One of the results of Non-Euclidean geometry is the notion of curved space. A surface may have positive curvature (elliptical) like that of an orange skin, or negative curvature (hyperbolic) like that of a saddle. Cosmologists use these concepts to decide whether space itself is positively or negatively curved. This has a profound effect on whether the universe is open or closed, and on its eventual fate.
However, I wonder about the following:
A point of zero dimensions cannot be curved. A one dimensional line is either straight or curved. A two dimensional surface is either flat or curved, but the curvature, as described above, can be of two types -- positive or negative. These are distinct types of curvature that a line of one dimension cannot have, just as curvature itself is a property that a line can have, but that a point of zero dimensions cannot.
So now let's look at the progression. Zero dimensions have zero types of curvature; one dimension have one type of curvature; and two dimensions have two types of curvature. This seems to suggest that three dimensional space should, correspondingly, have three types of curvature (in four or more dimensions, of course).
I have no idea what these three types of curvature could be. One wouldn't simply be adding a third type to the already existing positive and negative curvatures, just as a second type of curvature wasn't simply added to the singular curvature of a line to make positive and negative. The three types would probably be more of a bifurcation (trifurcation?), where both positive and negative curvatures would fit into all three types.
I'm not sure if this idea even makes sense, but if there is any truth to it, I think it would be valuable to pursue. If cosmologists knew only about the singular nature of one dimensional curvature, they would miss the many profound implications that the dual nature of the curvature of surfaces admits. Might there be even more profound implications to a three-fold curvature of space?Return to the Loose Cannons -- Science subject list
The next time you are near a globe of the earth, try turning it upside down so that you can see it as though for the first time. When I tried this (just out of the blue, one day) something that I hadn't noticed before about the land masses immediately struck me.
I would have thought that, statistically, the shapes of the land masses would be random in orientation. Of course, I realize that they tend to fit together as a result of the breakup of the Pangean supercontinent, and that the Pacific Ocean is nearly a hemisphere in itself.
But what I noticed was this: many of the larger land masses narrow and come to a point (or at least are "pointier"), but incredibly, they nearly all point south! One would expect such characteristics to be random and point in all directions equally, but I was surprised at how pervasive this southward-pointing phenomena was.
Examples are North America (coming to a point in Central America), Greenland, South America, Africa (this is more rounded on the bottom, but still narrower than the bulging top of the continent), Saudi Arabia, India, Malaysia; even Europe itself along with Norway and Sweden sweep downward in a southwesterly fashion.
Many smaller examples can be seen in Florida, the Baja peninsula, Italy, Greece, the Kamchatka peninsula, and the Alaskan peninsula. The only major counter-examples are in Antarctica, where land masses must, of necessity, point northward, and the Cape York peninsula in Australia.
I have queried various geophysical sites on the internet about this observation, and have received responses that range from "interesting, but probably not significant," to "total accidental alignment." Perhaps, but it looks so distinctively unrandom.
Does anyone have any pointed remarks about this observation?Return to the Loose Cannons -- Science subject list
Both musical notes and visual colours arise from waves of particular frequencies, and this fact makes me interested in their relationship to each other.
A note one octave higher than another note has double the frequency of the lower note. When we hear a note that is an octave up the scale from another note, we recognize that it sounds the same, only "higher" in some sense.
This leads me to wonder what a colour an octave higher (double the frequency) than another colour would look like in comparison. Would it look the same, only "higher" in some sense? I looked into this a little, and discovered that our range of vision in terms of frequency of light we can see is almost exactly one octave. This would seem to explain why a spectrum of visible light forms a colour wheel so nicely. As one gets to the high violet end, it begins to look like, and merge back into the octave on the low red end.
There are various intervals of notes that sound pleasing to the ear. This makes me wonder if those intervals of colour would be pleasing, in some way, to see. How about a "perfect fifth" of colour? And how would it change as the "key" changed? What about "chords" of colours? Does a "minor chord" look different than a "major chord" (I assume that a minor chord of colour probably wouldn't look "sad")? How about chord progressions? Does the standard I-IV-V have a "look"? Does a scale of colours have any visual appeal to it? I can think of many questions along these lines, but I don't know how to go about answering them.Return to the Loose Cannons -- Science subject list
Time is thought of, especially in explanations of relativity, as a dimension in the "time-space continuum," representable on a graph by one of the axes. Motion is represented as a line whose slope is associated with the velocity.
What I notice in most discussions about time, is an attempt to take time from around us, and by analysis lay it down on the table in front of us to observe and study. I can't help but think that the thing in front of us is merely the shed snake-skin of time, while the snake itself slithers and coils behind us.
Relativity says that time slows down in a moving frame from the point of view of an observer in another frame; likewise, the observer in the first frame sees the other frame's time running slow. The conclusion is that neither is correct absolutely. Each is correct from his own point of view. I would not disagree with this. But surely the fact that the two observers can compare and measure time with each other means that there is something about time that is absolute and common between them. I realize this is perilously close to the concept of the ether that Einstien forever abolished (or at least rendered useless to talk about), but it seems to me that something fundamental about time is not being recognized.
Whenever we try to talk about time, the very attempt causes it to slip behind us again. Even the word "whenever" in the previous sentence presupposed time before I was able to speak about time. Thus I have immediatly lost something in my attempt to analyze the analysis of time (the same problem applies to the words "thus" and "immediately" that occurred earlier in this very sentence -- and also the word "earlier". . . oh well).
It occurs to me that this problem is really just a particular example of The Negative Inverse of Non-Existence, but the problem of talking about time comes up in many of my conversations, so I have listed it as a separate topic.Return to the Loose Cannons -- Science subject list
A rocket burns fuel to create the thrust that accelerates the rocket. As the fuel burns, the mass of the rocket decreases and thus the acceleration is itself accelerated. What I want to know is, how far would a rocket go that is composed entirely of solid fuel burning at a constant rate with a constant thrust? As the mass approaches zero, the acceleration approaches infinity, and finally, at some point in time the rocket disappears. I want to know the limit of the distance it traveled before it disappeared. My guess is that it would travel an inifinite distance (consider this an idealized mathematical problem, avoiding any consideration of discrete atomic or molecular limits); but it has been too long since college for me to do the integration.
A modification of the problem is to ask how the answer might change by taking relativistic effects into account. In that case the speed of light would limit the velocity and acceleration, but time would slow down and mass would increase and distances would be shortened (to outside observers, at least). How would the combination of these effects affect the answer?Return to the Loose Cannons -- Science subject list
When I was a kid, I wondered why there wasn't a big hole in the ground under or around large trees, since they must be turning dirt into tree material as they grew. Later, I realized that lots of water also goes into trees. Later still, I realized that carbon dioxide from the air is being used by trees also. Now my question is, what are the percentages by weight for an average full-grown plant, of the material it gathers from air, water, and nutrients from the soil?Return to the Loose Cannons -- Science subject list
General Relativity describes gravity (via matter) as distorting or curving space. The standard image is of a pool table with a stretchable rubber sheet as the playing surface with the billiard balls creating depressions in the sheet. The rubber sheet represents space, the balls represent mass, and the distortions in the sheet are the gravitational effects of the mass
I have wondered, though, if the Theory sees matter as units distinct from space. Or could it be that the distortions themselves are the matter? Or is it possible to even distinguish between these two cases?Return to the Loose Cannons -- Science subject list
If I mentally "hum" an ascending scale in my head without physically making a sound, I find that I can keep ascending indefinitely without reaching a threshhold. I know that I could never hear notes that high, let alone actually hum them, but my mind lets me "hear" them in my head anyway. I have realized that I must be somehow adding lower octaves in gradually so that they "take over" without me noticing. This makes me wonder about the following:
Imagine a "chord" that combines all the audible octaves of the note "C," with the intensities adjusted so that no single octave note stands out above the rest. Then create similar chords for the other six notes of the scale (D, E, F, G, A, and B). The C "chord" that follows will sound the same as the original C "chord," since it will have the same composition. Now if one keeps cycling through these seven chords, will there be a sense of ascending notes, yet somehow not really ascending? What will it sound like?Return to the Loose Cannons -- Science subject list
(Note: The idea for this subject and its proposed explanation was conceived by my wife, Angelee.)
If you look at a map of the United States, you will notice that, in general, states on the east coast are small, and that the states tend to get larger as you travel west. One possible explanation is that as time went on, advances in transportation allowed people to travel longer distances and still be "connected". As the westward progression advanced, this permitted larger and larger areas of land to become states.
But this is only a conjecture. Does anyone have any other ideas or accepted definitive explanations of this observation?Return to the Loose Cannons -- Science subject list
Several years ago, a friend named Mike, and I were at a planetarium show. The lecturer stood at a podium covered with complex-looking controls and knobs. We could see the giant ant-like projector (it has a name, but I can't think of it just now) in the center of the room that sprayed star images across the domed ceiling and traced the paths of the sun, moon, planets, and stars over any desired time frame and duration.
It was all very impressive, but Mike and I were most intrigued by one thing in particular. As the lecturer talked, a little arrow of light would point to the star or planet in question. Then, as the lecturer changed the subject, seemingly at random, the arrow of light would soar instantly across the sky-dome only to stop precisely at the image of the astronomical object now being discussed. This occurred many times throughout the lecture. Always, the arrow would zip, without wavering, to its intended spot, as if pre-programmed.
In the darkened room, with only the star images overhead and that little arrow of light visible, we both separately and gradually developed a profound respect for the amazing dexterity of this lecturer, whom we pictured in our minds as nimbly working the knobs that controled the arrow. This perceived talent was not unlike that of the boy in the old cartoon, who controlled the giant flying robot, Gigantor, with only a hand-held box sporting two joysticks; or perhaps like that of being able to quickly draw smooth, perfect circles or figure-eights with an Etch-a-Sketch. Instead, maybe there were a whole set of buttons, one for each star, like a large computer panel whose positions he had memorized for instant touch-recall in the darkened room. How many years of practice had gone into creating this lecturer's stunning abilities?
When the show was over and the lights came on, we walked over to where the lecturer stood. We discovered that he was holding a small flashlight with an arrow shaped opening (probably formed with masking tape) allowing the holder to easily project the arrow image anywhere on the ceiling.
We both still get a chuckle whenever we think about our overblown imaginings, and I have since wondered at this tendancy to over-estimate the technology involved in certain situations.
There was an episode of Candid Camera in which a man sat at a desk whose drawers would not all stay closed at the same time. When he would close one, another one would pop open. Again, one imagined a complicated internal mechanism of gears and pulleys interconnected between the backs of the drawers. When they revealed how the desk worked, it merely had flaps on rods that extended through the wall into the next room, so that a hidden operator could simply rotate the rods, Foosball style, to pop the drawers open at will.
Here is a yet another example of a mechanism for which I don't currently know the "simple" explanation: Female readers may be unfamiliar with the newer urinals that are becoming more frequent in men's restrooms. They have an automatic flushing feature whereby one just walks away and it flushes by itself. This feature is handy enough in itself, but the interesting part is that if one tries to fool it by not really "doing" anything and then walking away, it will not flush! How does it know?
The technology overkill method I picture in my mind is of some kind of advanced imaging device that sends an electronic message, something like "Image recorded and analysis completed. Positive identification of penis confirmed. Initiation of flushing procedure authourised." Of course I know that the mechanism must be much simpler, but I am not terribly inclined to poke my head in an examine it.
Readers of this page could make three types of contributions. First, of course, is the answer to how the automatic urinals work. Second, if there is not already a word to describe this tendency to ascribe complex mechanisms in place of simpler explanations, I would like to come up with a good candidate, not unlike the popular Sniglets. Third, I would like to record here any other examples, humourous or otherwise, that readers may have run across.Return to the Loose Cannons -- Science subject list