Trefoil from 2005

Playing with Maple back in May 2005 I found this pretty trefoil which I encountered again today digging through some backup CD's looking for something else. Hope you enjoy it.

May Trefoil
(click on image to see it larger)

Squares on a grid

Over on Yahoo!Answers someone asked how many squares could be drawn on a 8 by 8 grid. I didn't have a clue so I wrote a program to generate them all and count as it went. The answer turned out to be 540. The image of all those squares is quite nice so I thought I would share it here:

Polyhedron from the inside.

Several people liked the swimming tadpoles, so here is another animated graphic. This one I created. It it is based on the snub dodecahedron. A stellated version of the entire polyhedron was projected from the center to a rectangle (like a Mercator projection of the earth), while the object rotated. Here it is:

Swimming Tadpoles

This is a test to see if tadpoles can swim in this blog.
But why are there 4 blank lines at the top?






End of test. It appears that they can. They were created by Patrick Snels whose other art and design work you can find by clicking here. If you can explain, or tell me how to remove the 4 blank lines above the image, please do so.

Rita's Dog Blog

Rita is our oldest dog and recently she and I had a heart-to-heart. It seems she has started her own blog here: http://rita314.wordpress.com. Her blog will include all dog-related stuff, especially dog-art. Actually this is good because it should free me up to write about other things, including Mexico, other art, geometry or whatever else strikes my fancy. Rita's most recent post is about portraits of dogs, click here to see it.

So, to make it very clear, there will be no more dog-art on this blog, instead it will (by mutual agreement) be on Rita's blog, see above.

Land of the Free

Today I learned a startling fact: 1 out of every 133 people in the United States of America is in prison, incarcerated, behind bars! This is the highest incarceration rate of any country in the world. By comparison Mexico imprisons 1 out of every 505 people. The incarceration rate of the United States is nearly 4 times higher than that of Mexico.

What does that say about the 'land of the free'?

You can view an interactive world map showing incarceration rates of different nations by clicking here. Sources are given on that graphic.

More Dog Art

In the labyrinths of lives and civilizations long past there were in Mexico and other parts of the Americas some who revered the dog. This first photo is a ceramic dog made perhaps 1900 years ago. I took the photo in a museum in Colima, Mexico in April of 2004. Probably it depicts a kind of hairless dog, known as the Techichi or Escuincle which is believed to be a relative of the Chihuahua and possibly also of the Mexican Hairless also called the Xoloitzcuintle. Apparently these dogs or their ceramic representations were sometimes buried with their masters, to guide on the long trip to another world.

Pre-Hispanic dog in Colima, Mexico

In Xalapa, capital of the state of Veracruz, in Mexico there is a wonderful museum of anthropology. I took the next two photos there in July, 2002. They are also prehispanic representations of dogs -- unfortunately I did not record the civilization or period they came from.

Ancient Mexican Dog

I wish I knew why this poor dog was hunched over and whether it was functional or ceremonial or decorative or what -- but I do not.

The next photo, also from the Museum of Anthropology in Xalapa, proves that the wheel was known in the Americas before Europeans came. Apparently this dog was a child's toy and so far as anyone knows the wheel was not used otherwise.

Prehispanic Mexican toy dog

On the edge of the city of Colima, in the state of Colima, Mexico is a wonderful statue of two dogs dancing. To me they seem to be kissing as well. Here is a photo I took of that statue in April, 2004:

Dancing Dogs -- Colima -- Mexico

These dogs are really quite large, maybe 6 or 8 feet tall, I wish I could remember better. They are modeled, I believe, after a much smaller archaeological find dating back to 150 AD approximately.

Back in August of 2003 I was in St. Louis for my daughter's wedding and had a chance to visit the St. Louis Art Museum. They allowed you to take no-flash-no-tripod photos then and I took a this picture of what I think is a well known painting of two sighthounds:

Two sighthounds. Artist = ???

Unfortunately I didn't write down the artist's name or the date of the painting. Please email me ( jbuddenh at gmail dot com) or comment if you have any information about this painting and I will update this blog entry.

The next image is a photo I took at the Texas Clay Festival in Gruene, Texas back in October, 2005. I don't remember the artist's name. Here it is:

Dog art at Texas Clay Festival, October, 2005
(ceramicist unknown )

By the way, I put 49 photos of the Texas Clay Festival of 2005 out on the internet and they are still there, just photos, no text. To see them click here.

In the year 2004 my wife and I drove across the United States and spent a couple of days in the Silver City, New Mexico area. We heard there would be a festival called "Fiesta de la Olla" in the small mountain town of Pinos Altos near the 7080 ft high continental divide north of Silver City. This festival celebrates (among other things) the wonderfully decorated Mexican pottery made in the village of Mata Ortiz, Chihuahua, Mexico. So we went to see the fiesta and I took some photos including this one which should give you an idea of the Mata Ortiz pottery:

Mata Ortiz pottery

Later we wandered a bit around the town and I came across this wonderful sculpture of a dog in someone's yard:

Folk Art, Pinos Altos, New Mexico (July 17, 2004)

Some of you, like me, may be old enough to remember when the Soviet Union shot off Sputnik 1 on October 4, 1957. This was the first artificial satellite to orbit the earth and it shocked the U.S. out of the complacent feeling that we were the most technologically sophisticated nation on earth. It precipitated changes in the teaching of science and mathematics and was surely one of reasons for John F. Kennedy's 1961 decision to put a man on the moon. You may also remember that Sputnik 1 was unmanned but that shortly thereafter on November 3, 1957, the Soviet Union put the first 'astronaut' into space and that astronaut was a dog named Laika (original name кудрявка). That poor dog met a terrible fate. At least we have this painting (which I snagged from some commercial site, if memory serves) to remember her by:

Laika the Space Dog

My next dog art image is by Irish printmaker Tim Mara(1948-1997) and is one of my favorites. One Tim Mara site said that images can be used 'for educational purposes only', so I hereby declare this an educational blog. Here is the image:

A print of Tim Mara

Last (and in some sense least :)) let me conclude this dog art post with a painting of the well known Colombian artist Fernando Botero (born 1932):

Dog Turning a Corner -- Fernando Botero--1980

This concludes today's posting on dog art. Since my collection has swollen to over 250 images, I reserve the right to someday post again on this topic.

Four dogs and a Cat

We used to have one dog and one cat, but now we have four dogs and a cat. Too many, and yet it has opened my eyes. I used to think one owner one dog. Not any more. Now I think dogs' personalities only blossom -- they only really become full dogs -- when they can socialize and play with other dogs, and curiously they become more interesting for their owners then too. Here are our four dogs:



Rita

This is Rita, a foundling of nine years ago from San Antonio, Texas. Click on the image to see it larger.





Happy

This is happy our sweet Mexican mutt just under one year old. She was dumped as a puppy in the market in Coatepec, but my wife rescued her. Click on the image to see it larger.






Cosi

This is Cosi our other lovable Mexican mutt about one year and four months old. A neighbor rescued her as a puppy and then gave her to us. Click on the image to see it larger.





Giaco

There was an incredibly sad weak street dog, so thin and weak we thought he would die. But we fed him outside our gate. The idea was that maybe we could get him healthy and then give him away. After about one week of care I took the picture you see on the left. The weather turned cold and we started to cover him at night -- then, finally, we started to let him sleep on the porch. We called him Giaco, pronounced "Jocko" and short for the famous sculptor Alberto Giacometti.







The sculpture by Alberto Giacometti, 1951, and our dog Giaco's namesake.





Giaco, healthy and happy again.

Giaco got to be a really nice and pretty dog. We kept him. He is our fourth dog. But enough is enough, don't bring puppies or needy dogs to our doorstep.

Click on image to see it larger.








Louie

And this is Louie, our cat a foundling from San Antonio, Texas, and now five years old.





And you may as well see a recent photo of Esther, my wife, and me with Pico de Orizaba looming in the background. For more about this picture click here.





Most of the above photos came either from Rita's flickr site or mine. At least one cast of the Giacometti dog is at MoMA and there is a good chance the image originated there. You can find more information about the sculpture at the Hirshhorn museum, click here.

My next post will be soon and will feature part two of dogs in art.

Squares and Wallpaper

One of the nice things about writing a blog is that you can write about whatever you want. So today a few words about wallpaper patterns and squares. I haven't forgotten that I promised to write more about The Great Swede. In due course, hopefully, I will.

Now for squares: back in the early 20th century some folks puzzled over whether it was possible to disect a square into smaller squares, each a different size. An easier problem would be to disect a rectangle into unequal squares. Fairly early on, in fact it was in the year 1903, the same year my parents were born, Max Dehn proved that it sufficed for the squares to have whole numbers for the lengths of their edges. Then in 1925 a fellow named Moron (OK not quite, make that Zbigniew Moroń(1904-1971)) came along and found a solution for the almost square rectangle of dimensions 32 by 33. Nowadays it is known that there are infinitely many different almost square rectangles (length minus width equals 1) which can dissected into unequal squares. Moroń's 33 by 32 example was the basis of a ceramic tile I built a few years ago. Here is a photo of it:

Ceramic tile based on Moroń's dissection

For a while it was suspected that such a dissection was impossible for a square. But in 1939 R. Sprague published the first example of a 'squared square' as such things came to be called. At nearly the same time, in the years 1936-8, four youngsters, then undergraduates at Trinity College, Cambridge became nearly obsessed with this problem. They found squared squares and developed a whole theory involving electrical networks for working with this and related problems, eventually published as: The dissection of rectangles into squares. R. L. Brooks, C. A. B. Smith, A. H. Stone and W. T. Tutte. Source: Duke Math. J. Volume 7, Number 1 (1940). These youngsters all went on to be well known in mathematics-related fields. A more thorough discussion of all this can be found here.

The examples of squared squares at this time were far from the simplest, and no one knew what the simplest example would be. Finally the simplest, a dissection of a square into 21 unequal squares, was found in March 1978 by AJW Duijvestijn, published as 'Simple perfect squared square of lowest order', J. Combin. Theory Ser. B 25 (1978), no. 2, 240-243. You can see a colored picture of it in my previous blog post called Hommage à A.J.W. Duijvestijn.

Recently I found and tried out some free recursive postscript software and made this image based on Duijvestijn's order 21 squared square:

OK, enough about squares. Now a tiny bit about wallpaper. Wallpaper often contains repeating patterns and mathematicians have studied these and determined that every repeating wallpaper pattern belongs to one of 17 possible types. To see examples of these 17 different types and much more information just click here. I just wanted to share a wallpaper pattern I recently created with free software called 'kali' which has been available on the internet at least since 1995. Here is my image, now on flickr (a larger version is here):