STEMantics

Service and Volunteer Work

Humanitarian Engineering in Koh Chraeng and Other Adventures in Cambodia

Grant Goldenberg January 28, 2018

As some of you might already know I recently returned from a design summit in Cambodia hosted by Engineers Without Borders (EWB) Australia. Getting back into the swing of life hasn’t been easy after such an incredible academic, cultural, and emotional experience. Even though my trip was merely a month ago, it feels like an eternity has passed, and there isn’t a day that goes by that I don’t think about the locals I met, friends I made, and memories that have impressed upon me lifelong lessons I will cherish forever.

My journey began with a sixteen and a half hour flight from JFK to Taipei where I waited a few hours for a four and a half hour flight to Phnom Penh International Airport. This lengthy trip in itself was a life experience for an inexperienced traveler such as myself. The combination of jetlag, time change, and flying halfway across the globe to a land with which I was unfamiliar was exhausting in itself.

Arriving blurry-eyed in Cambodia I was met by a few Australian university students and team leader of the design summit. It was here that I got my first real taste of Cambodia and the exceptional individuals with whom I would be working. With our luggage stacked in a motorcycle-driven rickshaw taxi (tuk-tuk), we squeezed ourselves into the remaining space and took a short drive to our hotel in Phnom Penh. On this eye-opening ride, I learned two things, if the Khmer (Cambodian) people were as kind and generous as I had heard, then the Australian university students I was working with were giving them a run for their money. Within approximately half an hour I had made about three lifelong friends who I was ecstatic to work with on the summit. The second thing I learned was how scary it was to drive in Phnom Penh. I had heard stories about the clogged streets of New Delhi and thick traffic of Hong Kong, but I wasn’t sure how it would compare to Phnom Penh. The roads were a free for all of pedal bikes, cars, motorcycles, and tuk-tuks. With no traffic lights, road lines, or seemingly minimum age to drive (it wasn’t uncommon to see young kids riding without helmets or parental supervision) you feared for your life at every intersection; the ultimate game of chicken.

A typical street in Phnom Penh lined with vendors and shops.

Upon arriving at the hotel, I met the whole of the design summit team. Here, I started to become somewhat used to their style of speech (there is a nickname for everything; swimsuits are “bathers,” sunglasses are “sunnies,” flip-flops are “thongs,” etc.). We met our roommates, grouped up, and explored the city. Phnom Penh is hard to picture but if I had to describe it would be best characterized as a combination of the winding streets, buildings, and vendors of Cairo, mixed with the traffic of New Delhi, and pungent smells of New York City.

At first, I must admit I was taken aback at the deteriorating state of some of the buildings, roadways, and markets. But, as with most things, a quick history lesson of Cambodia and its government put this all into perspective. Phnom Penh is by no means a perfect city. Though it boasts some of the kindest and most generous people I have ever met, markets that offer everything from Yeezy knockoffs to wooden and marble chess sets, fantastic cuisine, all combined in a scenic setting, it definitely falls short on a few fronts. For one, there are places where raw sewage runs in large open sewers around the city. The country suffers from contaminated tap water which must be boiled. Western toilets are not very common across the nation and air-conditioning is a luxury that many can’t afford. Furthermore, illnesses such as malaria and dengue are all too familiar in cities such as Phnom Penh. Despite these critical issues you see many people using cell phones, watching online videos, or using Facebook. A naive foreigner such as myself thought some of the sanitation issues would take priority. How could necessities such as clean tap water and closed sewers be less important than Facebook?

A few critical experiences during my trip cleared up this confusion. The first was my trips to S-21 and the Killing Fields. A trip to Cambodia would not be complete without paying respects to the country’s dark past. During the 1970’s, Cambodia was shaken to its core by extreme changes in its political stance due to the Communist Party of Kampuchea, more commonly known as the Khmer Rouge. With the Vietnam War raging next door and a half million tons of U.S bombs being dropped on the country since the Viet Kong’s Ho Chi Minh trail cut into Cambodian territory, anti-Western sentiments developed among the Khmer population. This animosity would eventually lead to the demise of the pro-American government under Marshal Lon Nol and put the Khmer Rouge’s Pol Pot (short for “political potential”) in power. Almost immediately labor camps and prisons, such as the Killing Fields and S-21, respectively, were constructed at the beginning of the ill-fated Khmer Civil War. Genocide and human rights atrocities ensued which saw intellectuals such as teachers, doctors, politicians, etc. physically abused, and tortured. Unfortunately, the majority were executed. This mass “cleansing” aimed at producing a rural agrarian nation, would end up brutally exterminating almost a third of the Khmer population.

My trips to S-21 and the Killing Fields revealed the destruction and evil perpetrated in graphic detail. At S-21 we visited the cells of former laborers who were cuffed and restrained to iron grid beds often being starved and left to die. Many of the cots from the cells remain today mangled and bent with blood and decomposing stains beneath showing the struggle and suffering of these innocent civilians. In the next building, black and white mug shots line the wall documenting just some of who had been lost. Walking through these halls, I was deeply touched to the point where at one point I couldn’t speak. Yet one image among the thousands reduced me to tears. It was an image of a young doctor. The picture showed him smiling in his cap and gown just having graduated medical school and excited to begin a career in medicine. This bright young mind was unfortunately brutally exterminated by the country that had produced and educated him all because he was an intellectual hoping to attend to the illnesses of his people.

The Killing Fields only furthered my understanding of the tragedy that was perpetrated by the genocide. As I walked the trails that cut through the Killing Fields, I was surprised to see cloth sticking from the ground. In the beginning, I thought it to be the woven cloth that covers the roots of trees or is planted beneath newly placed sod and grass seed. To my dismay, I learned that these were in fact blindfolds sticking out of the Earth. When laborers were executed at the Killing Fields, they were blindfolded and hacked with machetes. In the words of our tour guide, “a quick death by a bullet was too expensive.”

I learned that when the Khmer Rouge came to power, they evacuated Phnom Penh allowing its infrastructure to crumble. Upon the end of the civil war when the Khmer People returned, the city lay in tatters. To make matters worse, Cambodia’s contemporary legislative issues have not been entirely resolved. Although a President is now in power, he is seen as unbelievably corrupt in the eyes of Cambodia’s citizens. Freedom of speech is not seen as acceptable in the eyes of the government, and thus the people’s inner thoughts remain silent.

Due to the genocidal killing of the majority of intellectuals in the country, education has suffered dramatically resulting in an economy mainly dependent on farming and small businesses and allowing more significant issues such as infrastructure to fall to the wayside. As a result, it’s not uncommon to see areas where crumbling roads, buildings, and old sewers remain in use, all too costly to repair remnants of the war. Yet despite this misfortune, the Khmer people are an unbelievably resilient and generous people, always smiling and willing to share what they have.

As engineers, we are trained to identify problems and then design and construct thoughtful sometimes ingenious, methods and solutions to solve them. In Cambodia, there are many infrastructural and economic problems which could be solved with high-tech solutions, but these solutions are only worthwhile so long as they can be maintained by the people who use them. Thus, the idea of humanitarian engineering; seeking solutions that are appropriate for the local population and their circumstances. Understanding the perspectives and needs of the local people was the focus of what we were after. We understood that the problem that was to be addressed must be relevant to those who suffer from it, and the solution must be maintainable by these individuals.

After a week of workshops and preparation in Phnom Penh focusing and reviewing ideas critical to humanitarian design, we split off into two big groups and headed for Kratié Province. Here we stayed on a small island in the Mekong River known as Koh Chraeng. Groups of twos and threes were selected and assigned a homestay on this rural agrarian island, the beauty of which is awe-inspiring and at times seems unreal. We stayed in small one-room homes elevated on large wooden supports which had thatched or wooden walls, a planked floor similar in consistency to a wooden deck. The floor had small planks and large spaces between planks for passive cooling, while the rooves were constructed of clay tile. Though definitely minimalist in style, these homes were very cozy and featured a small open kitchen, often a hammock, and even electricity using a car battery powered by a solar panel.

Team Ibis

During our time in the homestay, we attempted to talk with our homestay families in Khmer. During this period we conducted interviews with farmers, school teachers, residents, and the village chief to gain a deeper understanding of the issues that produced problems for them in everyday life and were important to the local residents. Eventually, we identified problems and then presented our ideas and designs to the residents and listened to their opinions, thoughts, and criticisms.

My homestay family and group.

My group decided to focus on storage issues relating to the construction and design of the locals’ homes. After many interviews with the villagers, we discovered a significant problem concerning storage of crops during the wet season. During the wet season, it is common that severe flooding occurs thus requiring elevated, living spaces. However, we learned that there was no designated space allocated for crops during the wet season often causing families to store massive amounts of produce (sometimes up to a couple feet of crops) in their homes. This prompted one such family to sleep outdoors during the wet season, a dangerous endeavor considering the mosquitoes, scorpions, and other animals and insects that are part of the local environment.

The interior of a raised home in Koh Chraeng.

Our team came up with a design primarily constructed of bamboo, a resource abundant on the island, to create a raisable pallet system. The design consists of bamboo planks laid across a square wooden frame (imitating the passive cooling design of the floor to aerate crops) with pullies attached to the sides. Each corner has a protruding wooden guide that cuts into notches on the wooden supports of the home. This allows for a customizable, raisable pallet system attached to the floor of the house that boasts stability and can be tied off at any height. Also due to the large number of supports beneath the home, many pallets can be constructed beneath the floor, increasing the surface area for storage. Furthermore, pallets can also be hung from other pallets producing a drawer-like system. We presented the design to local residents and leaders who were intrigued with the apparatus and our storage solution.

Raisable pallet storage solution (constructed in SolidWorks by Alvin Tsang).

My trip to Cambodia with EWB Australia was a life-changing experience that I shall cherish forever and undoubtedly will never forget. Not only did it open my eyes to what others struggle with on a daily basis, it gave me the opportunity to help the Khmer people through the application of humanitarian engineering principles. The experience provided a great opportunity to learn how powerful engineering can be and how its focus is the betterment of people and their lives.

I would like to express special thanks to EWB Australia for giving me the opportunity to take part in their summit. It is uncommon for the program to take international students, so I am genuinely honored they took a chance on me. Pictures of my trip can be found below. Photo creds go to Chelsea Jane, Ali Arman Khurshid, Logan Spiers, and Antony Maubach.

Mathematics

A History of the Moving Sofa Problem

Grant Goldenberg August 22, 2017

Abstract

Mathematician Leo Moser proposed the following question in 1966, “What is the largest area region which can be moved around a right-angled corner in a “hallway” of width one?” Such a simple question would appear to have a fairly straightforward solution. However, as of 2017 the maximum area capable of translating the corner of width one unit has not been confirmed. It’s simple yet unbelievably complex nature has attracted many mathematicians such as Daniel Romik, Joseph Gerver, John Hammersley, Phillip Gibbs, Kiyoshi Maruyama, and Nicole Song. The goal of this post is not to present any new solutions but to consolidate and distill the existing solutions into a single reference so those interested in the problem do not have to toil reading dozens of papers. However, if the reader is interested in understanding specific concepts about a particular solution, I highly recommend refferring to the literature that corresponds with the solution in which you are interested. Sources will be cited at the end of the post making it easy to find the literature that corresponds with your solution of interest.

Introduction

Conditions of the Problem

There are a few basic conditions that must apply to shapes of area $T$ tested within the described hallway associated with Leo Moser’s original dilemma:

An area proposed to be a sofa $T$ , must be a bounded region within a plane whose boundary is a closed curve.

Figure 1: Labeled unit width hallway (N. Song)
The $90 \textdegree$ hallway is the set of points $(l,w)$ defined in such a manner that $l\leq1$ and $w\leq1$ and such that either $l\geq0$ or $w\geq0$ . $(l,w)$ will be used to describe points within the hallway or to refer to coordinates associated with hallway. Thus, $(l,w)$ will be named hallway coordinates. The region that we call a hallway is a region bounded by the union of four rays: $l=0$ , $l=1$ , $w=0$ , and $w=1$ . If we are to construct a traditional hallway, $w=1$ refers to the top wall, $w=0$ , is the bottom wall, $l=1$ is the right wall, and $l=0$ is the left wall. The origin $(0,0)$ describes the corner found within the hallway.
Sofa $T$ must have an axis of symmetry $S$ .
When sofa $T$ is not in the corner, axis of symmetry $S$ may be oriented perpendicular to the direction of the hallway.
Figure 2: Axis of symmetry coincident with $y$ -axis (L. Moser)
As sofa $T$ moves around the corner, the axis of symmetry $S$ becomes coincident with the line through the inner and outer vertices of the corner, the $y$ -axis.

Some Basic Sofas

The Unit Square

Before looking into the more complex sofas associated with the moving sofa problem it is important to recognize some of the most basic sofas that can be used to solve the problem, as they will provide some key intuitions when describing later sofas. As described previously we have already described the hallway in which the sofa resides using four infinite rays. However, we should notate this mathematically. The hallway can be described using the following:

$L_l = \{ (x,y) \} \in \Bbb{R}^2 : x \leq 1 , 0 \leq y \leq 1 \} \\ L_w = \{ (x,y) \} \in \Bbb{R}^2 : x \leq 1 , 0 \leq y \leq 1 \} \\ L = L_l \cup L_w$

Now that we have our hallway in mathematical notation let’s start playing around with some sofas. The most obvious sofa to start with is the unit square. Let’s say we put our unit square in the corridor of one of the hallways. According to the definitions of our hallway $L$ , the max length of our sofa can be is $1$ , otherwise, it will not be capable of traversing the corner of hallway $L$ . We can also determine that the max height is $1$ otherwise it would exceed the defined height of the corridor. Note our unit square meets all the conditions sets earlier: it is bounded by a region, fits within the confines of our hallway, including when it switches from translating horizontally to vertically in the corner, has multiple axis of symmetry, is perpendicular to the corridor at all times, and has an axis of symmetry $S$ that becomes coincident with the line through the inner and outer vertices of the corner in our hallway. Now that it has been confirmed that it fits the requirements of a sofa lets measure its area. We know that in a square $l=w=s$ . Indeed both $l$ and $w$ equal one. Thus, if $s=1$ , and the area of a square is $s^2$ than $A_s=1$ .

Figure 3: Unit square sofa (D. Romik)

Figure 4: Unit width one sofa traversing hallway (D. Romik)

The Semicircle

However, surely there are other sofas that can achieve a larger area. Consider a semicircle of radius $1$ . For a semicircle to negotiate the corner within the hallway $L$ it must first translate horizontally, rotate by touching the inside surface of the hallway, and then continue by translating vertically. However, how can we prove that the semicircular sofa indeed traces the inside and outside corner without exceeding the boundary of the hallway? Employing Thale’s theorem, which states that if $A$ , $B$ , and $C$ are points on a circle where segment $\overline{AC}$ is the diameter, then $\angle ABC$ is a right angle, we can indeed state that the semicircle can rotate around the corner.

Figure 5: Thale’s theorem (Wikipedia)

It will do so touching the inner side in three places, while the remaining arcs of the original semi-circle each touch the outer side of the corridor in two further places. Furthermore, the sofa also meets all the conditions required for the area to be a sofa.

Figure 6: Semicircle sofa traversing hallway (D. Romik)

Now that we have confirmed, that the semicircle fits around the corner in the hallway, can we notate it mathematically and calculate its area? If we want to describe the the movement of this sofa we must remember that we are translating this sofa in two directions horizontally and vertically. However, what if we were to keep the sofa in a set location and rotate the actual pathway around the sofa? This would allow us to clearly locate the points in the hallway which come in contact with the sofa as well as the rotation path it would take around the corner. The movement of the sofa through the corridor is fairly straightforward, so we will be focusing on the portion of the hallway where the sofa translates, rotates monotonically from an angle of $0$ to $\pi/2$ . As a further note we will associate time $t$ , with an angle $\theta$ . As time passes the angle of rotation of the hallway changes. Thus, the parameter, time, will represent the angle of the hallway.

If $T\subseteq\Bbb{R}^2$ is a sofa, and $L$ is the hallway of $90 \textdegree$ , a path of rotation for sofa $T$ is a continuous pathway $\vec{M}:[0,\pi/2]\rightarrow\Bbb{R}^2$ in such a manner that $R_t(T)+\vec{M}(t)\subseteq L$ for all $t\in[0,\pi/2]$ . Here the concept of a rotation path is defined. In simple terms we say that if sofa $T$ is a subset or equal to the two dimensional real numbers and $L$ is the hallway of $90\textdegree$ , than $\vec{M}$ has a domain of $[0,\pi/2]$ and codomain of the two dimensional real numbers assuming that we consider our hallway $L$ is a subset or is equal to the translation of the rotational matrix and rotation path at time $t$ , where time $t$ must be in the domain of $[0,\pi/2]$ . This allows us to say that sofa $T$ can be rotated in hallway $L$ if $T$ has a rotation path. As I mentioned before however we want to rotate the hallway. From the point of reference of the sofa we are rotating the hallway $L$ in the opposite direction that the sofa $T$ would be rotated, thus we say the rotation path that the hallway carves out is in fact $\vec{M}(-t)$ .

Now that we have all the correct parameters we can define our sofa $T$ . Let $T$ be a sofa, $\vec{M}:[0,t]\rightarrow\Bbb{R}^2$ be a continuous pathway. The sofa of the greatest area with said parameters is $T=\bigcap_{t\in[0,\pi/2]}R_{-t}(L-\vec{M}(t))$ . In other words sofa $T$ is the intersection of all sets between $0$ and $\pi/2$ . These sets are defined as the rotational transformations that result when the difference is taken between the hallway $L$ and the rotation path $\vec{M}$ .

Now that we defined the maximum area associated with a sofa of specific parameters, let’s calculate the area. If $L$ is the hallway, $\vec{M}:[0,t]\rightarrow\Bbb{R}^2$ than the path is $\vec{M}(t)=(0,0)$ . Rotating around the hallway according to $\vec{M}(t)$ we will get our semicircular sofa of radius $1$ defined by $x^2+y^2\leq1$ and $x\geq0$ . Even though the semicircle must rotate around the corner implementing translational motion in the vertical and horizontal direction, its boundaries are strictly defined.

Figure 7: Intersection of $x^2+y^2\leq1$ and $x \geq 0$ (G. Goldenberg)

If we graph the inequalities, a defined area is produced. This area falls between the equations $x^2+y^2\leq1$ and $x \geq 0$ . It is clear to see that the radius of this circle has a radius that measures one unit, and we also know this area is semicircular, thus we can calculate the area quite simply:

$A_{semi}=\frac{\pi r^2}{2} \\ r=1 \\ A_{semi}=\frac{\pi(1)^2}{2} \\ A_{semi}=\frac{\pi}{2}=1.57079632679...$

In this case it was not necessary to use set theory or derive a system of equations, but in some of the more complex sofas it will be, and now we have a solid understanding of how sofas traverse the corner of the hallway.

Famous Sofas

Hammersley’s Sofa & Upper Bound

As stated earlier there have been many mathematicians who have been intrigued by the moving sofa problem, and one of these mathematicians was John Hammersley who provided many mathematicians with insight into the moving sofa problem. Hammersley is most well known for the sofa he proposed as the sofa of largest area that can traverse the Moser’s hallway. Though a better sofa was later discovered, Gerver’s in 1992, Hammersley’s sofa would provide the foundation of Gerver’s discovery.

Figure 8: Hammersley’s sofa traversing hallway (D. Romik)

Hammersley’s sofa is unique as it is bound by six curves. A Hammersley sofa can be constructed if radius $r$ , which is required to construct the sofa, falls in the interval $(0,1)$ . The six curves that bound Hammersley’s sofa are as follows:

A semicircular arc with radius $r$ spans from $0$ to $\pi$ , with its center at the origin $(0,0)$ . The arc starts at $(r,0)$ , and ends at $(-r,0)$ .
A line segment that connects $(r,0)$ to $(r+1,0)$ .
The arc of a quarter circle that begins at $\pi/2$ and ends at $0$ . Its center is located at $(r,0)$ . The quarter circle begins at $(r+1,0)$ and ends at $(r,1)$ .
A line segment that begins at $(r,1)$ and ends at $(-r,1)$ .
A quarter circle arc that begins at and ends at $\pi/2$ and whose center is $(-r,0)$ . This arc spans from $(-r,1)$ to $(-r-1,0)$ .
A line segment that starts at $(-r-1,0)$ and ends at $(-r,0)$ .

Unlike the other shapes we have seen, this shape doesn’t seem as if it should translate the corner. How can it be proved that it indeed fits? If we have a radius $r$ we can claim claim that the points on our rotational path are $r(-\cos t,-\sin t)$ . Essentially we are scaling the points on the rotation path by the length of our radius $r$ . Why can this be stated? The sofa only touches the top wall, corner, and right wall as it rotates around the corner, thus we merely must prove these three portions are tangent to the corner and employ Thale’s theorem.

Figure 9: Labeled Hammersley’s sofa (N. Song)

The line tangent to the corner is easier to prove since if we only have to show that point $O$ is the midpoint of segment $\overline{AB}$ . Based on our visual, we can state that points $\odot A$ and $\odot B$ have coordinates with the following value: $(-2r\cos t,0),(0,-2r\sin t)$ . Assuming that we are following convention where time $t$ represent angle $\theta$ , $t$ will represent the angle with which the sofa rotates around center of rotation $O$ . Then $\angle{ACO}=\angle{OAC}=t$ . This can be concluded since both $\overline{OA}$ and $\overline{OC}$ are radii of the the arc $\stackrel \frown{AB}$ . Thus the coordinates of $O$ are in fact $r(-\cos t,-\sin t)$ . From here we just need to check that both quarter circles that are in contact with hallway $L$ are tangent to the corner. However, the quarter circles are the same arc just reflected over a vertical axis of symmetry. Thus if we prove one is tangent, than we can prove the other. It turns out that both exterior arcs are tangent. This can be concluded since both arcs are quarter circles of radius one rotating about point $\odot A$ . $\odot A$ is also translating along the $x$ -axis as it was just proved that there is always a line tangent to the corner. Via Thale’s theorem, we can confirm that Hammersley’s sofa can traverse the hallway $L$ .

Now that we have proved Hammersley’s sofa let’s get its area. The simple way to think about its area is by reverting to our investigation of the semicircular sofa. Hammersley’s sofa is simply a variation of the semicircular sofa. He splits the semicircle in half, adds a rectangular block in between the the two quarter circles and removes a small semicircle from the center of the added block. It turns out the empirical formula for said shape is $A_{Hamm}=\pi/2+2r-\pi r^2/2$ . This assumes the radius $r$ falls in the interval $[0,1]$ . Using some basic calculus we can determine the max area of said equation by taking the derivative and finding its max:

$\frac{dA}{dr}=2-\pi r \\ 2-\pi r=0 \\ r=\frac{\pi}{2} \\ A_{max}=\frac{\pi}{2}+\frac{2}{\pi} \\ A_{max}=2.20741609916...$

Hammersley also contributed to the moving sofa problem by determining the upper bound for the area of a sofa declaring it was $2\sqrt{2}$ . He was able to do so with the following method:

Figure 10: Determing upper bound of maximal area (Mathematics Stack Exchange)

If we consider the situation when the sofa is rotated by $\pi/2$ , in order for the sofa to pass through the straight portion of the corridor, the sofa must be able to lie within the confines of the dotted lines. This represents the straight corridor. We can translate the dotted lines vertically but one must fall within the inner and outer corners. If it does not meet this requirement the sofa will either vanish or break into pieces. The max area is found when the topmost dotted line comes into contact with the inner corner. If $h$ is used to denote distance from the outer corner to the upper dotted line, the shaded region is defined in the following manner:

$A(h) = \begin{cases} 2h+1 & 0\leq h \leq\sqrt{2}-1 \\ 2\sqrt{2}+2+2\sqrt{h}-h^2 & \sqrt{2}-1\leq{h}\leq{\sqrt{2}} \end{cases}$

$A(h)$ reaches its max, $2\sqrt{2}$ , at the upper limit, $h=\sqrt{2}$ .

Gerver’s Sofa

Mathematician Joseph Gerver is the current mathematician whose sofa holds the record for the largest area. His sofa is actually quite similar when compared to Hammersley’s sofa as the both share the same Shepard piano shape, however Gerver optimized his sofa using set theory, system of equations, and differential equations to determine the absolute maximum area that could fit around the corner.

Figure 11: Labeled Gerver sofa (J. Gerver)

Remarkably, Gerver’s sofa is bounded by 18 different curves. Parts $V$ , $XIII$ , and $XVIII$ are straight segments. Parts $I$ , $VI$ , $XI$ , $XVII$ are circular arcs of radius $h$ . Parts $II$ , $III$ , $VII$ , $XI$ , $XV$ , and $XVI$ are involutes of circles and $V$ and $XIV$ are involutes of involutes of circles. Involutes, sometimes called evolvents, are curves produced using another given curve via connection of an imaginary taut wire to the given curve and tracing its free end as it is coiled onto the given curve.

Figure 12: Example of an involute (Wikipedia)

Gerver’s sofa negotiates the corner in the moving sofa problem differently than other sofas, such as Hammersley’s. As Gerver’s sofa negotiates the curve from $0$ to $\pi/2$ , it does so in a number of steps:

In the first stage of rotation, Gerver’s sofa touches the hallway $L$ in parts $XII$ and $XVII$ . It continues to move horizontally until it reaches the hallway $L$ or part $VII$ of the sofa. Then hits point $\odot G$ .
In the second stage the sofa touches the hallway along four different parts of the sofa $XI$ , $VIII$ , $IV$ , and $XVI$ , and then hits $\odot G'$
In the third stage the sofa comes into contact with hallway $L$ at parts $X$ , $V$ , $IX$ , and $III$ and then complete stages two and one in reverse respectively.

Now that we have discussed its movement lets calculate its area. Gerver in his paper generates a system of equations which he derives by attempting to locally optimize the shape of the sofa. The derivation of these equations is quite tedious and difficult to follow, however Gerver produces four equations which are in terms of constant $A$ , $B$ , $\phi$ , and $\theta$ . These equations allow us to determine the values of the terms which determine the local optimalities of the curves of which the sofa is constructed. While studying Gerver’s sofa, we will use his notation to avoid confusion between his notation and the notation used in this post:

$A(\cos \theta - \cos \phi) - 2B\sin \phi + (\theta - \phi - 1)\cos \theta - \sin \theta + \cos \phi + \sin \phi = 0 \\ A(3\sin \theta + \sin \phi) - 2B\cos \phi + 3(\theta - \phi - 1)\sin \theta + 3\cos \theta - \sin \phi + \cos \phi = 0 \\ A\cos \phi - \left ( \sin \phi + \frac{1}{2} - \frac{1}{2} \cos \phi + B\sin \phi \right)=0 \\ \left (A + \frac{\pi}{2} - \phi - \theta \right) - \left [B-\frac{1}{2}(\theta - \phi)(1 + A) - \frac{1}{4}(\theta - \phi)^2 \right] = 0$

I used GNU Octave to solve the system of equations. My calculated values matched those found in Gerver’s paper and other available literature:

$\phi = 0.03918 \\ \theta = 0.68130 \\ A = 0.09443 \\ B = 1.39920$

Gerver defines $r(\alpha)$ , $u(\alpha)$ , $D_{u}(\alpha)$ , and $s(\alpha)$ in the following manner:

$r(\alpha) = \begin{cases} \frac{1}{2} & 0 \leq \alpha < \alpha \\ \frac{1}{2}(1 + A + \alpha - \phi) & \phi \leq \alpha < \theta \\ A + \alpha - \phi & \theta \leq \alpha < \frac{\pi}{2} - \theta \\ B - \frac{1}{2} \left ( \frac{\pi}{2} - \alpha - \phi \right )(1 + A) - \frac{1}{4} \left (\frac{\pi}{2} - \alpha -\pi \right)^2 & \frac{\pi}{2} - \theta \leq \alpha < \frac{\pi}{2} - \phi \end{cases}$

$u(\alpha) \equiv \begin{cases} B - \frac{1}{2}(\alpha - \phi)(a + A) & \phi \leq \theta - \frac{1}{4}(\alpha - \phi)^2 \\ A + \frac{\pi}{2} - \phi - \alpha & \theta \leq \alpha < \frac{\pi}{4} \end{cases}$

$D_{u}(\alpha) = \frac{du}{d \alpha} \begin{cases} - \frac{1}{2}(1 + A) - \frac{1}{2}(\alpha - \phi) & \phi \leq \alpha \leq \theta \\ -1 & \theta \leq \alpha < \frac{\pi}{4} \end{cases}$

$s(\alpha) \equiv 1 - r(\alpha)$

If we want to calculate the area beneath the curves we must now define our functions:

$f_{1}(\alpha) \equiv 1 - \int^{\alpha}_{0} r(x)\sin(x)dx \\ f_{2}(\alpha) \equiv 1 - \int^{\alpha}_{0} r(x)\sin(x)dx \\ f_{3}(\alpha) \equiv 1 - \int^{\alpha}_{0} r(x)\sin(x)dx - u(\alpha)\sin(\alpha)$

If we now plug in the appropriate intervals for each set of equations and complete some $u$ substitution in the last of the three previous equations we get:

$A_{1} = 2\int^{\frac{\pi}{2} - \phi}_{0} f_{1}(\alpha)r(\alpha)\cos(\alpha)d\alpha + 2\int^{\theta}_{0} f_{2}(\alpha)s(\alpha)\cos(\alpha)d\alpha \\ A_{2} = 2\int^{\frac{\pi}{4}}_{0} f_{3}(\alpha)[u(\alpha)sin(\alpha) - D_{u}(\alpha)\cos(\alpha) - s(\alpha)cos(\alpha)]d\alpha \\ A_{1} + A_{2} = 2.2195...$

In Summary

The moving sofa problem is the quintessential example of a problem that may look simple on the outside, but is unbelievably complex. It’s simplicity and real life applications toy with your mind. To date no one has found a sofa with a larger area, and although his solution is only a conjecture, it seems probable that Gerver may have found the sofa of maximal area, however, the search continues.

It is worthy to note that the moving sofa problem has also inspired a few other spin-off problems. In 2016 Daniel Romik conjectured he had found the sofa of maximal area in a problem called the ambidextrous sofa problem. This problem is similar to the moving sofa problem, but the sofa must be able to negotiate two right angled corners where the hallway takes on a z-like shape. Nicole Song constrcuted a spin-off problem where the corner angles of the hallway(s) in the problem were smaller or larger than $\pi/2$ . Other mathematicians such as Kiyoshi Maruyama and Phillip Gibbs have experimented with different methods of shape extraction by using numerical sequences and angular steps and limits respectively. Though the problem may never be solved, as long as there are mathematicians there will be curious individuals attempting to solve the moving sofa problem.

Figure 13: Ambidextrous sofa traversing hallway (D. Romik)

“Pivot! Pivot! Pivot!”

—Ross, “The One With the Cop,” Friends

References

Wikipedia contributors. Moving sofa problem. Wikipedia, The Free Encyclopedia. Online resource: https://en.wikipedia.org/w/index.php?title=Moving_sofa_problem&oldid=791382852. Accessed August 21, 2017.

D. Romik. Differential equations and exact solutions in the moving sofa problem. (2016), 8. Online resource: https://www.math.ucdavis.edu/~romik/data/uploads/papers/sofa.pdf. Accessed August 21, 2017.

D. Romik. MovingSofas: A companionMathematica package to the paper problem “Differential equations and exact solutions in the moving sofa problem.” Online resource: https://www.math.ucdavis.edu/~romik/publications#companionfiles.

D. Romik. The Moving Sofa Problem. Online web article (2016): https://www.math.ucdavis.edu/~romik/movingsofa/. Accessed August 21, 2017.

Numberphile, Daniel Romik. The moving sofa problem. Online web video (2017): https://youtu.be/rXfKWIZQIo4. Accessed August 21, 2017.

Weisstein, Eric W. Moving Sofa Problem. MathWorld–A Wolfram Web Resource. Online resource: http://mathworld.wolfram.com/MovingSofaProblem.html. Accessed August 21, 2017.

P. Gibbs. A computational study of sofas and cars. (2014), 1-3. Online resource: http://vixra.org/pdf/1411.0038v2.pdf. Accessed August 21, 2017.

J. L. Gerver. On moving a sofa around a corner. (1992), 268. Online resource: http://newweb.tlsh.tp.edu.tw/mediafile/2042/fdownload/216/219/2015-4-14-2-7-23-219-nf1.pdf. Accessed August 21, 2017.

J. Baez. Hammersley sofa. (2015). Online resource: http://blogs.ams.org/visualinsight/2015/01/15/hammersley-sofa/. Accessed August 21, 2017.

Multiple contributors. What’s the upper bound for sofa problem? (2016). Online resource: https://math.stackexchange.com/questions/1847453/whats-the-upper-bound-for-sofa-problem. Accessed August 21, 2017.

Steven Finch. Moving sofa constant. (2002). Online resource: http://web.archive.org/web/20080107101427/http://mathcad.com/library/constants/sofa.htm. Accessed August 21, 2017.

K. Maryuyama. An approximation method for solving the sofa problem. (1971), 15-23. Online resource: https://archive.org/stream/approximationmet489maru#page/n7/mode/2up. Accessed August 21, 2017.

N. Song. A Variational Approach to the Moving Sofa Problem. (2016), 12-22. Online resource: http://math.bard.edu/belk/projects/NicoleSong.pdf. Accessed August 21, 2017.

Multiple contributors. Modelling the “Moving Sofa”. (2017). Online resource: https://math.stackexchange.com/questions/1787466/modelling-the-moving-sofa. Accessed August 21, 2017.

L. Moser. Problem 66-11: Moving furniture through a hallway. SIAM Rev. 8 (1966), 381.

Multiple contributors. The moving sofa problem. (2014). Online resource: https://plus.google.com/+johncbaez999/posts/DXb7RuQH8wc. Accessed August 21, 2017.

Multiple contributors. The moving sofa problem. (2014). Online resource: https://www.reddit.com/r/math/comments/2145c0/the_moving_sofa_problem/. Accessed August 21, 2017

A. P. Goucher. Complications of furniturial locomotion. (2014). Online resource: https://cp4space.wordpress.com/2014/01/09/complications-of-furniturial-locomotion/. Accessed August 21, 2017.

Material Science & Chemical Engineering

Creating Advanced Nanomaterials Using the Properties of DNA

Grant Goldenberg July 21, 2017

Aperiodic 2D Lattice in the Shape of the Sierpinski Triangle

Last week I had the pleasure of attending the 91st Colloid & Surface Science Symposium held at The City College of New York (CUNY). The three-day symposium is held annually by the Colloid and Surface Chemistry Division of the American Chemical Society and showcases the most recent advances in colloid and surface science. The conference also features integrative material science research that incorporates other scientific disciplines such as biophysics, environmental science, and biotechnology. Graduate students, postdocs, professors, and vendors attend from all over the world, setting an international stage for the presentation and discussion of new advances in the field of colloids and surface science.

This year’s symposium featured approximately 500 oral and poster presentations that attendees could choose to explore. I attended many lectures in the areas of electrokinetics and microfluidics, emulsions, bubbles and foams, particles at interfaces, and applications of colloids and surface science in medicine.

It was truly an amazing experience for someone starting their journey in the world of material science because I had the opportunity to hear detailed lectures on topics ranging from the treatment of thromboses using microbubbles to the application of polymer ionic liquids for interface control and hybrid design.

When I read papers in journals and science-based magazines, I often have questions about the details of the research. Unfortunately, my questions often go unanswered. The beauty of a symposium such as the one I attended is that it not only provides an opportunity to hear, firsthand, about groundbreaking research, but it also provides a venue to have questions answered by the person who can best explain their findings, the researcher who first completed the work. So, when Robert Macfarlane from MIT presented his groundbreaking research on Polymer- and DNA-Directed Assembly of Nanocomposites I was ecstatic.

The goal of material science is to discover and design new materials that can solve specific technological dilemmas both by examining the properties of established materials and by using chemistry, physics, and engineering to construct novel materials. Developing new processes to manufacture these materials is an integral part of materials science research. Nanocomposites are materials that have recently garnered a significant amount of attention. These composites are an important class of materials that integrate at least two different phases and achieve physical characteristics unavailable in single-phase materials. However, assembling the composites in a controlled manner is a difficult task.

Robust DNA-bonded Nanocomposites

During his lecture, Macfarlane explained that he had created an adaptable process to precisely construct nanocomposites using the sequence dependent recognition properties of DNA, a simple yet robust solution to the complexity of the nanocomposite manufacturing process. Macfarlane presented work that showed that DNA can be used as a nanoscale binder to link individual nanoparticles together. The idea is to create vast arrays of bound nanoparticles to construct superlattices that achieve custom crystal geometries. The physiochemical properties of a biologic material, DNA, were used to engineer the construction of nanoparticle composites elegantly. Click here to Macfarlane’s full paper.

In my opinion, what is remarkable is that a single fundamental biologic principle was used to solve a complex problem. Thus Macfarlane’s innovative solution reminded me of a quote by Albert Einstein, “It can scarcely be denied that the supreme goal of all theory is to make the irreducible basic elements as simple and as few as possible without having to surrender the adequate representation of a single datum of experience.” Incorporating a single powerful biologic tool avoided surrendering to the complexity of the engineering problem.

The concept that DNA, the complex molecule that is the essential building block of all living things, could also be used as the building block of new nanocomposite materials that have the potential to travel to space, create a variety of building materials, and generate new technologies, is certainly an exciting development.

Sources

STEM Person of the Month

STEM Person of the Month: Dr. Natarajan, a Champion of Astrophysics, Women in STEM, and Gender Equality

Grant Goldenberg July 5, 2017

Priya_WSF_photo

The world of STEM is a place of wonder; it provides a venue for individuals of all different backgrounds to come together to solve the world’s most pressing problems. As some of us begin to explore the expansive environment of STEM, we become aware of individuals who are truly unique. They are often incredibly bright and ambitious, but more importantly, they are insatiably curious, and natural leaders.

I have had the pleasure to meet and read about a number of these individuals through my rather short yet compelling journey in the fields of science and technology. In fact, I feel such a passion for the work some of these individuals complete on a daily basis that I feel that I must honor them by sharing their efforts. Thus, I will be starting a section of my blog titled STEM Person of the Month. Without further ado, I will introduce my very first “STEM Person of the Month,” Dr. Priyamvada Natarajan.

Dr. Natarajan is a professor in the Departments of Astronomy and Physics at Yale who attended MIT for undergraduate studies and research, receiving multiple awards for her work. Dr. Natarajan received her Master’s Degree from MIT’s “Program in Science, Technology, and Society” and completed her graduate work in theoretical astrophysics at the University of Cambridge. She would become the first woman at Trinity College to receive a “Title A Fellowship” in Astrophysics from the prestigious University.

Her current research interests include cosmology, gravitational lensing, and black hole physics. A rather interesting study Dr. Natarajan has participated in recently used clusters of galaxies as “astrophysical laboratories.”

Her most recently published articles include Unveiling the First Black Holes with JWST: Multi-wavelength Spectral Predictions, published in The Astrophysical Journal in April of 2017, and Feedback Limits to Maximum Seed Masses of Black Holes, published in The Astrophysical Journal Letters in February of 2017.

Though her credentials and research accomplishments are extensive, these are not the only reasons I chose to highlight Dr. Natarajan. I decided to select her as the “STEM Person of the Month” since she has made a significant impact on social issues that pertain to the world of STEM by working to break boundaries and impassion young minds. In particular, Dr. Natarajan has actively worked to promote STEM and its many fields to women. As the chair of Yale’s Women’s Faculty Forum, she has been both an activist for women in STEM and a proponent of equality for females in the workplace.

Dr. Natarajan has also been an advocate for Women’s rights and gender equality. She has worked to establish gender parity at Yale through events such as the “Parity as Practice: The Politics of Equality” and “Contesting Gender Inequality” conferences.

I got to meet Dr. Natarajan two years ago during a college tour I took visiting Yale University. I had at that time begun to develop an interest in the topics of astrophysics and astronomy. Every time, I would look at the relatively starry sky of North Jersey I would become entranced, not just by the sky’s astonishing beauty, but my realizing that some of the light that I saw was light years old. So when I planned to visit Yale and found out that Dr. Natarajan was teaching a course on cosmology, I had to ask if I could sit in on her class. As I wrote to Dr. Natarajan, I remembered being somewhat apprehensive about writing to someone so esteemed in the science community. However, the kindness, warmth, and passion of the letter I received and the opportunity to sit in on her lecture proved to me that Dr. Natarajan is truly inspirational.

Dr. Natarajan’s hugely positive impact in the field of STEM, her research accomplishments, her commitment to supporting women’s rights and gender parity, and her dedication to teaching young students compelled me to name Dr. Natarajan as the first “STEM Person of the Month.”

For more information about Dr. Natarajan click here.

Sources

Electronics

Exploring Wireless Power Transmission Using a Low Voltage Slayer Exciter

Grant Goldenberg June 24, 2017

Let’s Make Some Music!

And for you internet trolls and memers…

Credit for both videos goes to Fabrício H. Franzoli at Franzoli Electronics. For more Telsa coil content check out his YouTube channel here and his Facebook group here.

Background.

When most of us think about the scientific concepts of electricity and power a few images often come to mind, home appliances with rubbery black plugs, the familiar three-pronged wall sockets that line the walls of our homes, or maybe even an image of an incandescent light bulb. It is certainly true that today power is distributed via wires and cables, however, every so often it’s important to expand our horizons by looking to the past. Surprisingly the groundwork for the extraordinary display of electricity and sound you just watched was laid over 100 years ago…

In 1900, an ambitious and now rather famous inventor and electrical engineer, Nikola Tesla, approached American financier and banker James Pierpont Morgan (J. P. Morgan) for the funds to develop a pair of wireless towers, one constructed on each side of the Atlantic.

Nikola Tesla (1890)

The goal of Tesla’s endeavor was to transmit textual messages, voices, and facsimile images (fax images) to vessels at sea to gain such information as instantaneous stock quotes from the New York Stock Exchange. Although requesting $100,000 in funds and receiving a greater $150,000 from Morgan, Tesla became overly ambitious and greedy attempting to scale up his project to include global transmission of wireless power in hopes of beating out rival Guglielmo Marconi’s radio-based telegraphic communications system. This decision would prove to be fatal, as the funds required to build Tesla’s improved project would skyrocket to a staggering $450,000 (about $12,164,846 using a dollar estimate from 2016). Despite Morgan’s offer to invite other investors (a result of Morgan’s unwillingness to provide the greatly ballooned funds required by Tesla), Tesla was not able to get the necessary funding, and thus, Wardenclyffe Tower, the first tower constructed by Tesla in Shoreham, NY, was abandoned, fell into disrepair, and was demolished in 1915.

Wardenclyffe Tower (1904)

If Tesla had not been so greedy with regards to his endeavor we may have had noncommercial versions of a wireless text, voice, and image transmission system possibly as early as 1902 or 1903. However, the greater question that many of you are likely asking, (I know I was itching to ask the same question) “How was he planning to do it?” The invention employed has become a household name around the world, the Tesla coil.

A Tesla coil in its simplest form is something called a resonant transformer. For those of you that don’t know what a transformer is, it is a basic electronic device that transfers electrical energy via induction through a pair of coils usually wound around an iron-based or magnetic object. In the case of a resonant transformer, the point of the transformer is not to transfer but to store energy for short periods of time. The number of coils on the resonant transformer determines its overall capacitance, its ability to store charge. The summed coils act as a resonant circuit (LC circuit) meaning they are able to store electrical energy at a specific frequency. If the resonant transformer is driven by a radio frequency oscillator, an electronic circuit that can produce an electromagnetic signal in a periodic fashion, very high voltages (in some cases as high as millions of volts) can be generated with a relatively low current.

So how did Tesla actually use the Tesla coil to transmit wireless power? Tesla found in his experiments that if a Tesla coil could be used to generate high voltages, then another coil tuned to the same frequency as the first could intercept the signal over relatively long distances. Even more fascinating for long range applications, Tesla hoped pulsed currents could be directed into the Earth itself and made to resonate at the proper frequency using a “grounded Tesla coil.” Tesla hoped that the electrical potential of the Earth itself would resonate at a standing wave worldwide. This would allow alternating current, current that travels in the form of a sinusoidal wave (reverses direction), to be distributed globally and intercepted using a simple capacitor based antenna tuned to the resonant frequency.

There are some drawbacks however to Tesla’s idea. First off, induction occurs over relatively short distances. As a result, there is not much hope for long range transmission via induction. Second, as we will see later with my low voltage slayer exciter if a very small capacitive metal surface is brought close to a Tesla coil high voltages can run through the surface. This is a great safety hazard as Tesla’s endeavor could have become more than a wireless transmission of power, but also death. Human beings are not the best conductors, but at high voltages we certainly conduct. Thus if any living being were to get too close to a wireless power emitter or receiver they could be killed due to the high AC voltage across their body as well as the electrocution that would result from arcing electricity (Tesla’s coils were apparently capable of arcing as much as 136 ft). Lastly, Tesla’s wireless transmission of power was also questioned for its economic feasibility. Worldwide, wireless and electric power would be wonderful, but how could individual usage be measured and billed? As most homeowners know, utilities are not free, and they certainly weren’t free in Tesla’s time. Many corporations and investors frowned upon Tesla’s idea as a result of its lack of economic feasibility.

Unfortunately, Tesla’s ideas for wireless transmission of power were not well received during his time period and unfortunately aren’t of much use today in terms of their wireless power transmitting capabilities (ideas relating to Tesla’s experiments with wireless communications are used quite extensively in modern technology). However, this does not discourage a hobbyist engineer and scientist, such as myself, from making a lower power version. Tesla coils, in my opinion, are of infinite value since they make great educational demonstrations. They are complex to make and tedious to perfect but provide key insights into many Physics concepts, particularly in electricity and magnetism, that are often touched on in high school physics and more deeply explored at the university level. The Tesla coil’s complexity should not scare one away from exploring and making a low power version, a slayer exciter. Tesla coils are fascinating to learn about and are among the coolest novelty items to own. They truly look like something out of a science fiction movie when constructed properly, streaming electricity in all directions, lighting fluorescent and incandescent light bulbs, and literally as well as metaphorically shocking your friends.

Building a Slayer Exciter

Before we get started I would like to make very clear that the slayer exciter is NOT my invention, and I DO NOT take credit for its invention. I have merely built a version of my own. That being said I firmly believe this project is a good choice for those interested in getting started in electronics. It’s fairly simple to build, requires minimal soldering, and uses materials easily ordered online or found in old electronics in your garage.

Before we go any further if we plan to build a Tesla coil we should first consider its components:

A voltage supply transformer (T). This transformer takes an input alternating current (AC) voltage and steps it up to a greatly increased voltage value. Such high voltage is required to create a short arch between the two electrodes of the spark gap.
A primary capacitor (C1). This capacitor is connected in conjunction with the primary coil (L1) to create a tuned circuit.
A spark gap (SG). The spark gap acts as the main circuit that provides a pulsed electric signal to the primary circuit.
A primary coil (L1). This is the primary winding of the resonant transformer referred to previously.
A secondary coil (L2). This is the secondary winding of the resonant transformer referred to previously. Together the primary and secondary coils (L1 & L2) create the recognizable components of the Tesla coil. Note the Tesla coil is air-cored and does not use an iron or magnet-based choke like other common transformers.
A top load (TL). The top load can be thought of in similar terms to an antenna with a great surface area. It is constructed of a fairly conductive metal usually in the shape of a toroid, ball, or disk. This optional component decreases the chances of premature discharges from the secondary coil (L2).

So what is a Slayer exciter? A slayer exciter is an electronic version of a Tesla coil in which the spark gap (SG) is replaced with a semiconductor, a material that has a conductivity between that of a conductor and an insulator such as a transistor, MOSFET, or IGBT. Furthermore, slayer exciters feature a built-in feedback loop, a loop that automatically adjusts the resonant frequency of the coil within the oscillating electrical signal circuit. This loop allows the primary and secondary coils (L1 & L2) of the Tesla coil to be tuned to the proper resonant frequency so that extensive manual tuning is not required. This is what differs a slayer exciter from a solid-state Tesla Coil (SSTC), which is is similar in design but does not feature a feedback loop and must be tuned manually. This tuning is often completed using an oscilloscope, electric function generator, and meticulous mathematical calculation of the parts of the Tesla coil prior to its construction.

Now that we know how a slayer exciter works, let’s build one. I have made a schematic for a basic slayer exciter:

The following parts are required:

R1 – 10kΩ Resistor
C1 – 1000µF Capacitor
Q1 – TIP31C NPN Power Transistor (I Highly Recommend Buying or Creating a Heatsink)
D1 – 1N4148 Standard Diode
L1 – 7 Loops of 22AWG Solid Copper Wire
L2 – 1000 ~ 1200 Windings of 30AWG Enameled Copper Wire Around 6cm Diameter PVC Pipe
Battery – 12 to 24V (Bench Supply or Modified Wall Adapter Recommended)
(Optional) Top Load – Can be easily constructed from two bases cut from the bottoms of two soda cans. For a more handsome top load, recycle the platinum platters from an old mechanical hard drive!

I recommend breadboarding the circuit prior to soldering the components onto a protoboard or PCB in case you make any mistakes.

The circuit functions in the following manner:

Voltage flows through the capacitor smoothing the input voltage and tuning the primary coil.
The resistor drives the base of the transistor.
The transistor turns on and drives current into the primary coil. This current is limited by the limited available base current.
The created magnetic field drives the secondary coil of the slayer exciter.
The voltage across the secondary coil wants to grow larger, but the minor capacitance on the output resists the change, against the rise of the output end. In return, the voltage on the other end of the secondary drops, pulling the base of the transistor low.
The diode prevents the base voltage from falling more than 0.7 V below ground. This pushes the output end of the secondary.
The transistor switches off so the magnetic field begins to fall.
The base voltage then begins to rise turning on the transistor and the cycle restarts.

Credit goes to YouTuber, blogger, and electrical engineer ElectroBOOM for the numbered steps regarding the workings of this circuit. Check out his YouTube channel here and his website here. As of writing this blog, he has recently amassed ONE MILLION subscribers!

Results

When the construction of the coil and its circuitry is complete, we can play with wireless transmission of power by bringing capacitive metal surfaces such as the electrode of a fluorescent or incandescent light close to the top load or secondary coil in order to light them!

Before I show off some fancy photos, I’d like to take the time to give special thanks to my partners in crime (classmates) on this project: Henry Hunt, Christina Lee, and Nicholas Carrero!

Here are the promised images of the finished product: