Featured Images

In Canada, for an extensive period of time, the game Monopoly (a trademark of Hasbro; then, of course, of Parker Brothers) was sold with relatively conventional wooden game pieces, rather than metal tokens. As this set of pieces is relatively obscure outside Canada (although for brief periods wooden pieces, some of which were similar, were used in the U.S.) I have pictured it here.

I believe this set of pieces was also used with sets for the game Clue, also a trademark of Hasbro at this time.

The above image has been split, with a part in black and white, to prevent fraudulent use in online auctions.

A little mathematical artwork I've come up with; I call it The Electric Dodecahedron.

I was a bit disappointed that no Archimedian solid includes both octagons and pentagons. And so, after some thinking, I came up with this shape:

Start with the 30-sided rhombic triacontahedron, and put an octagon (shown in green) on each of its diamond-shaped faces. Where three obtuse angles of the diamonds meet, an equilateral triangle (shown in yellow) is formed. Then, by suitably joining other corners of the octagons, a decagon (shown in red) with alternating long and short sides, still having pentagonal symmetry, is formed. The shape is less elegant than an Archimedian solid, as there are left-over isosceles triangles (shown in white) with narrow bases present as well. There are 30 octagons, 20 equilateral triangles, 12 decagons with alternating sides, and 60 narrow isosceles triangles in this shape. Since only the octagons and decagons are large shapes, this could form the basis of a somewhat imperfect 42-sided die, having pentagons and bilaterally-symmetric, but not regular, hexagons as faces. Perhaps if they ever make the Hitchhiker's Guide to the Galaxy into an RPG, they might use such a shape. Having seen this interesting site, however, I thought perhaps it might have already been done, but it seems not to have been.

The shape depicted, however, is indeed one that has been previously known; it is designated as I(*,1,4,e) in a paper on "symmetrohedra" by Craig S. Kaplan and George W. Hart.

Speaking of useful shapes for dice, since 7 times 20 is 140, and 5 times 12 is 60, and 140 plus 60 is 200, one could start from the shape of a soccer ball, following this principle:

and obtain another imperfect die, but one that it seems could be made quite close to a regular design, with 200 faces.

Here's an image of the graticule of a very clever circular slide rule for vector addition. Basically, take polar-coordinate graph paper, and bend it around into a circle... this is the Jensen aircraft computer.

This image shows how five sets of three great circles can be disposed symmetrically about the sphere, using the "soccer ball" Archimedian polyhedron as the base.

To the right is a picture of the cylinder of a steam engine. A valve lets steam in on one side of the cylinder, and lets spent steam out on the other side of the cylinder. Since the piston moves from one end of the cylinder to the other as it is pushed from one side, and then back again pushed from the other, the motion of the valve has to lead that of the piston by 90 degrees in phase.

If you remember pictures of steam turbines, you will recall that there are many layers of vanes, each one larger than the one before. This is to continue extracting energy from the incoming steam as it gradually loses pressure. Reciprocating steam engines, particularly when used on ships, could do the same thing by having three cylinders, in each of which the steam lost part of its heat and pressure, before the spent steam was pulled from the last one by the condenser. The picture at the right illustrates such an arrangement. The smaller the change in the pressure and temperature of the steam in each cylinder, the closer the cycle approaches reversibility, which is the condition for the maximum theoretical efficiency, that of the Carnot cycle.

All the cylinders must operate in the same phase, simply because as the wheel turns, the piston is at a position, and moves at a speed, proportional to a trigonometric function of the phase. The cylinders are illustrated at phases spaced by 120 degrees to show more of the operation of the engine.

But the real secret of how a steam engine works is not visible anywhere on the diagram at right, and usually is not mentioned in most short descriptions of how a steam engine works!

Steam engines on steam locomotives just let the spent steam out of a smokestack as they went along, and frequently obtained new supplies of water. But most steam engines recirculated the water that served as their working fluid.

If you have a hot boiler, out of which steam is coming at a high pressure, then that pressure is the same in all directions everywhere in the boiler! So how do you push the recirculated water back into the boiler?

The short answer is, with a pump. But the next question that might come to mind is how you avoid, then, using up all the energy your steam engine produced in trying to pump the water back in to where it came from.

And the secret of how a steam engine works is this: steam made from water takes up about 2,000 times as much volume as the water it came from. Since the amount of work required to cram something into an area at high pressure is proportional to its volume, because steam takes up so much more volume than water, pumping the water into the boiler only uses up a tiny amount of work compared to what the steam engine produces, even if the steam engine is only 4% efficient in the first place, as was true of the first modern steam engines as invented by James Watt! As the increasing size of the cylinders indicate, though, unlike water, steam is compressible; but even one part in 500 instead of one part in 2,000 is still inconsiderable enough.

A special kind of paper tape, having room for only two holes across, was used for recording Morse code. However, this tape came in two very different forms, which has led to some confusion.

On the top, the Wheatstone form of this tape is shown. This tape used a hole on the top to represent the start of a dot or dash, and a hole on the bottom to represent the end of a dot or dash. This kind of tape, moving at a uniform speed through an automatic sender, would produce dots and dashes in the ideal 1:3 ratio.

On the bottom, the form of tape used for sending messages in cable code is shown. Here, a dot is represented by a current in one direction, and a dash by a current in the opposite direction, or down another wire, each taking an equal time. In a sense, cable code isn't "really" Morse code, but it used the existing International Morse Code as its basis, rather than inventing a new code. When in use, this tape also moved at uniform speed through transmitting equipment, which made such equipment incompatible with any attempt to use it to transmit conventional Morse code.

Both tapes are shown with the name "Morse" encoded on them. Of course, Morse himself only invented the American Morse Code, not the International Morse Code; it might be possible to coax the latter from a Wheatstone device, but a code with different lengths of dashes is quite incompatible with cable code.

From the Book of Ezekiel, chapter I, verses 16 and 17:

The appearance of the wheels and their work was like unto the color of a beryl: and they four had one likeness: and their appearance and their work was as it were a wheel in the middle of a wheel.

When they went, they went upon their four sides: and they turned not when they went.

In a mechanical computer mouse, the ball that moves when the mouse is moved on a surface is within wheels that sense its vertical and horizontal movement; a mouse, unlike a toy truck, is not rotated when one changes from moving it up and down to moving it from side to side. Could the Prophet Ezekiel have been describing something based on a similar principle?

Of course, a normal wheel, rather than a ball, that can itself turn, being gimballed within a wheel, is another possibility; that which the wheel bears does not turn, but the wheel itself can.

And (at one time) finally, here is

the message encoded in the stones of the Great Pyramid: "The Pharaoh is cheap and wants a 15% discount whenever he buys cloth".

In the illustration below, first you see a line with a ? as the prompt, and then a BASIC statement with line number entered by the user.

This shows how a simple character-oriented display on a terminal works, with the screen divided up into a grid of characters, all of equal width, as would be made by an ordinary typewriter (as opposed, say, to a proportional-spacing IBM Executive typewriter).

Many terminals supported the APL language, but in general, those video terminals (as opposed to printing terminals, which could simply overstrike on the paper) that did so were designed by one computer manufacturer for its own computers, and had special characters representing the overstrikes that its dialect of APL used.

The next group of lines shows, however, that if a display had two pages of text memory, and a suitable character set (presumably, two copies of the ROM containing it would be needed), then the use of an OR gate could allow arbitrary overstrikes. The idea had been thought of, but there was not enough demand, apparently, for any terminal manufacturer to make such a terminal.

The third group of lines shows that I feel they were not imaginative enough. If one has two pages of text memory, were one to have a somewhat extravagant character set (including superscript characters, and italic characters in both normal and superscript forms) and overstrikes at an offset of one-half of the height of a line (a characteristic, in fact, of most typewriters, so that one could select 1 1/2 line spacing), then one could actually display, without going to a full graphical display as is used with today's graphical user interfaces, mathematical formulas in all their glory, as I show with the definition of a Bessel function, which allows me to show both a sum and an integral.

In the example shown, it is assumed that there are only superscript characters, and subscripts are achieved by using those characters in the row half a line down. This would conflict with the fraction bar when superscripts are used in the numerator expression, so that is not really a reasonable assumption.

Some character (rather than graphic) terminals, such as the HP 2645, did provide a special character set for displaying equations, but they did so with conventional character spacing.

Just in time for Hallowe'en: a trapezohedron whose faces are not trapeziums, but trapezoids - even if somewhat unsymmetrical ones:

Using only symmetrical trapezoids, one can construct a distorted version of a cube or rectangular prism with two identical faces and two parallel faces, and one can also join two such shapes by their largest face to form a 10-sided solid whose faces are symmetrical trapezoids. The diagram below shows the view of such a shape when looking directly at one of the two parallel faces:

Note how the two tilted trapezoids on the sides, although symmetrical, do not appear to be; thus, on the right side of the diagram, a geometric construction is shown as an aid in overcoming the optical illusion.

Actually, quite a bit of freedom is available in designing such a shape... allowing, for example, an 18-sided solid, or a 14-sided one as shown below.

And, incidentally, the base plan is the plan from the diagram above, just turned upside down (and with a slightly different width), but in this form, the optical illusion of asymmetry is even stronger.

And while I'm on the topic of H. P. Lovecraft: the state of Massachusetts is justly proud of being the home of Harvard University, in the city of Cambridge, Massachusetts. Harvard's rival, Yale, is located in New Haven, Connecticut. In many of his horror stories, H. P. Lovecraft, an author from Providence, in Rhode Island, adjacent to both Massachusetts and Connecticut, made reference to a ficticious institution of higher learning called Miskatonic University. The Housatonic river is the only river in the area with a similar name, and that river flows through the south-western corner of Massachussetts, and continues through Connecticut to the sea... its mouth being beside New Haven. However, Miskatonic University is located in the ficticious city of Arkham, Massachusetts, said to be modelled on Lovecraft's native Providence, Rhode Island by some, and on Salem, Massachusetts by others. In The Dunwich Horror, both Miskatonic University and Harvard (in the Widener Library) are said to posess copies of the Necronomicon. As it happens, Providence, Rhode Island is home to another Ivy League college, Brown University.

As for the other members of the Ivy League, Dartmouth College is in Hanover, New Hampshire; Columbia University is in New York City, Cornell University is in Ithaca, New York, Princeton University is in Princeton, New Jersey, and the University of Pennsylvania is in Philadelphia, Pennsylvania.