Jeff Yoshimi @jyoshimi@qoto.org

Last week I saw the Conway et al. book "The Symmetry of Things" for the first time. Its a beautiful book.

The book I coauthored, on a different topic, will be available this June. Its titled "The Open Dynamics of Braitenberg Vehicles". Something it has in common with "The Symmetry of Things" is that it is intended for readers with a wide variety of mathematical backgrounds. There is also a lot of geometry. I will be getting into topics from the book in future posts.

https://mitpress.mit.edu/9780262548199/the-open-dynamics-of-braitenberg-vehicles/

This April 17 through 19 Barnes & Noble is offering Rewards and Premium Members 25% to 35% off on all pre-orders. A couple of their websites are here:

https://www.barnesandnoble.com/w/the-open-dynamics-of-braitenberg-vehicles-scott-hotton/1144089449?ean=9780262548199

https://www.barnesandnoble.com/b/25-off-the-list-price-on-all-pre-order-books/ebooks-nook/_/N-2vsyZ8qa

In addition to Bernoulli duality there are many interesting pairings that involve hypotrochoids and epitrochoids. The hypotrochoid here is the same as the earlier one with wheel ratio 3/7 and arm ratio 8/9. A collective term for hypotrochoids and epitrochoids is centered trochoids.

https://en.wikipedia.org/wiki/Centered_trochoid

The rolling disks of the centered trochoids here have the same center for all time. Instead of the arms being attached at their tracing points they are attached at the common center of the disks. The tracing points meet instantaneously at the points of extremal radius and the curves are tangent to each other at these points. As with Bernoulli duality the arms are collinear when the tracing points are at the points of extremal radius.

Both disks complete the same number of revolutions when the tracing points reach their starting position. During this time the rolling disk for the epitrochoid spins 10 times.

It looks as though the 7 loops of the epitrochoid meet tangentially but actually there is a tiny gap at these places.

The joy of finding an amazing bookshop, spending hours browsing the shelves, discovering new books, carefully deciding which ones to take—and having the magic destroyed by someone yelling at you: “Who are you, and what are you doing in my house?”

#books

Evolution of Similarity-Biased Social Learning.

New preprint with Alejandro Perez Velilla. A long time in the making. Feedback welcome!
https://osf.io/preprints/socarxiv/j7yas

The next two videos show just one of the agents' hypotrochoid-like paths. A combination of geometric and numerical methods can be used to illustrate how a mechanism a little more complicated than for a hypotrochoid can better approximate the hypotrochoid-like path. At each moment in time there are 12 points along the path that have been selected numerically so that they can be be partitioned into the vertices of nearly equilateral polygons in two ways.

An equilateral polygon is one whose edges all have the same length. A regular polygon is an equilateral polygon with the same angle at all of its vertices. An equilateral triangle is necessarily regular. In contrast there are infinitely many shapes a rhombus can have. They can be distinguished by the ratio in the lengths of their diagonals. A square is a rhombus whose diagonals have the same length.

The 12 chosen points can be partition into the vertices of 4 nearly equilateral triangles and into the vertices of 3 nearly equilateral quadrilaterals. The coefficient of variation for the edge lengths within each triangle and within each quadrilateral is, on average, about 6%. The triangles do not change shape very much from the small variation in their edge lengths. The variation in the shape of the quadrilaterals is more noticeable because the diagonals alternately expand and contract.

The change in shape of the path from a hypotrochoid to the Braitenberg path can be largely accommodated in the two polygon partitions by allowing the diagonals of the square to oscillate in length. This causes the speed of the vertices to vary more dramatically.

Each quadrilateral acts like a 4 bar mechanical linkage. We can attach a vertex of a triangle to a vertex of a quadrilateral to build a larger linkage that can draw an approximation to the Braitenberg path. The two polygons counter rotate around their shared vertex.

In the video the polygon's shared vertex is colored red. Points have been chosen at each moment in time to approximate the centers of the polygons. These points along with the path's center have been colored cyan. At the beginning of the video the shared vertex is located at the point of maximal radius on the right side of the path. When the shared vertex returns to its initial position the center of the quadrilateral has wound around the center of the path 8 times and the common vertex has wound around the center of the quadrilateral 3 times. During this same time the center of the triangle has wound around the center of the path 3 times and the common vertex has wound around the center of the triangle 8 times.

When the shared vertex is at a point of extremal radius the 3 centers, and the shared vertex, are nearly collinear. When the shared vertex is at a point of maximal radius the triangle is inside the quadrilateral and when the shared vertex is at a point of minimal radius the triangle is outside of the quadrilateral. This is reminiscent of Bernoulli duality even though the paths made by the Braitenberg vehicles are the result of sensory feed back between the agents.

This video shows again how the two construction methods together resembles a mechanical differential. Their wheel ratios are reciprocals. Once the tracing point returns to its initial position the larger rolling disk has revolved 8 times around the center of the hypotrochoid and spun 3 times around its own center. And during this same time the triangles have revolved 3 times around the center of the hypotrochoid and spun 8 times around its own center.

The quadrilaterals' diagonals oscillate in a fairly precise fashion. The period of oscillation is the same for all 3 quadrilaterals but their phases are shifted. When the lengths of the diagonals of a quadrilateral have switched from their initial values the quadrilateral is nearly congruent to its initial shape. So it nearly returns to its initial shape twice during one period in its diagonals' oscillation.

In the video the quadrilaterals change color and pause each time they come close to returning to their initial shape. When the quadrilaterals reach their 1st return to their initial shape (about 2 seconds) they appear as if they had been simply rotated around the center of the path by 2/11 of a turn in the anticlockwise direction, but actually the vertices were cyclically permuted in the clockwise direction during the quadrilaterals' motion. When the quadrilaterals reach their 2nd return they appear as if they had completed 4/11 of a turn and their diagonals have returned to their initial lengths. The 4th time the quadrilaterals return to their initial shape they appear as if they had completed 8/11 of a turn with the vertices in the correct order. On the 11th return to their initial shape the quadrilaterals return to their initial positions, but with their vertices cyclically permuted. On the 44th return the vertices are back at their initial positions.

During the time it takes for the 12 vertices to return to their initial position there are 44 moments where the quadrilaterals look as if they were incrementally rotated by 2/11 of a turn. Altogether the quadrilaterals appear to have made 8 revolutions around the path's center even though their shapes were changing as they moved.

It turns out tipping is customary on the Moon.

The epitrochoid and peritrochoid construction methods taken together also resemble a kind of mechanical differential where one of the rolling wheels drives the other through the linkage made up of the two arms. As with hypotrochoids the turning ratio for an epitrochoid or peritrochoid is the number of times the rolling wheel revolves around the center of the fixed circle to the number of times the rolling disks spins around its center.

In this example once the tracing point has completed tracing over the entire curve the smaller rolling disk has revolved 3 times around the center of the fixed circle and spun 8 times around its own center. So its turning ratio is 3/8. At the same time the larger rolling disk has revolved 8 times around the center of the fixed circle and spun 3 times around its own center. So its turning ratio is 8/3. As with hypotrochoids the arm ratios and turning ratios for the two construction methods are reciprocals.

This video shows the two construction methods together. Again the two construction methods together resemble a kind of mechanical differential where one of the rolling wheels drives the other through the linkage made up of the two arms. The turning ratios are the same as before. Once the tracing point has completed tracing over the entire hypotrochoid the smaller rolling disk has revolved 3 times around the center of the hypotrochoid and spun 4 times around its center. So its turning ratio is 3/4. At the same time the larger rolling disk has revolved 4 times around the center of the hypotrochoid and spun 3 times around its center. So its turning ratio is 4/3.

Here is an example of a hypotrochoid to illustrate Bernoulli's double generation theorem. It shows how the same hypotrochoid can be generated in two different ways.

A hypotrochoid can be constructed by rolling a disk along the inside of a larger circle as shown in the attached picture and in the first video. The rolling disk is shown in gray. The gold colored line segment inside of the rolling disk is called the arm. We think of the arm as being attached to the rolling disk. One of its end points is positioned at the center of the rolling disk. It traces out a circle called the deferent (not shown). As the disk rolls the other end point of the arm traces out the hypotrochoid shown in sea green.

There are two numbers that are often used to characterize a hypotrochoid. They are the wheel ratio and the arm ratio. The wheel ratio is the ratio of the radius of the rolling disk to the radius of the fixed circle. The arm ratio is the ratio of the arm's length to the rolling disk's radius. Two hypotrochoids with the same wheel ratio and arm ratio are geometrically similar.

Hypotrochoids are often divided into three classes based on their arm ratios. If the arm ratio is less than 1 then the tracing point is in the interior of the rolling disk. If the arm ratio is 1 then the tracing point is on the circumference of the rolling disk. If the arm ratio is greater than 1 then the tracing point is outside of the rolling disk. A hypotrochoid with an arm ratio less than 1 is called a curtate hypotrochoid. A hypotrochoid with an arm ratio equal to 1 is called a hypocycloid. A hypotrochoid with an arm ratio greater than 1 is called a prolate hypotrochoid.

Wow, PLoS' first logo, from the year 2000!

In 2005, I joined the advisory board PLOS ONE and it started publishing in 2006. #scientificPublishing #openaccess

2022 was a tough year. Sorrows from the war, other painful losses, a broken wrist that never healed. But let me celebrate some successes. Somehow several publications came out and I am especially proud of them. The main one is this volume

Please bear with me as I learn basic #mastodon #etiquette 😆 I welcome tips on how to settle into this new (to me) community and help make it a nice place 🙂

if you're a public funder, or just love some hardworking scholar, wouldn't you feel wonderful giving the gift of prestige this Christmas?

It's not too late to get for Christmas a publication in Nature Communications for the low price of $5890! (On 1 January 2023, the APC will increase to $6290 / £4590 / €5190) https://www.nature.com/ncomms/open-access#apc