Calibration

Here is a warning for those who are considering owning  a 3D printer.

This is what it takes to calibrate the dual extruder on the Lulzbot TAZ5. Each of the small cubes takes over an hour to make, so this was around 2 weeks worth of work.

Not to speak of just extruding 100 mm of filament over and over again to make sure the measured extrusion rate was just right.

And printing post-calibration prints to get everything working just right:

And some finer adjustments on a longer print.

USA SEF

On April 16, 2016, a series of my 3D prints were featured at the SIAM booth at the USA Science and Engineering Festival. Here are a few photos and a movie of the nested platonic solids by Jack Love and I which was the feature puzzle of the exhibit.

Earning my engineering badge

Our very nice pull out sliding kitchen drawers have a weak spot in the brackets holding them at the back. 

I have just earned my engineering badge by designing a replacement bracket in OpenSCAD. 

 STL file in meshlab:

 Here it the bracket in action:

My first non mathematical design!

Slide rules and other calculation instruments

Slide rules and other calculation instruments

Curated by Evelyn Sander, George Mason University

Please come visit this exhibit on display at George Mason University in the math department display case in the atrium on the first floor of Exploratory Hall.

For centuries, the slide rule was the leading device used for multiplication, division, and other operations (though adding machines were separate devices). The use of these devices ended  in 1972 when Hewlett-Packard calculators became the one-in-one go to device for all four basic arithmetic operations. 

This exhibit includes slide rules and other non-electronic calculating devices that would have been an integral part of the daily life of any scientist, mathematician, or engineer prior to 1972. 

Timeline

• 1614 John Napier discovers the logarithm function
• 1620s Invention of the slide rule, credited to Edmund Gunter, Edmund Wingate, William Oughtred 
• 1845 Invention of specialized nautical slide rule
• 1930s Invention of specialized aviation slide rule, still sometimes used as a backup device
• 1969 Astronauts in moon landing brought slide rules for last minute calculations
• 1972 Hewlett-Packard calculator makes slide rules obsolete

How slide rules work

The logarithm function has a powerful property: it converts multiplication into addition and division into subtraction.

The slide rule is a clever set of two logarithm tables such that by carefully matching up the numbers, multiplication reduces to an addition problem. Specialty slide rules include other number tables, such as  trigonometry, fuel efficiency, or prices.

Items in our collection

Russian abacus

On loan from Mary Nelson

The abacus has been used for millennia, and unlike the rest of Europe and the United States, Russians continued to abaci for basic arithmetic, until Soviet production of electronic calculators in 1974. 

Circular slide rule

On loan from Thomas Wanner

A circular slide rule is smaller than a linear slide rule, thus fitting better in a pocket. 

Slide rule and belt holster

On loan from David Walnut

This slide rule fits into  a leather holster, worn on the belt. 

Cylindrical slide rule

Donated to the GMU Department of Mathematical Sciences

Keuffel & Esser Model 1741 Thacher Cylindrical Slide Rule

This premier instrument’s cylindrical design makes it possible to get accuracy equivalent to a 59 foot conventional linear slide rule.  The US patent is dated 1881. Device sold from 1883 from1950. (Its box is also depicted below.) 

Linear pocket slide rule

On loan from Karen Crossin

This is a classic slide rule of the type that an engineering student would be expected to carry. It was used by Howard Wheatley. 

Retail store slide rule

On loan from Donald Nadeau

A specialty slide rule from the 1930s,  used for calculating retail markups and other 

calculations for operating a retail business.  

Adding Machine

On loan from Melissa Talbot

An adding machine was used whenever adding rather than multiplication was required. This includes anything from sales to data collection. This Clary adding machine (circa 1950) was used by Larry Burslie to do taxes for the people of Fertile, MN until the mid-1970s.

Gaps in our collection

(which we would love to fill)

Astronaut Buzz Aldrin brought this slide rule to the moon for last minute calculations. Purchased by a  serious collector in 2007 for $77,675.00.

Astronaut Buzz Aldrin

Aviation slide rule

World War II Load Adjuster slide rule

For further reading see: 

http://www.npr.org/sections/ed/2014/10/22/356937347/the-slide-rule-a-computing-device-that-put-a-man-on-the-moon (NPR article on the history of slide rules and their current classroom uses)

http://www.oughtred.org (organization of slide rule collectors)

http://sliderulemuseum.com (images of slide rules from outside the collection come from this site) 

http://americanhistory.si.edu/collections/search/object/nmah_1131290 (this gives more information about the cylindrical slide rule)

The “Not-so-Center” of Mass

This post is by Student C, Math 493.

What is the “Center of Mass”? Consider a system of n objects in 1 dimension. Then the center of mass follows this set of relations. Let Xi be the position of the ith object in our one dimenion. Then the center of mass is located:

X = sum(m_i,x_i) / sum(m_i)

But what does this concept of center of mass actually represent?

The center of mass of an object should be obvious. Let's redefine the center of mass. The center of mass is a place where, if you put your finger on that spot, you should be able to balance the object on your finger. Similar to the bird below:

Notice how this is at a state of equilibrium when placing your finger at the center of mass.

So what is not the center of mass? Is that where I can place my finger on the object and have the object fall off my finger? That is, if I were to place my finger on the wing of the bird, in theory, it should fall off my finger. Many readers, one would hope, may have had experience with these birds. To which you would say that the bird would simply fall to the ground if you tried to balance it on anywhere but it's beak. The reason for this is simply because you are able to, experimentally, calculate the center of mass of the bird and place your finger in such a way that the bird balances on your finger.

With this in mind, let us try to expand this notion of where we can place our finger. What if I placed my finger completely off the bird. Could it balance? Could such an object even exist? One where it's center of mass is not on the object. Intuitavely, this makes no sense as there is no purchase for my finger to go in order for the center of mass to exist. In comes a very simple object:

Note here, that neither one of the objects are being placed at their center of masses. Or, these two objects are balancing on their not-so-center of masses. So it seems that the solution here was to add another object.

This then begs the question. Can more than two objects have this relationship? What about an arbitrarily large amount of objects? What about an infinite number of objects? The best answer to this question is by challenging infinity itself and asking another very deep-seeded question- is there a center of mass for the universe? And if so, where is it? Consider this “proof” by contradiction

Suppose for a moment that there was a center of the universe. Besides the fact that we would have to re-write all physics text books, what would happen if I placed my finger there? Am I balancing the universe? Can I move the universe? And if I don't place my finger there, is the universe in a state of unbalance? In this manner, we have reached a contradiction.

One might conjecture that the center of the universe would be at the “location” where the big bang originated. This is a closer approximation. The fallacy here, however, is assuming that the big bang occurred at a location. To think about how this is done. Consider trying to find, in 3-dimensions, a location for this big bang. Under the assumption that all objects are radially moving away from a central point, we can follow these paths back to the “origin”. However, the most surprising thing happens when you do this. Wherever you started, when doing this process, is where this process will end up. Coming to the conclusion that the center of mass of the universe is everywhere at once.

Citations:
Center of Mass equation - https://en.wikipedia.org/wiki/Center_of_mass

Bird photo - https://en.wikipedia.org/wiki/Center_of_mass

Motivation - http://map.gsfc.nasa.gov/universe/uni_shape.html

Appendix
For the wine bottle holster. The equations that were used was assuming that we had a uniform rectangular prism. That is:

xcm = sum(mx) / M

ycm = sum(my) / M

zcm = sum(mz) / M

Next, using the same equations, we removed the triangular prism and cylinder from the rectangular prism. This generates the center of mass of just the holster. Experimentally, one can find the center of mass of the wine bottle and adjust how much/little you remove from the rectangular prism to make the center of mass an equilibrium point.

Iterated Function Systems

The following prints of iterated function systems are on display at GMU.

Iterated function systems are a system of multiple affine functions. The limiting attractor of each iterated function system is a distinct fractal.  These prints show the process of iteration for a variety of iterated function systems in two and three dimensions. The bottom layer is a starting polygon, and each layer is a subsequent iterate. The top layer is a good approximation of the system's fractal attractor.