Based on a photograph of a sunset over a lake, Professor Robert J. Vanderbei of Princespine University proved that the Earth is round and gave the radius of the Earth.
Written by | JI Yang (Institute of Semiconductors, Chinese Academy of Sciences), GAO Guoqiang (Harbin Institute of Technology (Weihai))
Knowing that something can be made is the biggest hint. As long as you know that something can be made, even if you don't know exactly how it is done, many people can make it. In the history of scientific development, such things are not uncommon.
For example, Galileo Galilei invented the telescope. In fact, Galileo was not the first person to invent the telescope. In May 1609, Galileo Galilei heard by chance that someone in the Netherlands had invented a "magic mirror" that could see distant landscapes. Instead of visiting or buying this "magic mirror", he took matters into his own hands and, in less than 3 months, created two replicas of the instrument and used it to look up at the stars. He observed the moon's highlands and the shadows cast by craters, discovered sunspots, and discovered Jupiter's four largest moons. With the help of telescopes, Galileo also discovered the rings of Saturn, the rotation of the Sun, the waxing and waning of Venus and Mercury, the sun-moon and moon-to-day balances of the Moon, and the fact that the Milky Way is made up of countless stars.
The following example takes place in modern times.
On July 5, 2008, on the shores of Lake Michigan, Robert J. Vanderbei, a professor at Princeton University, took a photo of the sunset as the sun was about to set below the horizon, leaving a reflection on the lake's surface. Professor Van der Bick realized that using this photograph, combined with some secondary school physics and mathematics knowledge, it would be possible to prove that the earth is round, and that the radius of the earth could also be obtained.
Fig.1 Sunset over Lake Michigan, USA. Photo taken by Professor Van der Beigh of Princeton University on July 5, 2008.
01
Basics
The physics required here is that light travels in a straight line, and when light is reflected by a mirror, the angle of incidence is equal to the angle of reflection.
The mathematics required here are: the Pythagorean theorem, where the sum of the squares of two right-angled sides is equal to the square of the hypotenuse, and the sine theorem of triangles, which states that the length of the sides of a triangle is proportional to the sine of the opposite diagonal.
Here's the takeaway: The surface of the water is flat like a mirror and reflects the sun's rays. However, the "flatness" of the mirror or the surface of the water is partially flat, while the surface of the earth is curved, and you and I stand in two different places, and your "flatness" is "oblique" to me. For parallel incident sunlight, the inclination of the water surface relative to the sunlight is different in different places.
This analysis is straightforward, but somewhat cumbersome. Below we will write out the analysis process in detail, so that readers of "Mathematics, Physics and Chemistry for Middle School Students (Junior High School Edition)" can also understand. We encourage readers to try to deduce for themselves first, and if they get stuck, come back to our explanations. At the beginning, we only knew the conclusion, so we deduced it ourselves. The difficulty of this question is not in derivation, but in recognizing that this picture contains all the information needed to draw conclusions.
02
Models and derivations
The schematic diagram of our analysis (Fig. 2) is similar to that of Professor Van der Beech, except that for the sake of illustration, it is marked with some new symbols, and the process of analysis is slightly different. The details are as follows:
Fig.2 Schematic diagram of the sunset photograph over the lake.
03
data analysis
Prof. Van der Bey analyzed the data of the photographs (Figure 3): the diameter of the Sun is 317 pixels, the height of the Sun on the horizon
Fig.3 Data analysis of the sunset over the lake by Prof. Fan Debei.
The radius of the Earth can be obtained as
degree. However, this is a bit too precise, and this simple model doesn't have to be so precise, right?
04
discuss
In short, the Earth is so big that to know if it's round, or even to measure its radius, you need to travel a long way.
However, Professor Vanderbey of Princeton University recognized that it only takes a picture of a sunrise or sunset by the sea (or a very large lake) to prove that the Earth is round, and that the radius of the Earth can also be obtained. And the reason here is very simple, and middle school students can understand that the difficulty is that you first have to realize that you can get all this information based on this photo.
In hindsight, this analysis is very simple, but the first person to realize the truth is still very remarkable, and the key is that he thought about it seriously. Someone may have thought about it before, or even written it out. Important things are always discovered many times.
On the one hand, there is nothing new under the sun, and on the other hand, new concepts, new methods and new measuring instruments bring new understanding, and it is sometimes rewarding to think deeply about the phenomena that are "taken for granted" every day.
People should have long recognized that the earth is round, but it is always a little bit off.
For example, in the Spring and Autumn Period and the Warring States Period, we know: "Confucius climbed Dongshan and Xiaolu, climbed Mount Tai and made the world small" ("Mencius: Devoted to the Heart"); "I try to look forward to it, it is better to climb the heights of the broad-mindedness" ("Xunzi Persuasion"). Now it seems that what is behind this is obvious: ascending to the top has a wide view, because the horizon recedes. Of course, this is only a qualitative understanding, and Mencius and Xunzi did not infer that the earth is round.
There are also quantitative ones. In the chapter "On Heng and Speaking of the Day", Wang Chong of the Eastern Han Dynasty said: "Try to make a person walk the torch on the road at night, it is easy and riskless, go to people for ten miles, and the fire will be extinguished." "On the plain, a man walks at night with a torch, and when he goes about ten miles, you don't see the torch. There is a quantitative data of "ten miles" here, and the height of the torch from the ground is about one or two meters, so it is estimated that the diameter of the earth is about tens of thousands of miles. It's a pity that Wang Chong didn't recognize this, but used this as evidence for the "Gaitian Theory" to refute the "Hun Tian Theory" (which we now seem more reasonable).
Of course, there have been people who have long guessed that the earth is round, such as Shenzi and Zhang Heng in ancient China, and Pythagoras and Aristotle in ancient Greece. It is said that in the 3rd century BC, the ancient Greek astronomer Eratosthenes calculated the circumference of the Earth for the first time based on the sunlight shining at noon and the distance between the two observation sites. In 726 AD, the Tang Dynasty astronomer Yixing Zen Master presided over the national astronomical geodetic survey, using the height of the North Pole and the length of the day in summer to calculate the length of one degree of the meridian and the circumference of the earth.
This article may also answer a common question: Why can't textbooks be more detailed? The answer is, it could have been written in more detail, but the book became too thick.
Regarding the analysis of the sunset by the lake, there are many factors and details that have not been considered. Think about how they affect the outcome of the analysis.
For example, if you take such a photo on the same day and around the world, the results will be different, and even if you take such a photo in the same place on different days, the results will be different. Because the Earth revolves around the Sun (on the ecliptic plane), and the Earth's axis of rotation is not perpendicular to the ecliptic plane. So, if you take a picture of the same place for many days, or take a picture of many places on the same day, you can get a lot of information, including but not limited to: the earth is round, the radius of the earth, the central angle of the axis of rotation relative to the ecliptic plane, the latitude of the earth where the photo was taken, and so on.
Of course, you may also recognize that there are similarities to sundials. You can also get this kind of information by looking at the sun and shadowing the sun in the same place, observing it in the same place for a year, or recording many places on the same day. The ancients didn't have cameras, but they observed and thought.
05
A limerick poem
This book is divided into teachings:
Walking by the lake in the evening, suddenly a flash of inspiration,
There is a sun in the sky, and the water reflects the image.
From this we can see that the earth is a ball,
Using geometry knowledge, get the radius of the earth.
bibliography
[1] Robert J. Vanderbei, The earth is not flat: An analysis of a sunset photo, Optics & Photonics News (OPN), 34-39, November 2008.
[2] https://vanderbei.princeton.edu/FRS_131/lectures/lec3.pdf
This article is supported by the Science China Star Program
Producer: Science Popularization Department of China Association for Science and Technology
Producer: China Science and Technology Press Co., Ltd., Beijing Zhongke Xinghe Culture Media Co., Ltd
Special Reminder
1. Enter the "Boutique Column" at the bottom menu of the "Huipu" WeChat official account to view a series of popular science articles on different themes.
2. "Back to Park" provides the function of searching for articles by month. Follow the official account and reply to the four-digit year + month, such as "1903", to get the article index in March 2019, and so on.
Copyright Notice: Personal forwarding is welcome, and any form of media or institutions may not be reproduced and excerpted without authorization. For reprint authorization, please contact the background in the "Huipu" WeChat public account.