Wiggles, part 1

Just a little ongoing story to give you something to play with until the next blog post.



SMBC on Strong AI

Roger Penrose has no fear of this ever happening. I have my doubts, which is why I always keep an ethics guide in my bedstand.

Strong AI

Simple simplicity answer


This week’s problem: “Two or more spots are placed anywhere on a circle’s circumference. Every pair is joined by a straight line. Given n spots, what is the maximum number of regions into which the circle can be divided? The above figure shows the answers for 2, 3 and 4 spots. Find the answers for 5 and 6 spots, and if possible, the general formula.”

(Cut-up proof)

For points 2-4, the circle is divided into 2, 4 and 8 regions. It looks like the formula might be 2^(n-1). But, for 5 points you get 31 regions, not 32, and for 6 spots it’s 57 regions. The general formula is:
n + (n // 4) + ((n – 1 // 2),

where in this case the fractional parts (//) are treated as factorials for the combinations which may occur. “(It equals m!/k!(m-k)!)

Written out, it’s (n^4 – 6n^3 + 23n^2 – 18n + 24) / 24.
For 0 to 10 points, the sequence goes 1, 1, 2, 4, 8, 16, 31, 57, 99, 163, 256.

Today’s Saturday Morning Breakfast Cereal comic:

Science can solve many things. Ignorance is not one of them.

Printing Press Assembly Photos

My original intention was to run the instructions for building the printing press next week, after the review of the magazine. Instead, I’m uploading two blog entries in one day, today. If you haven’t read the one on the Newton magazine article on Penrose Tiles, click the “previous” button.

There’s about 40 pieces to the Gakken mini printing press kit, plus the 2 letter sets, ink bottle, screwdriver, and eye dropper. The only other tool you’ll need is a scissors or diagonal cutter for cutting the letters out of the mold frame and trimming off the flash. Figure 30-45 minutes for building the kit, and up to another hour for cutting out the letters. You may want to have a tray ready for holding the letters to avoid losing them; or take a sheet of foam core, rule a 9×9 grid on one side, punch holes at the intersections and use that for holding the letters in the same positions as in the mold frame. You don’t really need to check if you’re missing any pieces, but laying everything out flat on a table may make it easier to find the parts you need for each step.

(All rights belong to their owners. Instructions from the magazine used here for assembly purposes only.)

Ok, getting started. Get the left and right arms, plus one of the regular screws. Position both arms so the ends of the 2 cross beams for the right arm fit into the matching left arm cross beams, and hold them together by tightening down the screw inside the middle cross beam.

Take the left and right frame pieces, and fit them together as shown in the photo.
I messed up the numbering. Put screws in the cross beams marked #1 and #2, plus the unmarked one, and tighten them down snugly. Don’t put a screw in the cross beam marked #3 – you’ll need that free to squeeze the arms assembly in next.

Stand the frame up as shown in the photo. You can see the screws in place in what’s now the back side of the frame. Pull the cross beams at point #1 (numbering from the previous photo) apart just enough to let you slide the arms assembly into the frame. The two nubs on the ends of the arms will go into the matching wells in the frame. When you’re done, the arms should be able to rotate up and down inside the frame.

Now, put a regular screw into cross beam #1 of the frame and tighten that down snugly.

Take the letter tray holder (my kit had the the letter tray already in position in the holder; you can take the letter tray out if you like) and flip it over to be face down, with the tabs pointing to the frame. See the two little tabs on the frame the lower arrows are pointing at? Take the letter tray holder and hook the tabs at the end of the holder behind the tabs on the frame, and rotate the holder to an upright position. The screw holes at the back of the holder will line up with the matching tab holes on the frame (see below photo).

Like this.

Turn the frame around, and take two screws and screw them down into the tray holder until they’re snug.

Locate the three metal shafts, and the shaft board. You’ll need the shorter and middle-length shafts right now.

With the back of the frame facing you, position the shaftboard so that the two pins at the side point to your right. Put the far end of the shaft board inside the frame such that the short shaft will run through the middle hole of the frame, through the shaft board, and then out the other side of the frame, as shown above. (Note, I have the screwdriver propping up the shaftboard just to help me get a better photo.)

Turn the frame around again, and run the medium shaft through the holes in the middle of the arm assembly.

Now, this is a slightly tricky part. You want the two linkage arms and two retaining caps. If you play with the linkage arms, you’ll see that they can be folded into different shapes, some more useful than others. Also, the arms are marked “1” and “2”.

With the back of the frame facing you, put linkage 1 to the right side and linkage 2 to the left. Fold the arms to match the photos. I’m calling the two joints in the middle of the linkage “elbow 1” (e1) and “elbow 2” (e2). The end of the linkage that just has the hole is the “tail” (t), and the piece with the spring and rotation thing is the roller holder.

Turning the frame counterclockwise a little so you’re looking at the left side, take linkage 2 and hold it so that the tail is aimed down at the table, and the spring roller holder is kind of tucked in between the other two arms. Slide the linkage arm onto the medium and short shafts, with the medium shaft going through e1, and the short shaft fitting into the end cap of e2.

Turn the frame to get to the right side, and mount linkage arm 1 onto both shafts in the same way. Then take the two retaining caps and put them on either end of the medium shaft. You don’t have to push them on very far. Just enough that the linkage arm elbows don’t shimmy around too much as the linkage moves back and forth. You can adjust the retainers later when the press is fully assembled (see below photo).

Take two of the regular screws and put them in the screw holes of the mounting caps at the elbows “e2” of the linkage and tighten them down so the linkages are firmly secured to the short shaft. (Push the linkages together so they’re all the way on the short shaft before tightening the screws.)

These guys here (shown before being tightened down).

You now want the long metal rod, the stirrup, the stirrup shaft, the collar and the two remaining retainer caps. Slide the pin of the stirrup into the matching slot on the stirrup shaft (doesn’t matter which side of the stirrup faces forward or backward).

Slide the long metal shaft through the linkage tail (t) on the left side of the frame, through the collar (with the thicker disk end closest to the linkage arm), through the lower holes of the shaft board, then out the other side of the shaft board and through the tail (t) of the right-side linkage arm.

Hold the shaft in place by sliding a retainer cap onto the left end of the shaft.

The caps are going to look like this on the left side of the frame.

The stirrup shaft will then fit over the right end of the metal shaft, and the two plastic pins of the shaft board below. Hold the right side of the metal shaft in place with the other retainer end cap.

Notice that the tail of the right-side linkage arm is located between the shaft board and the stirrup shaft. You can push the back retainer caps so that they fit snugly on either side of the press frame.

Take the washer-head screw and use it to fix the stirrup shaft in place on the shaft board. Tighten it down snugly, but be careful to not strip out the plastic threads.

Like this.

Get the ink plate and set it face down. Take the stopper and attach it to the plate with a regular screw.

This way. Tighten down the screw until it is snug and the stopper doesn’t wiggle at all.

Almost done.

Turn the ink plate right side up, and position the two fingers at the front of the plate so that they fit under the top edge of the letter plate. Rotate the back edge of the ink plate downward.

Looking at the back of the frame, you’ll notice that the stopper kind of hits the top cross beam of the frame. The idea is that you’re going to put a piece of cloth over the ink plate when it comes time to start printing, and the edges of the cloth are going to be tucked under the side edges of the ink plate, so that when you push the plate down, the stopper will snap past the frame cross beam and hold the ink plate firmly in place, simultaneously trapping the cloth sheet in place. This will make more sense later, when you get ready to start printing. The point is that to clean the cloth afterward, you’re going to need to pull the back of the ink plate up, pulling the stopper off the cross beam and letting you fully remove the ink plate from the press.

For right now, push the ink plate down so the stopper snaps into place below the crossbeam, as shown in the photo above.

Find the roller, and the two roller end caps. Push the end caps firmly into place at either end of the roller.

Like this.

Push the stirrup down to bring the roller hand ends of the linkage arms up to where you can see them. It will help to have the hands rotated to be pointing up and to the back of the press as shown in the above photo. Snap the roller into place in the hand pieces.

Like this. Note that the roller end caps have two small spacers that slide back and forth on the roller shaft. Make sure those spacers are between the roller hand pieces and the roller itself. But really, there is only one way the roller will easily fit onto the linkage arm hands.

Rotate the hands forward, and now you’re ready to start printing. Push the stirrup down, and pull it up, to test the movement of the arms and the roller. If everything moves smoothly, the roller should roll back and forth across the ink plate, down over the letter plate, and to the bottom of the frame. If you keep pushing the stirrup down, the press will push against the letter plate. If you have letters in the plate, and a business card blank in place on the press, you’ll make an imprint on the card.

Find the card holder plate, the felt sheet and the two felt holder fingers.

Put the felt sheet in the indent in the card holder plate, and push the finger pieces into the holes at either side of the sheet to keep it in place. Note now that there’s a lip on the holder plate at one of the edges of the felt sheet. You’re going to set your business card blank on that lip when it comes time to do the printing, so that lip indicates “down” on the card holder plate.

Snap the card holder plate onto the medium-length rod at the front of the machine, with the lip at the “down” position.

Like this.

If the letter tray is already in the press, lift it out. Use a scissors or diagonal cutter to cut all the letters out of the mold frame. Remove any excess flash from the sides of the letters. Put the letters you want into the tray. When you’re done, slide the tray back into place in the tray holder plate.

And that’s it for the kit assembly portion. Again, pull and push the stirrup to check the roller movement across the ink plate and the letters, and that the card plate moves forward to press the card against the letter faces (the type). If necessary, loosen or tighten the retainer caps on the short and medium-length rods to prevent shimmying or avoid jamming of the arm linkages against the frame.

I haven’t had time to do any printing myself, so I don’t have instructions ready for that, yet. In the meantime:
The black “ink” that comes with the kit is actually a water-based paint. Pull the back of the ink plate up so the stopper comes off the crossbeam. Take the piece of white cloth and lay it flat on the ink plate, and tuck the edges under the plate. Put the ink plate back in place on the printer and snap it down again, keeping the cloth sheet flat on top. Use the eye dropper to wet the cloth, and squeeze ink onto the sheet. Push and pull the stirrup to run the roller over the ink until it’s evenly smeared across the sheet. This may take a couple minutes, and it may help to just push the roller against the cloth sheet directly with your hand. Put a blank card on the card holder, and push the stirrup down farther to ink the type. Run the roller over the type a few times to make sure the letters are evenly inked.  Finally, push the stirrup all the way down, to press the type against the blank card. Release the stirrup and check the card to see if it printed the way you want. If not, re-ink the type and press again, or maybe press a little harder. Don’t press so hard as to damage the press. Practice a few times until you get it right. If necessary, experiment with the amount of water you put on the ink cloth, to avoid the “ink” being too thick and too watery.

Direct akken youtube video link

To print a card using the same letters more than once, use the little extraction tool to push the letters out of the tray, and reposition them as needed. Print again, and repeat as necessary. When you’re done, unsnap the ink plate and remove the cloth sheet and the roller.  Soak the sheet and roller in a bowl of warm water to clean off the paint before it has a chance to dry.

Newton Science Magazine, Jan. 2018, part 1

(All rights belong to their owners. Images used here for review purposes only.)

Newton is Japan’s answer to Scientific Amercan, but with kind of a National Geographic vibe. It tends to focus on hard science with a softer, math-lite approach aimed at casual adult readers, accompanied by LOTS of graphics, photos and simplified illustrations. Past feature articles have been on finding a new dimension, how the brain works, and dinosaurs. The current issue covers AI and machine image recognition. Each issue also has a panoramic photo fold-out spread of various locations worldwide, and a few astronomical photos.

There are two articles I’d like to comment on from the Jan., 2018, issue. The first is a little overview of Foucault’s Pendulum. The article provides a brief summary of Leon Foucault’s life, and how he set up his pendulum in the Paris Pantheon to demonstrate that the Earth rotates. Most of the discussion of the mechanics mirrors what’s in the English wiki page.

To some extent, the reason for mentioning this article now is that it kind of addresses the rising influence of the Flat Earthers. If the Earth can be shown to rotate at different speeds at different latitudes, then it is a globe. Not that a hardcore flat earther is going to be swayed by logic, math, or physical proof. But still, the thought is there.

The idea is that if you have a sufficiently big pendulum that has a sufficiently heavy bob, then as the pendulum swings the Earth will rotate under it. Preferably, you want the bob to swing for at least a few hours to gather enough data points to work with. If the pendulum is at the North Pole (3-a), from the viewpoint of observers standing near the pendulum, the swinging pendulum will make a full 360 degree clockwise rotation in 24 hours. That’s a displacement of 360/24 = 15 degrees from the original starting path of the pendulum’s swing per hour. (At the South Pole, the rotation is the same but counterclockwise.) When you’re at the equator (3-b), there’s no rotation.

The math is: omega = 360 * sin(theta)/day

Where omega is the angular speed of the pendulum rotation in clockwise degrees per day, and theta is the latitude of the pendulum (North Pole is 90 degrees; equator is 0; South Pole is -90 degrees). Tokyo is at 35.69 degrees, so omega is 360*sin(35.69)/24 = 8.75 degrees per hour, and it would take the pendulum 41.1 hours to make a full clockwise rotation. (For comparison, at a latitude of 30 degrees, omega is 7.5 degrees per hour and a full rotation takes 2 days.

If the Earth were flat, or placed on a cylinder, the pendulum wouldn’t rotate. The only explanation for the changing values of omega at different latitudes is that the Earth is a globe, more or less.

Gakken Little Printer Kit

Finally, the Gakken website got updated (last update was for the scratch art kit in January). The new pages advertise the printer kit, showing examples of the cards you can print with it (basically, the little name cards you’d put on the dining table to show where people are to sit for a party), highlights from the magazine (the history of printing, examples of fancy printing by professional artists) and the downloads page (PDF of the assembly instructions (40 pieces total, including the screws and small bits), and operating instructions). 3,500 yen (approx. $32 USD) not including tax. The editors suggest a 30-minute assembly time, which may not include trimming the flash from the edges of all the letter blocks.

You only get two block sets, one for one each of the Japanese hiragana letters, and one for one set of the upper and lower case alphabet, plus numbers. So, unless there’s a way to by more block sets, or if you can make your own on a 3D printer, there’s going to be a very tight limit on what you can print with this. I’m assuming the idea is that you print in multiple passes.

The kit hits the shelves on mainland Japan on the 15th. Kagoshima won’t get them until 2 or 3 days later.

Answer for regression of the answer for the article on regression with the answer…

For the cross-stitch curve, how long is the final perimeter, and how large is the enclosed area?

If the cross-stitch is built to extend outward from the center of the square, the perimeter is infinite, but the area is twice that of the original square. If the stitch extends inward towards the center, the perimeter is still infinite, but the area goes to zero.


When I first started going through the Gardner Colossal book, I made notes of things that I wanted to follow up on, or include in the blog entries. One of which was David Cope’s EMI (Experiments in Musical Intelligence). I’m not sure I’ll be able to do much with this program in the short-term, so I wanted to put the link in the blog now, in case anyone wants to check it out.

Answer Rotations

Answers for rotation and reflection:
1) Oliver Lee, age 44, lives at 312 Main Street. He asked the city to give him the license plate 337-31770 for his car. Why?

Rotated 180 degrees, the plate reads “OLLIE LEE”.

2) A basket contains more than 6 eggs, which are either white or brown. If x is the number of white eggs and y is the number of brown eggs, then what values are needed such that the sum of x and y turned upside-down is the product of x and y? That is, how many eggs are there?

x = 9 and y = 9. x + y = 18. x * y = 81. Turn 18 upside-down and you get 81.

If the problem allowed any number of eggs, then x = 3 and y = 3 (6 and 9) would also work.

Colossal Gardner, ch. 13

Hypercubes. George Gamow and Martin Gardner both put a lot of effort into describing higher dimensions, since the idea of curved (or warped) space is tied to universe systems of 4 dimensions or more. While String Theory never shows up in Gardner’s book, there is one mention of M-Theory in the addendum for the Hyperspheres chapter, which speculates on 10 or 11 dimensions, most of them wrapped up in a really small space. If M-Theory ever pans out, it could relate gravity to the strong and weak forces, so there’s a reason to consider 4 dimensions and up kind of seriously. On the other hand, Gamow and Einstein argued that our universe is closed in such a way that a spaceship traveling in a straight line out away from our sun would eventually return from the opposite direction. Gamow’s discussions indicate that there might be a left-for-right mirror flip along the way, so that the pilot would be able to put a left-handed glove on his right hand (or that he might be upside-down compared to when he left).

Most of the discussions of hypersolids seem to come down to explaining what a tesseract is, and how it would look to 3D observers. If you’re not familiar with the idea, first take a point on a piece of paper. If you extend the point in one direction, you get a line segment – with 2 points, 1 line, 0 squares, 0 cubes and 0 tesseracts. If you take that line segment and extend it in a direction perpendicular to the segment, you get a square – with 4 points, 4 lines, 1 square, 0 cubes and 0 tesseracts. Extend the square in a direction normal to the plane and you get a cube – with 8 points, 12 lines, 6 squares, 1 cube and 0 tesseracts.

(Rotating tesseract from a 3D perspective, from the wiki article.)

The trick is to then extend the cube in a direction normal to the 3D space, which in effect makes 8 cubes that overlap without touching each other (there’s the original cube, 6 cubes extending from the 6 faces of the original cube, and the final cube you get when you stop pushing in the 4th direction) – with 16 points, 32 lines, 24 squares, 8 cubes and one tesseract.

You can calculate the number of faces, etc., for a hypercube of any dimension “n” by using the simple binomial (2x+1)^n. So, for a 4D cube, write out (2x + 1)(2x + 1)(2x + 1)(2x + 1):
(4x^2 + 2x + 2x + 1)*(4x^2 + 2x + 2x + 1)
(4x^2 + 4x + 1)*(4x^2 + 4x + 1)
16x^4 + 16x^3 + 4x^2 + 16x^3 + 16x^2 + 4x + 4x^2 + 4x + 1
16x^4 + 32x^3 + 24x^2 + 8x + 1
(where the x^4 coefficient gives the number of points, and the x^3 coefficient gives the number of squares, etc.)

To go to a 5D hypercube, just multiply the above equation by (2x + 1) again.

The rest of Gardner’s article is just an examination of what the hypercube would look like to us if we use 3D projections or slices. He goes on to give examples of hypercubes in art, including Dali’s Corpus Hypercubus, and Heinlein’s “-And He Built a Crooked House” short story.

(Dali’s Corpus Hypercubus, from the wiki entry.)

Gardner closes by saying that Heinlein’s idea of the house dropping out of 3D space could have a possible analogue in giant stars that undergo gravitational collapse, according to J. A. Wheeler. The density of a quasar could be great enough to curve space to the point where the mass drops out of space-time “releasing energy as it vanishes.” This might explain the enormous amount of energy emanating from quasi-stellar radio sources.

If the longest line that can fit into a unit square is the diagonal, with a length of sqrt(2), and the largest square that can fit into a unit cube has an area of 9/8 and a side of 3/4 * sqrt(2), then what’s the largest cube that can fit into a tesseract?

(All rights belong to their owners. Images used here for review purposes only.)