Can the sum of interior angles of a triangle be more than or less than 180 degrees?

September 19th, 2016
Can the sum of interior angles of a triangle be more than or less than 180 degrees?

What is the sum of the interior angles of a triangle? Most of people would answer 180 degrees, but physicists and mathematicians want to point out something with the question and answer. 

Normally, if you draw a triangle on a paper, then the sum of three angles must be 180 degrees. 

This is not an accurate description, but the geometry on a flat plane is called the Euclidean geometry. For instance, “parallel lines do not intersect" is one of the statements included in this geometry.

Now, what will happen if it is not on a flat plane? Namely, if you draw a figure on a curved surface, you may encounter some results which are not our commonsense. Such geometry is known as a non-Euclidean geometry. 

A typical example is a sphere. So let's imagine drawing a figure on the surface of the Earth. For humans’ point of view, it looks a flat plane on the Earth. However, it is a sphere macroscopically. 

Let's start with the North Pole A in the figure. Draw a line straight down to the equator. Then, change the direction at the intersection B with a right angle. Keep drawing a straight line along with the equator. The distance should be one forth of the circumference of the circle made with the equator until point C. Finally, change the direction with a right angle, and draw a straight line toward the North Pole as shown in the figure. 

You can see a large triangle on the Earth; then, look at each angle. The angles of B and C are 90 degrees, respectively. In this example, the angle A is also 90 degrees.

Therefore, the sum of the interior angles of the triangle becomes 270 degrees. In other words, the sum of the interior angles of a triangle in non-Euclidean geometry is not necessarily equal to 180 degrees. 

In fact, this geometry is used for Einstein's theory of general relativity. The mass of stars deforms the space, and light goes on the curved surface. Thus, you have to take the above discussion into account to describe such mechanism. 

Simply put, the general theory of relativity reinterprets the dynamics as the strength of gravity equivalent with the curvature of space. 

Speaking of relativity, it still sounds like a scientific fiction. However, the Global Positioning System (GPS) is an application of the theories of relativity, and GPS could not be made without such modern theories of physics.

What we should learn from the history of the US education; and how the future education should be done

July 28th, 2016
What we should learn from the history of the US education; and how the future education should be done

The US government had to be aware of the significance of science and engineering education especially after World War II. In 1957, the former Russia, Soviet Union, launched the first satellite, Sputnik, as their national project. The nation pride of US was damaged and the national security was also considered at risk then.

This situation forced people to think of improving science education. Thus, the US Department of Education was founded as an independent agency.

With tremendous amount of funding and a number of the projects, the consistent efforts on education resulted in the fact that human landed the moon in 1969 through 1972.

However, in 1980’s, there was a report of overall decline in the quality of science education. In addition, due to the Cold War, it was again considered as nation at risk for US national security.

The numerous projects were launched and they improved the higher education in the US into the world top level. However, the current state becomes much more complicated. 

Due to globalization of finance, industries, and work forces, a lot of resources are transferable among countries. In addition, energy and environmental issues give us completely different frameworks to consider a new management of countries and communities. 

It may be time to establish a different type of quality education based on science for the next generation which provides another chapter of security for human being and the world. Quality of education will not only enhance nation’s security, but will improve human’s mindset.

Physics Question and Answer: What is light?

July 25th, 2016
Physics Question and Answer: What is light?


How do you explain the nature of light from physics point of view? Why do the Sun and the Moon appear to be the same in size to us when you look at them from the Earth? Is it just an accident?


When you see it in a large scale, light is an electromagnetic wave, it travels through space as a transverse wave. So called white light looks white to naked human eyes because it contains all wavelengths of seven colors of a rainbow. We can only perceive colors in spite of the fact that there are various electromagnetic waves in the world. In fact, we are surrounded by invisible light that technology has made us able to use in everyday life. For example, remote-control, radio wave, mobile phone, etc. In a smaller scale, light acts as gathered particles. There is a machine with attached feathers that was designed to turn when light shines the feather. The machine is called the radiometer. In a much smaller scale, it has been known in physics to determine the energy of light, which is not based on the intensity of light; rather, it is determined by its wavelength. 

Also, the way the Sun and the Moon appear to be the same in size to us. Scientifically, we can only say that it is an accident; however, we can not exclude some unknown reason behind it. It is interesting to see that the Earth is the only planet within our solar system that has a satellite as big as the Moon. And if you'd notice, the Moon turns once in its axis while it revolves around the Earth in a month so that showing only the same surface towards the Earth at all time making other side hidden to see from the Earth.

If the distance between the Sun and the Earth had been slightly different, neither of human life nor any form of life on Earth could not have happened. 


Yasuo Koide & Hiro Shimoyama

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