Since ancient times, food has not only been a source of human matter and energy, but also a profound science. As a discipline that explores the relationship between humans and food, Gastronomy itself is a pluralistic discipline, which contains medicine, agriculture, chemistry, biology, geography, but also history, philosophy, anthropology, psychology, sociology, and mathematics, which everyone likes, is also inextricably linked to food.
Mathematics, as a highly creative discipline, has artistic properties in itself in a sense, and the beauty of mathematics and the beauty of food complement each other. In order to let more readers (foodies) realize the charm of mathematics, today we will take you to explore the secrets of food in mathematics and reveal the mathematics on the tip of the tongue for everyone.
△ In "China's Little Master", the steel rod master has already learned to use the golden section to make roasted wheat.
Thought Question: AQ: AB = ?
Image source: Anime "China Little Boss"
<h1 class="pgc-h-arrow-right" > fast food: sandwiches vs burgers</h1>
Food doesn't necessarily require ornate presentation and tedious steps, and everyday food can be delicious, such as the sandwich that represents fast food. Sandwiches are popular with mathematicians, such as the following theorem, which is called the "sandwich theorem":
Astute readers may have discovered, this is actually the pinch theorem or pinch theorem mentioned at the beginning of university calculus. It is called the sandwich theorem because the sandwich vividly expresses the characteristics of this theorem being "clipped".
△ Mickey's superb knife work ensures that the limit of the two slices of bread is the same, so the limit of the slices of bread between the two slices of bread is also the same
Image source: Mickey and the Magic Bean
Of course, a sandwich can't be just bread, according to the official definition of a sandwich, it is a snack that places meat, cheese or vegetables in the middle of the bread, plus seasonings and sauces. Hence the following theorem, the ham sandwich theorem. [1]
There are n measurable "objects" (tight sets, i.e., closed and bounded sets) in n-dimensional space that can be divided into two equally measured parts simultaneously using a (n−1)-dimensional hyperplane.
To understand this theorem, a few nouns need to be clarified first:
· Measure: In simple terms, the so-called measure is to map the set to a non-negative real number to define the size of the set. The strict mathematical definition is as follows:
· σ-algebra of the set X: Also known as the σ-field of X, it is a subset of the power set of X (the collection system that contains all subsets of X).
· Hyperplane: Refers to the subspace of n-dimensional Euclidean, where the codimensity is 1, or the (n−1)-dimensional subspace in n-dimensional space. For example, a line in a plane, a plane in space.
The ham sandwich theorem can be seen as an inference of the Bossouk-Ulam theorem ( any continuous function of n-dimensional spheres from n-dimensional spheres to Euclidean n- dimensional space must map a pair of pairs of metatarsal points to the same point ) . Applied to sandwiches, i.e. 3 measurable tight sets in three-dimensional space (ham sandwich = bread + ham + bread), they can be divided into two parts with equal measurements with a (3−1)-dimensional hyperplane (blade).
△ Image source: "Wei Gong Family's Meal today"
Of course, the choice of fast food is not only a sandwich, and the burger-loving friends can also find their favorite in mathematics - Hamburger moment problem
△ Image source: "Crayon Xiaoxin"
The Hamburger moment problem is the Hamburger moment problem, in which Hamburger refers to the German mathematician Hans Ludwig Hamburger (so this theorem seems to have nothing to do with the hamburgers we eat). To understand this problem, we must first understand the meaning of "moment". Moment translates in some physics literature as "motion difference" to indicate the shape of an object. The n-order moment of the continuous real function f(x) relative to the real number c is defined as mn,
But in fact, moments are also widely used in the field of probability. A clever reader may have discovered that if c = 0, n = 1, and f(x) is a function of probability density, then the 1st-order moment of f(x) relative to the value 0 is equal to the mathematical expectation of a continuous random variable:
Further, let c = E(x), change the value of n, we have:
· n = 2, m2 defines the variance
· n = 3, m3 defines the skew
· n = 4, m4 defines the peak state
The so-called "moment problem" refers to the possibility of having a metric determined by a metric μ moment sequence mn. thereinto
Depending on the support set of the μ, three well-known "variants" of the moment problem arise [2]:
· Hamburger moment problem: the support set of μ is (−∞, +∞)
· Stieltjes moment problem: the support set of μ is [0,+∞)
· Hausdorff moment problem: the support set of μ is bounded closed interval
Similar to other moment problems, the key to the Hamburger moment problem lies in the existence, uniqueness, and structure of the moment sequence of μ (i.e., how to describe the set of μ), which is explained in detail in the literature.[2]
<h1 class="pgc-h-arrow-right" > meal: pizza beef and sheep lobster</h1>
Sometimes diners choose fast food not out of taste preference, but because they have no choice but to do so. Let's walk into a relatively better-looking (relatively higher-priced) dish and discover the mathematical mysteries behind it.
Image source: One Piece
Pizza has become a popular food all over the world because it can be shared with everyone. However, friends who eat pizza regularly often encounter a difficult problem: how can I get more pizza? The pizza theorem in plane geometry may solve this problem [3-4]. The pizza theorem states that if you cut n knives around any specified point on a pizza so that the angles of the adjacent two knives are the same, and then alternately dye the cut pieces in a certain direction in two colors, then there are:
· When there is either a knife through the center of the circle or n is greater than the even number of 2, the area of both colors is equally large.
· If either knife does not pass through the center of the circle, then:
n When n = 1 or 2, or when n divides by 4 for 3, the part containing the center of the circle is larger
n When n is greater than 4 and divided by 4 more than 1, the part excluding the center of the circle is larger
This theorem can illustrate how two people can get more when they are sharing pizza. The proof of the pizza theorem is not difficult, nor does it use much knowledge beyond the field of calculus, but the steps are far more tedious than I could have imagined (I could even write a separate article to discuss). The following provides a special case of n = 4, which does not require a proof of the formula, and the complete proof of the theorem is shown in [4].
The picture above is n = 4, which is the case of a pizza with 4 knives. The proof idea can basically be summarized as a patchwork. Obviously, the area of the shadow part and the area of the blank part are equal.
Of course, in addition to pizza, beef and mutton are also essential classics. Obviously, the study of cattle and sheep in mathematics has never stopped, such as the classic problem of primary school mathematics "cow/sheep grazing" problem. However, today we are going to share another classic "cow/sheep/horse grazing" problem.
Image source: Shaun the Sheep
There is a circular meadow with a center of A and a radius of r. There is a little B at the edge of the meadow, with a sheep tethered (of course you can tie a cow if you like to eat beef). The rope that binds the sheep is R length. Q When is R long enough for the sheep to graze on half of the pasture?
The most direct idea of this problem is to use calculus to solve it. But if we can introduce equations
Solution x = 1.9056957293..., we can completely solve this problem with the help of junior high school knowledge.
Without losing generality, here we assume that the grassland radius r = 1, and define the following symbols
According to the cosine theorem,
Similarly
Using the geometry knowledge of elementary school, we can easily calculate it
Substitute the preceding expression
Let SEDBC = the area of half a meadow = pi/2, and solve the value of R. The problem here ultimately boils down to solving in the range of [0, pi].
It is also the only super-program knowledge point.
Using this theorem, cattle and sheep can grow more vigorously before they become delicacies.
In addition to the above ingredients, there are also many delicious ingredients in seafood, such as lobster, which we will introduce next. Lobster is a common name for the species of the arthropod phylum Phylloscopus Decapoda, and is widely loved for its delicious taste.
Image source: Spirit Eater
In graph theory, there is a special class of "trees" called "lobster graphs". To understand lobster diagrams, you first need to understand the concepts of "diagram" and "tree".
· Graphs: In graph theory, graphs are used to represent relationships between objects. A graph consists of vertices (or nodes) and the edges that connect those points. Figure G can be defined as a binary pair (V, E) consisting of vertex set V and edge set E. If you specify a direction for each side of the graph, the resulting graph is called a directed graph. Conversely, a graph with no direction on an edge is called an undirected graph.
· A tree is an undirected graph in which there is a unique path between any two vertices.
For a tree, we can pick one of the vertices and call it the root of the tree. Thus we define a rooted tree:
A rooted tree is called a rooted tree
For convenience, we also give the following definitions:
· Of the two endpoints of an edge, the node near the root is called the parent node of the other node.
· The node that is farther from the root in the two endpoints is called a child of the other node.
· Child nodes that do not have child nodes are called leaf nodes.
· The sum of the number of edges associated with a vertex is called the degree of that vertex.
With the above nouns, we can define the following two kinds of diagrams:
· A caterpillar chart (caterpillar tree) refers to each vertex of the diagram either on the central axis or only one edge from the central axis. Trees are caterpillar graphs if and only if all nodes greater than or equal to 3 degrees are surrounded by at most two nodes greater than or equal to 2 degrees.
· Lobster diagram (lobster tree) refers to the diagram where removing a leaf node leaves a caterpillar diagram. Because of its shape resembling a lobster, it got its name.
Caterpillar in mathematicians' eyes (top) vs lobster in mathematicians' eyes (bottom)
Image source: [5-6]
<h1 class= "pgc-h-arrow-right" > snack: custardly banana malagh chicken</h1>
After eating a meal, desserts after a meal are also essential. The first thing to introduce to you is the famous French dessert - blancmange.
Image credit: giphy
The above magic GIF also happens to reflect an important feature of French custard, self-similarity. Based on this property, we can speculate that french custard itself should have a parting structure. In 1901, Sadaharu Takagi, the founder of domain-like theory, gave a function.[7]
where s(x) is a trigonometric wave function, defined as
This is the famous French custard curve.
△ French custard curve, as the name suggests, is a curve that looks like French custard. Like other parting functions, the French custard curve is a function graph that is continuous everywhere but not different at all.
Image source: Homemade
Traditionally, French custards are made with milk or whipped cream with sugar, while almonds are added to add flavor. However, the improved dessert can be flavored by adding strawberries, bananas or other fruits. Speaking of bananas, it is a plant of the banana genus of the family Plantain family, and many delicious desserts are inseparable from bananas.
In mathematics, bananas can also be seen from time to time. For example, in mathematical optimization theory, the Rosenbrock function is also known as the "Rosenbrock banana function", which is often used to test the performance of the optimization algorithm. The Rosenbrock function is defined as[8]
The contours of the function are roughly parabolic, and their global minimum is located in a banana-type valley. This point is actually easy to find with the naked eye, but because the value in the valley does not change much, it is still difficult to find this minimum value through the algorithm.
△ Test gradient descent using rosenbrock function. where the red dot represents the global minimum value f(x,y) = 0, in which case (x,y) = (1,1).
Sweet and delicious snacks, salty and crispy snacks are also popular, such as Wheat Chicken.
△ The dipping sauce of McLe chicken has a lot of fans Image source: "Rick and Morty"
Mathematically, the coin problem (also known as the Frognius coin problem) has a variant known as the "McNugget number problem",[9] which also reflects the strong influence of the McNugget chicken.
Coin problems refer to the maximum amount that cannot be obtained by finding a coin that can only be obtained using a particular denomination. For example, the maximum amount that a $3 coin and a $5 coin can't get is $7 (so almost no central bank will issue a $3 coin). The maximum common divisor of the coin denomination is required here to be 1. Mathematical language can be described as , the positive integers a1 , a2 ,......,an and ( a1 , a2 ,......,an ) = 1 . Then for the set of positive integers {k1,k2,......,kn}, find k1a1 + k2a2 + ··· + knan can compose the largest of the numbers.
A variant of the coin problem is the so-called Wheat Chicken Number Problem. McDonald's is known to sell McDonald's in three sizes: 6, 9 and 20 (the 4 mcgons included in the "Happy Meal" are not considered here), and the number of mcGonagall chickens that can be composed of the above three specifications of mcgonah is called the mcgonah. The Problem of The Number of McLe Chickens refers to finding the maximum number of non-Mala chickens, i.e. the number of chicken nuggets that cannot be composed of the three sizes of Malak chickens mentioned above.
If we dig deeper, there are still many angles to the nature of the Wheat Chicken number problem that needs to be explored. For example, the existence of the maximum number of non-Malak chickens, the impact of the increase in the size of the Malak chicken on the maximum number of non-Malak chickens, and so on. However, for the problem at hand, we can solve it in a very intuitive way.
△Image source: homemade
In the figure above, the number marked in red is the non-Malle chicken number, note 44 ~ 49, these six connected numbers are all Mala chicken number, so the number greater than 44 is guaranteed to be the Number of Mala chicken. Therefore, the largest number of non-Malak chickens is 43, that is, when the number of McLe chicken nuggets exceeds 43, it must be composed of several boxes of 6 pieces, 9 pieces and 20 pieces of Maile chicken.
Mathematics, like gastronomy, has accompanied humanity through a long period of time. The beauty of food can only be experienced after tasting; the magic of mathematics can only be discovered after walking in. I hope that after reading this short article, you can also appreciate the commonalities between mathematics and food. Good food can live up to, and so does mathematics.
bibliography:
[1] Beyer, W. A., Zardecki, Andrew (2004), The early history of the ham sandwich theorem, American Mathematical Monthly, 111 (1): 58–61.
[2] Chihara, T.S. (1978), An Introduction to Orthogonal Polynomials, Gordon and Breach, Science Publishers.
[3] Upton, L. J. Problem 660. Mathematical Magazine. 1967, 40: 163.
[4] Mabry, R. and P. Deiermann (2009). Of Cheese and Crust: A Proof of the Pizza Conjecture and Other Tasty Results. American Mathematical Monthly. 116: 423–438.
[5] Harary, F. and Schwenk, A. J (1973). The Number of Caterpillars. Disc. Math. 6, 359-365.
[6] Mishra, D., Panigrahi, P. (2016). Some new classes of graceful Lobsters obtained from diameter four trees. Mathematica Bohemica, Vol. 135 (2010), No. 3, 257-278.
[7] Takagi, Teiji (1901), A Simple Example of the Continuous Function without Derivative, Proc. Phys.-Math. Soc. Jpn., 1: 176–177.
[8] Rosenbrock, H.H. (1960). An automatic method for finding the greatest or least value of a function. The Computer Journal. 3 (3): 175–184.
[9] Wah, Anita; Picciotto, Henri (1994). Lesson 5.8 Building-block Numbers. Algebra: Themes, Tools, Concepts. p. 186.
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Producer: Popular Science China
Production: Cast Snow Foreign Research Agency
Producer: Computer Network Information Center, Chinese Academy of Sciences