later.

At the end of the day Bob is again left with the painful realization that he baked too many breads or bought too little sandwich filling. When he is packing his stuff. A whiff of a strange cheese enters his nostrils and there is the old man again patiently standing in front of his market stand. "Sir, what will you be doing with those breads?" asks the old man. "You can have all of them and some cheese if you want." replies Bob. "I won't be needing any cheese good sir. I have my own cheese" replies the old man and he rushes off with the breads. "Well that is odd" says Bob to himself.

Photo by Sabine Schulte

Again a week later

Bob has again some left-over bread. As he starts packing, the cheese smell rushes into his nose and there is the old man. "Sir, what will you be doing with those breads?" asks the old man. A curious Bob replies, "All of this seems a bit strange. Why do you only accept my loaves of bread and not my fillings? Could you maybe tell me what is going on?"
"Well only because you are such a kind sir" replies the old man. The old man puts his hand in his pocket and pulls out a small strong smelling cheese cube with and edge of 1cm which looks something like this:

"This is the Infinity Cheese. It is a piece of cheese which gives an infinite supply of cheese slices. I will demonstrate it!"

The old man cuts the cheese along the following dashed line:

This divides the cube in two equal blocks. He places the two blocks next to each other



Observe that the cheese blocks will cover 2 cm² of a sandwich surface. He cuts along the dashed line in the previous picture which yields two new blocks. He places all the blocks next to each other:


Now they can cover a surface of 3 cm². He cuts along the dashed line in the previous picture and places the resulting blocks next to each other:



So the blocks can cover a total area of 4 cm². Repeating the procedure again gives



Now they can cover 5 cm². You could continuously repeat this process and cover more and more of the bread with cheese! " With this piece of cheese you could cover the whole world!" exclaims Bob. The old man responds " Yes, that is true. With this Great Cheese comes Great responsibility! The Infinity Cheese does not follow the laws of physics because

I can cut this cheese as finely as I desire at no level will I separate the underlying molecules since this Infinity Cheese is an infinity of cheese.

These wise words escape Bob since his mind is far away dreaming about the greatest cheese sandwich in the world. Because Bob is and will always be the Sandwich man.

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Conclusion/reflection

Each cheese cut preserves the volume but increases the total surface area. The number of cuts is equal to the surface area that the cheese slices cover. Informally, the surface area will go to infinity as the cuts approach infinity(in a more rigorous sense you would formulate this in terms of limits, see the Technical Appendix). Mathematically, we observed that

a three dimensional object with finite volume can have an infinite surface area.

Is this statement reasonable in our physical world? No. The thickness of the cheese slices becomes smaller and smaller but at some point you cannot cut any finer because the fundamental physics particles have a fixed size. In a sense physical matter is discrete whereas the generation of infinite surface area requires continuous matter.

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Technical Appendix

Bluntly put we have been looking at the limit:

The term in the summation represents the volume of the block resulting from the kth cut. The n in the summation represent the total cuts. The limit is equal to the total volume of all the blocks.

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Sources

Top photo from Pixabay

Pictures were made using inkscape and the equations were written using quicklatex

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Thank you!

Thanks for being so kind to read my post. You are awesome! Please follow me if you enjoyed it. If you have any questions just post them below and I will answer them. Or if you might have a nice topic you want me to cover also let me know below. :o)

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Photo by Mark Kent