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Why does a Japanese Sword have a Curvature?
Curious: Seminars by Young Researchers

• What is the mathematical theory? How is the steel hardened? What is new in MRI? Three young Japanese researchers in Chicago presented curious seminars about their studies at the Japan Information Center of the Consulate General of Japan in Chicago on May 24.
• The researchers have formed the Japanese Researchers Crossing in Chicago to help each other and held meetings to share their studies. They also have invited people who are interested in learning new things.

• This time, Teruhisa Koshikawa, graduate student in the Department of Mathematics at the University of Chicago, spoke about “The theory of numbers and geometry”; Yukihiro Shingaki, Visiting Scholar of JFE Steel at the Northwestern University, spoke about “The steel around you – the study of steel”; and Keigo Kawaji, Scientist (Staff), at Cardiac MRI, spoke about “The new technology in MRI.”

The theory of numbers and geometry

• Teruhisa Koshikawa had the experience of participating in the International Mathematical Olympiad. He became interested in mathematics when he was in middle school and has continued to study it until today. He says, “My goal is simply to understand what we haven’t known yet.”

He explained Pell's equation and if it had infinite solutions. The answer is yes.
Pell’s equation is written as x2 – 2y2 = ±1.
When x=1 and y=1, its value is -1. When x=3 and y=2, its value is 1. When x=7 and y=5, its value is -1.
How can you find the numbers of x and y? The answer is (-1 + √2) n. When n is 2, its expression is -3 + 2√2. When n is 3, its expression is -7 + 5√2. So you can find the solutions of x and y.

How can we utilize Pell’s equation? Shimpo’s reporter struggled to find an example. Take a look at the gray and white area in figure 1. Which is bigger?
The answer is drawn from “x=7 and y=5”. Its value is -1; therefore, the gray area is smaller than the white area.

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The steel around you 
the study of steel

• Yukihiro Shingaki is leader of a research team in JEF Steel and recently has studied at Northwestern University. His specialized field is electrical steel. Iron and steel are common materials around us, but he said that there were many subjects to study them.

• Character and applicability of iron/steel

• The character of steel varies. For instance, a bridge cable can hold 18 tons of weight per one square centimeter. When a high-rise building is built, thick steel stocks are stronger. However they take the floor space. He said, “Thus, further study is necessary to make stronger, harder steels.”

How is steel made from iron?

• Iron ore is mixed with coke in a furnace to get rid of oxygen from iron ore, and then impure substances such as silicon and phosphorus are removed by inspiring oxygen. When a degree of purity reaches a certain percentage, iron is called steel. Thus iron and steel are not the same.

• How can the steel be hardened?

• Metallic crystals are bound by free electrons. Their bond is not strong; thus, they are able to move in a certain direction. Their movement is similar to cards, which slide in a horizontal way. For this reason, steel changes its shape.
• To make stronger steel, different elements are mixed with steel in order to prevent the crystal slides. Reducing crystal size is another way to make steel stronger and harder. When a crystal moves, it hits a neighboring one, so it cannot move further. Shingaki said that researchers could utilize those steel’s characters to make stronger steel.

Solid solution and precipitation

• Salt melts in water and makes salt water when the temperature is relatively high. When the temperature is lowered, salt reappears in the water.
• Steel is the same. When melted steel is cooled quickly, it becomes a state of solid solution because iron atoms cannot move. When melted steel is cooled slowly, atoms move and create bigger crystals. When the temperature is lower, atoms move shorter distances and make smaller crystals.
• Utilizing those characters, they can make different types of steel. For example, a type of steel can be easily processed, but it becomes hard after the work. Another type of steel becomes hard in a moment when it was impacted by something.

Hardening sword

• When hagane (steel) is heated, its body-centered cubic lattice structure becomes a face-centered cubic lattice structure. When hagane’s temperature is lowered slowly, the structure returns to body-centered cubic lattice structure. When heated hagane is cooled very quickly, the structure cannot return to the original state and tries to make a similar structure by creating a body-centered cubic lattice between two face-centered cubic lattices. Consequently, hagane is hardened, and its cubic volume is increased.
• This phenomenon creates the curve of a sword. The blade edge of a sword is thinner than the other side, so the blade is cooled quicker and becomes face-centered cubic lattice structure, which increases its volume; therefore, a curvature of the sword is naturally created.

What is electronic steel?

• When a coil is electrified, a magnetic north pole appears. When a bar of steel is inserted in the coil, magnetic energy is strengthened. Each atom of iron plays a role of a magnet, and when each crystal lines up the same direction, magnetic energy becomes very strong. Therefore, a very strong magnetic field can be created by a small amount of electricity, and it works very well for one direction. It is called oriented magnetic steel sheet.
• When different angles of crystals are arranged to spread evenly in a steel sheet, magnetic characteristic works for every direction. It is called non-oriented magnetic steel sheet.
• Oriented magnetic steel sheets are used in electric transformers, which changes voltage. Shingaki said that electric steel was very useful as an energy saving material which had little loss of electricity.

Figure 1
Which is bigger, gray or white area?
The answer is drawn by Pell's equation























To make stronger steel, different elements are mixed
with steel in order to prevent the crystal slides.


Body-centered cubic lattice

Face-centered cubic lattice

When heated hagane is cooled very quickly, it tries to make a similar structure by creating a body-centered cubic lattice between two face-centered cubic lattices.

Teruhisa Koshikawa

Yukihiro Shingaki