I came to Silicon Valley in 1972. One thing I remember clearly up to these days is the visit of a place called "Mystery Spot" at Santa Cruz in 1973. It has been almost 40 years, I still remember seeing some strange things & phenomena in that place. However, I knew at that time, the whole thing was like a magic show. You were awed at what you saw & wondered how they could do things like that. The tour guide tried to convince the visitors it was all due to some magnetic field, some aliens from outer space buried heavy metals beneath or some unknown force around this mysterious spot. But deep down, you knew it was like a magic show, a full show of illusion & entertainment. The place is still open today. One thing different is that with the advancement of science, they throw in some new terms like Dark Matter, Dark Energy, Torsion Field to impress the visitors. Fortunately they haven't got the Lee's gospel of "信息場". The whole scenario was interesting that I describe in the next paragraph.
In some weekend, nice & sunny, you longed for a relax afternoon & drove to Santa Cruz. After a nice lunch, sightseeing the wavy coast & some begonia garden, you came to a shop ordering some beer to quench your thirst. After that, you still got some time to kill. Ah, you noticed the board nearby enticing you to visit the "Mystery Spot". What a nice idea this was. You hopped into your car & drove to the spot. A tour guide greeted you in front of the entrance friendly. Within a few minutes, several other couples with kids joined you. The guide then briefed us this mystery spot, its history & super-natural event was going to happen. We were guided to walk in a narrow path, the slope is so tiny you didn't notice it. Meanwhile, you saw two rows of junipers or some pine trees grown along the path. Here the poem of 李白 applies " 綠竹入幽徑,青籮拂行衣". Pretty soon, you were in front of a little house with a capacity of 20 people. By this time, you were so relax & also with the help of beer drank before, you were ready to believe whatever the guide told you. Inside this little room, with window of course, the guide showed you all bunch of strange phenomena about gravity that included water flowing from low spot to high place. Also a guided marble moved from low-left corner of the window toward high-right corner. All kids were marveled at what they saw & parents awed with their eyes wide open. Whole thing was very magical & entertaining. Of course the guide showed us other attractions & spent a lot of time to emphasize the mysterious & super natural field around the spot not yet discovered yet by science. Time went fast while you had a good time. After one & half hour, you found that you were in front of the entrance again. All visitors were happy & believed what the guide had said. At this time, the poem of 李白 applies again "我醉君復樂,陶然共忘机". You totally forgot you were a "台大電机系高材生".
I found it educational, because I thought I knew all the tricks & the truth. Here is the trick. While you were guided into the narrow path, you actually walked into a slightly uphill slope that you didn't notice. The trees planted along the path were not vertical to the earth surface (horizontal plane). Instead it is perpendicular to the hill. It gave you the feeling you were not walking on the slope. The little house was built the same way, perpendicular to the hill. So when you were in front the house, you were already tricked into thinking you are leveled with the earth gravity. The angle of the slope was big enough to perform the tricks, but small enough for you not to sense you were actually on the slope. Here is the lesson, our eyes & ears are not good enough under certain circumstances like the one I just described. The figure 1 shows actual plan you are in. The figure 2 shows the plan you think you are in. You think point 'a' is the same level of point 'b' & so point c is lower than 'a'. Actually point 'c' is the same height as point 'a'. So if you pick any point d between b & c, the water will flow from d to a (since d>c & c=a). But you still think b is the same height as a & b higher than d, so a is higher than d or d is lower than a, ie water flows from low to high.
Now we go back to this Magnetic Hill in Canada. I visited Montreal, Quebec & the surrounding area 15 years ago. I heard this Magnetic Hill of Moncton. But I figured this had to be something like Mystery Spot in Santa Cruz. I rather stayed & spent the time walking around the Plains of Abraham in Quebec. This was the battle field between James Wolfe & Louis de Montcalm in 1759. The battle of Quebec was the turning point of the history. Since then, French were driven out of North America & it led to American Revolution indirectly. Quebec is the only walled city in America. The wall is well kept but not that impressive comparing to China's Great Wall. Both Wolfe & Montcalm died in this battle. There is an obelisk below the fortress near St. Lawrence River commemorating both of them.
I think the whole scenario of Magnetic Hill is similar to the Mystery Spot except that the little house becomes a long hilly pavement. In order to demo it well, I add figure 3 & 4 to show the possible plan of the situation. The road can be separated in three sections with three different slopes. The slopes are possibly all uphills with the middle slope less hilly.
Here comes the interesting part when we look at the website of 林星雄(LSH). Per his own description, he is 1963 NTUEE graduate, one year senior to us. His career & track record are impressive. However, in this matter of Magnetic Hill, he seems way off the tangent of normal engineering thinking. I know we studied General Physics from 許照 & Applied Mechanics from 翁通楹. Since I don't know the whereabouts of 許照, perhaps we should just visit 翁通楹 (台北市青田街) to see what he says. He may ask you whether you believe what the website says. If you say yes, I bet 9 out of 10, he will kick you out & says, "How could you believe such a thing?" First he (LSH) believes what he saw but not simple instruments like Bubble Leveling or plumbing lines. He perhaps forgot his ears (crude leveling device). Since he trusts his eyes but not the simple instruments & forgets his ears, he can't explain the strange phenomena of Magnetic Hill. He has to elicit the help of Dark Matter, Dark Energy or String Theory, M-Theory, Universal Hidden Dimension, Parallel Universe etc. I think these are all unnecessary red herrings (misleading clues). The phenomena is really easy to explain with the Newton's Theory. The trick from LSH is that he doesn't trust the instruments because he argued that the instruments are also affected by those Dark Matter or horizontal gravity stuffs. In other words, he wants you to prove there is nothing like dark matter, hidden dimension or horizontal gravity exist. But isn't it true that the burden of proof is on him since he claims all these wield stuffs have something to do with the Magnetic Hill? LSH trusts his eyes but his argument of point 'L' is lower than point 'S' is not all that convincing.
I add Figure 3 & 4 that may depict the scenario of the Magnetic Hill. The actual terrain is as Figure 3 shows & Figure 4 is the road people think they are in. They think L is the lowest point on the road but actually as Figure 3 shows it is actually higher than the starting point S. The trick is that the guide tells you to proceed or drive down to the lowest point L. They never says to coast down (with neutral transmission) to point L. I bet you will never be successful by coasting down to L as the case of bicycle & horse carriage experienced uphills described in the website. It seems an independent test from GPS should resolve the problem. Unfortunately, even the newest GPS is not accurately enough to a few feet of R coordinate. If a modern geographer can measure Himalayas raises its peak 2.4 inches every year, why can't they measure the exact geographical location to a few inch accuracy? I think we have the technology to do it. But since nobody is serious enough to tackle this kind of magic show problem, it is just not worthwhile spending the time & money on the illusion of Gravity Hills. If there are people still thinking the whole thing have something to do with some horizontal gravity field around the mysterious spot, I will tell them the following: Give me enough money, I can create something similar anywhere in the world (except oceans where people can detect the real horizontal plane right away). As a matter of fact, if our classmates would like to build one in Taiwan, we can make it happen & may make money from it since we haven't heard anything like it in Taiwan.
One easy way to prove the point 'S' actually is lower than point 'L' is the following as shown in Figure 3 (in green color):
1. At point S & L, use water bubble to find horizon & raise a pole of 12 meters high.
2. Pick a point of 8 meters from the pole base of S & shoot a green laser beam to 'D' of Pole L. Green laser has more range & better quality for this purpose. Just be sure the laser beam is parallel to the horizon.
3. Measure the distance (D) from the laser beam on Pole L to its base.
4. If D>8 meters, it means L is lower than S. Otherwise L is higher than S.
I agree with AhMo on his statement: They are all TRIVIAL in principle but can be highly bewildering and entertaining in practice. This is the postscript. I just wonder perhaps there are some inherent problems in our NTUEE education. We have prof Lee & now we have LSH. They are all brilliant in their field & yet get side tracked to pursue something phantom & mysterious. Finally I heard that 李家同 is our another controversial NTUEE graduate, 1961. I have no comment on him as I know very little about him.
ps: You can expand the figures to see them in details.
6 comments:
Why are almost all big objects in the universe round & spherical? The answer is the "Gravity". When a star forms, the heavier objects attract each other & gradually form a center that pulls down all other lighter objects. Since there is only one center, its natural shape is a sphere. Planets & satellites all behave the same. For 6 billion years, the earth has gone through up & down with land & ocean. But it always works to a shape of sphere due to the gravity. Gravity exists & works all the time. Since the earth is nearly a sphere, the best coordinate system is Spherical Coordinate (r,theta,phi). Actually the longitude (M) is theta & latitude (L) is phi. The r is the height plus the radius of the earth R. In daily usage, H (height) is used instead of r, ie r=R + H. So the GPS is the device that measures (M, L, H). Most GPS give you the reading, but you don't know the accuracy. In most cases, you will be happy if are around 20 meters to the place you are looking for. Meanwhile, we don't really care about H. You may ask what has this to do with Gravity. Here comes the important concept. Since (M,L,H) coordinate system uses the earth center as the reference point (origin), it is the center of the Gravity, ie gravity makes the whole thing meaningful. When we are in the outer space, M,L,H means nothing. Also the directions like east, west, south & north become meaningless. In fact, gravity gives H extra meaning that detaches from the definition of r coordinate. Assume the earth is a sphere with its mass concentrates on the spot of north pole. Now what will happen? Since everything will fall to the north pole, all points on the earth is higher than the north pole. Then what is the meaning of H? It is simply a geographical location (GPS measures) without the meaning of Height. The meaning of Height derives from the Gravity. Water flows from high to low. The high spot has more potential than the low spot. This concept of Potential (gravitational potential) derives from the Gravity. Without Gravity there will be no high or low Potential. From this concept, regardless of M,L,H, if the water flows from A to B, A has higher potential and so A is higher than B, period. In our universe, it happens that the gravity makes big celestial objects spherical and so the Gravity H & Geographical H coincide. So to be precise, if water flows from A to B, then A is higher, regardless what your eyes tell you it is low or high. This reminds us one other concept, Distance. What is the shortest distance between A & B? The answer is "Follow the Light!", not "Follow the Eyes". The path of the light is the shortest path per Relativity. This begs the next question, which point is higher? Perhaps one answer could be "Follow the water!". If you build a pipe between A & B as Light travels & find that water flows from A to B, then A is Higher. With this concept, we completely bypass the mystery of Gravity Hills as the visual effect plays no part of the matter. In other words, what you see really has nothing to do with High & Low.
Thank 陳哲俊 for the info about 台東水上流 that water goes up. It reminds me a nostalgic trip in Taida when we (洪敏弘,陳正一,蔡宗元,陳呈祿...)travelled around the island. We passed 台東 but didn't hear anything about this 台東水上流. If we spend more time searching, we will find something similar almost in any country. If we use the new definition of Height in my first comment "Follow the Water!", we completely bypass the illusion problem. But then the world may not be that interesting anymore. If we put Height aside, sometime the direction gets problem too. For example, in California, we go to the Far East(遠東) by going west. That was exactly what Columbus did in 1492. Here comes a good old question I put in our blog long time ago, but nobody bothers to answer. Again the question:
How many points on earth (a sphere) satisfy the following condition:
From point A, go south 1 mile, then go east one mile, then go north 1 mile, it returns to the original point A.
I thought that there is only one point on earth, i.e.,the north pole to satisfy Mark Lin's descrition.
A. J. Chen
I thought that there is only one point on earth, i.e.,the north pole to satisfy Mark Lin's description.
A. J. Chen
Some comments were communicated via emails. The one below was from 沈運申 on 09/14/2011.
This was a classic question which I saw during middle-school days.
The North Pole obviously fit the condition.
But there are an infinite number of points near the South Pole
which also fit.
For example, you may draw a circle of one mile around the South
Pole (which is about one-third of a mile north of the South
Pole). Any point one mile north of that circle will fit the condition:
go south one mile to get on that circle, go east one mile to get
back to the same spot, and go north one mile to Point A.
You may then set Point A closer to the South Pole. After going
south one mile, going east one mile will make two full circles
before returning to the same spot.
You may continue with this logic so that you can make three full
circles, four full circles, ad infinitum, until you are exactly
one mile north of the South Pole.
So there are an infinite number of points on earth that fit the
condition.
The next one was from 陳哲俊 on 09/14/2011.
Dear Vincent:
When you go around the circle once, the radius of the circle should be around 1/6
not 1/3 mile ( 1/2/3.14 ). Of course, when you go around the circl twice, the radius will be around 1/12 mile.
Cheers,
A. J. Chen
The next one was from 王澤霖 on 09/15/2011.
Very interesting!
Tse Lin Wang
The next one was from AhMo on 09/15/2011.
All,
Soooooo trivial a question that I do not even bother to answer. J
But, you young fellows missed the point, “on earth” is not “on earth surface” and “go south” can be defined in multiple ways, thus your 2-d solution is just a subset: one of the 2d-hyper surface imbedding in 3d.
Best regards,
AhMo
A simple question very often becomes an interesting question, not just trivial, but intriguing. This particular one involves some basic concept of Directions & the practicality of two dimensional usage in a three dimensional world.
As I mentioned in the previous comment, there is no direction defined in outer space. Perhaps there will be one defined someday later. However, up to now there is just no need to do it in a meaningful way. It is as if to define a unit called AhMoTum, the product of mass times wavelength of its color. It might mean something but it is just not that useful or meaningful. For the same reason, we define direction on the earth (not in the earth) as two dimensional plane like xy coordinates and so comes with north, south, east & west. Since in our daily use, the distance we travel is very short compared to the earth size. In navigation & world geography, we do define things differently. For instance, the city location of San Francisco is 122/25/5W + 37/46/30N. The unit is Longitude + Latitude. It is actually part of the spherical coordinate system (r, theta, phi) without r. Why? We assume the surface of the earth has a constant r to the center of the earth. So when we travel on an airplane, the screen shows Longitude, Latitude & Altitude (delta r). In almost all our application, the direction inside the earth has very little meaning even you can define whatever you like. In our interesting question, it follows the definition of geographical direction: with north pole in the upper position, east is to the right of longitude, west is left of longitude, north is to the upper position of latitude and south is to the lower position of latitude. With this definition, there is only one direction South on the North Pole, only one direction North on the South Pole. So the North Pole definitely is a point meets the requirement of the question. Vincent's answer is also correct with those circle (on the earth) with circumference equal to 1/n mile (n=1,2,3...) near south pole. Write it in equation, 2*pi*r *n = 1 mile, or r = 1/(2*pi*n). If R is the radius of the earth, d is the surface distance from the south pole to the circumference of the circle & theta is the angle spanned between the south pole & the circumference of the circle, then r = R*sin(theta) & d = R*theta. r/d = sin(theta)/theta. Since the radius of the earth is about 4,000 miles, r < 1 mile, the angle theta is approaching 0, and so sin(theta)/theta is approaching 1. So essentially r is almost equal to d, the distance from south pole to the points on the circle. From this fact, AJ is correct in stating that the (surface) distance starts from about 1/6 mile from the south pole instead of 1/3 mile, ie 1/6, 1/12, 1/18 ...
In summary, the combined answer from AJ & Vincent is the correct answer ie 1 + Infinity (1,2,3,...) * Infinity (number of points on a line). The next question is how big is this infinity. As we know there are several types of infinity. Cantor worked on Infinity & came up with notation Aleph(0), Aleph(1), Aleph(2) etc. He proved that the number of all integer corresponds to Aleph(0) and the number of all geometrical points on a line corresponds to Aleph(1). Since the number of points in a x-y plane is the same as the points on a line (can be proved by one-to-one correspondence), our answer the points on the north pole & many points near the plane of south pole, the sum is still the infinity Aleph(1).
According to modern mathematics, the number of all geometrical curves corresponding to Aleph(2). But nobody so far has been able to conceive any definite infinite collection of objects that can be described by Aleph(3). It sounds as if the three first infinite numbers are enough to count anything we can think of. We heard the people of Hottentot can only count 1,2,3 and many. It seems we are just as primitive as Hottentots since we can only count infinity from Aleph 0,1,2 & no more.
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