WISRDwisrd.org/wp-content/uploads/2016/02/WISRD_Journal_1_1-.pdf · 2017. 4. 18. · The...

WISRD Research Journal

Volume I, Issue 1

Spring 2015

Wildwood Institute for STEM Research and Development

Editors: Owen Leddy, Caleb Zakarin, Scott Johnson

WISRD JournalEditors: Owen Leddy and Caleb Zakarin

Wildwood Institute for STEM Research and [email protected]@wildwood.org

Spring 2015

WELCOME to the first issue of the official journal of the Wildwood Institute for STEMResearch and Development. In the following pages you will find papers describing theresults of original research projects produced by our organization in its first year ofexistence as well as proofs and summaries of existing mathematical theory. Beyond the contentsof this journal, the institute’s ongoing research projects span areas including information theory,particle physics, and radio astronomy, among others. The editors and contributors of this journalwould like to thank the Wildwood faculty members who facilitate WISRD and the AdvancedTopics in STEM class - namely Joe Wise, Tim Sekula, and Scott Johnson. The success of WISRD’sfirst year of research is a credit to their guidance and support.

Contents

Proofs and Summaries 2The Brachistochrone Problem – Miana Smith . . . . . . . . . . . . . . . . . . . . . . . . . . 2Mersenne Primes – Max Caplow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Gödel’s Incompleteness Theorems – Caleb Zakarin . . . . . . . . . . . . . . . . . . . . . . 14

Original Research 19Differential Calculus of the Can Module – Noah Goldman . . . . . . . . . . . . . . . . . . 19Bayesian Reinforcement Learning – Owen Leddy . . . . . . . . . . . . . . . . . . . . . . . 29Creating Solid Potassium Nitrate-Based Rocket Engines – David Olin . . . . . . . . . . . 35

Curriculum Development 39Underwater Navigations as a Platform for STEM Curriculum – Dylan Vecchione . . . . 39

1

mailto:[email protected]:[email protected]

WISRD Journal• Spring 2015 • No. 1

The Brachistochrone ProblemMiana Smith

Wildwood Institute for STEM Research and Development (WISRD)[email protected]

I. Introduction

The Brachistochrone problem was one of the first problems presented in calculus of variations,which is the branch of calculus that concerns the minimization and maximization of functionals,which are essentially functions of functions. The problem was first solved by Newton, Leibniz,L’Hopital, Johann Bernoulli, and Jakob Bernoulli in the late 1600s.

The Brachistochrone problem asks to find the path that minimizes the amount of time it takesfor a point particle of mass m to travel from point P1 to point P2 in a uniform gravitational fieldwithout friction.

Figure 1: Diagram of system

II. Derivation

P1 is chosen to be (0, 0) and P2 is some (x1, x2)

v =dsdt

t =� dS

v

dS =�

dx2 + dy2

dS =

�1 +

dy2

dxdx

t =�

�1 + dy

2

dxv

dx

(Definition of velocity)

(Integrating for time of travel)

(Pythagorean theorem)

(Factoring out dx2)

(Plugging in the value of dS)

2– The Brachistochrone Problem – Miana Smith


The work done by the normal force of the track is zero as it is always perpendicular to thepath of the point particle. Due to the absence of friction in this system, the mechanical energy isconserved.

12

mv2 = mgy

v =�

2gy

t =�

��1 +dy2dx

2gydx

t =� �1 + y02

2gydx

F =

�1 + y02

2gy

t =�

F(y, y0)dx

(Conservation of energy)

(Solving for velocity)

(Re-writing with velocity)

(Re-writing with y’ notation)

(Defining the integrand to be F)

(Substitution)

To minimize the time of travel over the path, it is necessary to find the integrand that solves theEuler-Lagrange equation.

ddx

(�F�y0

) � �F�y

= 0

dFdx

=�F�y

y0 +�F�y0

y”

�F�y

y0 =dFdx

� �F�y0

y”

ddx

(�F�y0

)y0 � �F�y

y0 = 0

ddx

�F�y0

y0 � dFdx

+�F�y0

y” = 0

ddx

(F � y0 �F�y0

) = 0

�F�y0

=y0�

2g(y + yy02)

ddx

(

�1 + y02

2gy� y

02�

2g(y + yy02)) = 0

ddx

(1 + y02 � y02�

y + yy02) = 0

ddx

(1�

y + yy02) = 0

1�y + yy02

= C

(The Euler-Lagrange equation)

(Defining dFdx in terms of partial derivatives)

(Solving for �F�y y0)

(Multiplying the Euler-Lagrange equations byy’)(Plugging in �F�y y

0)

(Pulling ddx out by reverse product rule)

(The partial derivative of F with respect toy’)

(Plugging in the values of F and y0 �F�y0 )

(Combining like terms and factoring out1

2g )

(Simplifying)

(Integrating)



�y + yy02 = B

y + yy02 = A

y0 =

�A � y

y

dydx

=

�A � y

y�

dx =� � y

A � y

letz =yA

dzdy

=1A

x = A� � z

1 � z dz

letz = sin2�

2dzd�

= sin�

2cos

�

2

x = A�

�� sin2�2

1 � sin2 �2sin

�

2cos

�

2d�

x = A� sin �2

cos �2sin

�

2cos

�

2d�

x = A�

sin2�

2d�

x =A2

�1 � cos�d�

x =A2

(� � sin�) + k

(Defining a new constant)

(Defining a new constant)

(Solving for y’)

(Re-writing in dydx notation)

(Integrating both sides)

(Substituting z for yA )

(Differentiating with respect to y)

(Plugging in z and simplifying)

(Making a trigonometric substitution in or-der to simplify the integral)

(Differentiating with respect to �)

(Plugging in sin2 �2 and sin�2 cos

�2 )

(Simplifying)

(Simplifying)

(Substituting in the sine half angle formula)(Integrating)

By taking P1 to be (0, 0) the integration constant must also be zero because if 0 = A2 (0 � 0) + k,then k = 0.



x =A2

(� � sin�)

y = Asin2�

2

y =A2

(1 � cos�)

letA2

= r

x = r(� � sin�)y = (1 � cos�)

(Simplifying)

( yA = sin2 �

2 as defined by the earlier z-substitution)(Substituting in the sine half angle formula)(Re-defining the constant)

(Solving for x)

(Solving for y)

The period of this function is 2�r in terms of �, which is the circumference of a circle of radiusr, which is why the constant is defined as r. This is because x = r(� � sin�) and y = r(1 � cos�)describe the graph of a cycloid, which is generated by tracing a point on the circumference of acircle rolling in a straight line (see appendix A).

A graph of the solution with r = 1:

Figure 2: A graph of x = � � sin� and y = �1 + cos�

III. Conclusion

This solution (x = r(� � sin�) and y = r(1 � cos�)) minimizes the amount of time it takes for aparticle to travel from P1 to P2. No other path will allow a particle to travel faster between thetwo points than the cycloid. P1 can also be arbitrarily changed without changing the time it takesfor the particle to travel to P2. A method to empirically demonstrate this is to create a 3D trackthat models the cycloid and a few other curves and roll marbles down them in order to show thatthe cycloid is the fastest path.



IV. Appendix

The process of generating a cycloid:



V. References

1. "Cycloid." - from Wolfram MathWorld. Wolfram Research, n.d. Web. 11 Dec. 2014.

2. Susskind, Leonard, and George Hrabovsky. The Theoretical Minimum: What You NeedTo Know To Start Doing Physics. New York: Basic, 2013. Print.

3. "Brachistochrone Problem." - from Wolfram MathWorld. Wolfram Research, n.d. Web. 11Dec.



Mersenne PrimesMax Caplow

Wildwood Institute for STEM Research and [email protected]

I. Introduction

LA Mersenne prime is any number inthe form 2n � 1, which is prime. Thereare currently (as of November 2014) 48known Mersenne primes, with the largest,257885161 � 1, being discovered on January 25,2013. In fact, the top 10 largest primes cur-rently known are all Mersenne primes. Thisis mostly because of the relative efficiencies ofthe algorithms used to search for them, andthe use of distributed computing. The leadingresearch in finding these primes is the GreatInternet Mersenne Prime Search (GIMPS, cre-ated in 1996 by George Woltman), which usesdistributed computing to test for new primes.1

These numbers are often studied purely forthe sake of scientific research, but they alsohave many practical uses. These numbers arehighly sought after for use in cryptographyalgorithms and number generators.2 They arealso useful to finding "perfect numbers," or anumber where the sum of its factors is itself.3,4

II. Early History

One of the first major discoveries aroundMersenne primes came in 1536, from Hudalri-cus Regius, who showed that not all Mersennenumbers in the form 2p � 1, where p is prime,are Mersenne primes. He did this by showingthat 211 � 1 = 2047 = 23 ⇤ 89. The next bigdiscovery came when Pietro Cataldi predictedthat M17, M19, and M31 were also prime. Heincorrectly believed that M23, M29, M31, M37were prime. By 1640, Pierre de Fermat verifiedthat M23 and M29 where composite. MarinMersenne, the man that these numbers arenamed after, predicted, in 1644, that for thenumbers n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127and 257, 2n � 1 would be prime. Mersennelater admitted to not being able to test all of

these numbers himself, but neither could anyof his peers. It took 100 years for more progressto be made, when Leonhard Euler showed M29was composed, and M31 was prime. It wasn’tuntil 1947 that the complete set of Mx valuesunder 257 producing Mersenne primes wascompleted: 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107and 127.5 All of this changed in 1996 becauseof computer scientist George Woltman. Wolt-man decided to use computers to search forthese prime numbers. He also began to use dis-tributed computing, and created a system hecalled GIMPS. This means that any computerrunning a piece of software at home (calledPrime95) could participate in the project. Thecomputer would get tasks from a central server,and upon completion, it would send the resultsback to the server. This allowed for multiplecomputers across the world to work together,and eliminated the need for supercomputers.Furthermore, multiple computers could testone Mersenne number at the same time by test-ing different factors individually, minimizingduplicate processing. Later, software engineerScott Kurowski unveiled PrimeNet, which stillexists today. This is an update employed byGIMPS for distributing assignments to com-puters. All Mersenne primes discovered since1996 have relied on GIMPS.6

III. Basic Properties

Mersenne primes possess a number of uniqueproperties that make their study particularlyinteresting. The first thing that can be provedis that for 2p � 1 to be prime, p must also be aprime number. A version of this proof adaptedfrom Book of Proof by Richard Hammack, isincluded below.

"Theorem: If for some positive integer n,2n � 1 is prime, then so is n.

9– Mersenne Primes – Max Caplow


Proof. We will use contrapositive proof. Ifsome positive integer n is not prime, then2n � 1 is not prime.

Let a and b be positive integers. Let n = ab,to form the polynomial 2ab � 1. Then 2ab � 1 =(2a � 1) ⇤ (2ab�b + 2ab�2b + 2ab�3b + 2ab�4b +2ab�5b + 2ab�ab). Hence, 2n � 1 is compositefor any non-prime n, so n must be prime for2n � 1 to be prime."7

This is useful in the search for prime num-bers because it means that time can be savedsince any non-prime n does not need to betested. In addition, any prime number canlead to a possible Mersenne prime, which al-lows for large Mersenne prime candidates tobe easily discovered. However, this also meansnew Mersenne primes tend to be very large,and is why most of the largest known primesare Mersenne primes. While it is believed thatthere are infinitely many Mersenne primes, noproof for this theorem exists. This theorem isbased on a heuristic argument surroundingthe probability of 2n � 1 being prime approach-ing a particular log curve. This argument isbeyond the scope of this paper, but can befound here.8 However, the law of small num-bers, coined by Richard Guy explains whyjust because we know of a particular set ofnumbers, we do not know that the patterncontinues infinitely. We cannot be sure if thereare infinitely many Mersenne primes basedsolely on a heuristic argument, but only onconcrete proofs, for which there are none. Thelaw of small numbers states that "there arenot enough small numbers to satisfy all thedemands placed on them." What this means isthat sets of small numbers possess propertiesthat do not carry on indefinitely. One basicexample is that 1, 3, 5, and 7 are all prime, butalso happen to be the first four odd numbers.A possible conclusion is that all odd num-bers are prime, but this is obviously wrongsince 9 is not prime. However, "small" is veryrelative. Another example from The PrimePages is that gcd(n17 + 9, (n + 1)17 + 9) = 1for any positive integer n. For example,when n = 1, gcd(1 + 9, 131072 + 9) =

gcd(10, 131081) = 1 since 131081 is not di-visible by 2, 5, or 10, the factors of 10. Thiscarries on for all numbers n until the number8424432925592889329288197322308900672459420460792433. The takeaway is that numbers maybehave differently than we think as they be-come larger and larger. Patterns should not beassumed, but rather proven.9

IV. Trial Factoring

GIMPS uses various algorithms to search forMersenne primes, and rule out composite num-bers. The first method used the GIMPS soft-ware in finding Mersenne primes is trial factor-ing. It is a modified version of simply pickinga factor and checking if it divides the numberin question. The algorithm used is developedto run efficiently on computers. The algorithmstarts with a prime exponent to denote theMersenne number in question (as shown above,a Mersenne number can only be prime if theexponent is prime). Convert the exponent tobinary. As an example we will use 211 � 1,which is not prime, and check if 23 is a factor.In binary, 11 = 1011. Starting with 1 squarethe number. In this case, 1*1 = 1. Now look atthe top bit of the binary. This is the leftmostdigit, in this case 1. If the top bit is a one, mul-tiply by 2, otherwise keep the number. In ourcase, it is 1, so 1*2 = 2. Once performing thisoperation remove the top bit. This would turn1011 into 011. The resulting binary numberbecomes the new binary number for the nextrepetition. Now take the resulting number thatwas either multiplied by 2, or left the same, andcompute that number modulo 23 (the remain-der when divided by 23. In this case 23 is usedbecause we are trying to see if 23 is a factor.This number should be whatever the factor inquestion is). In this case 2 mod 23 = 2. Takethis resulting number, and repeat the entireprocessing using this as the starting numberinstead of 1. Once the binary string reachesthe last digit, stop. If the resulting number ofcompleting the mod 23 operation is 1, then thenumber is a factor. A proof for this is beyondthe scope of this paper.10 Another property of



Mersenne numbers is that factors q must bein the form q = 2kp + 1, where k and p arepositive integers. Since the smallest factor anumber is always a prime, it is efficient to onlytest a q that is also prime. Furthermore, q mustbe 1 or 7 mod 8 (again the proof is too complexfor this paper, but it is included in the previ-ous endnote). Therefore a list of all possibleq values under 40,000 is used by the GIMPSsoftware to test. These rules are applied first,eliminating the need to test 95% of factors bythe previously stated algorithm.

V. Alternative Factoring Methods

While the methods to test factors discussedpreviously are frequently used by the GIMPSsoftware, it also employs more powerful algo-rithms. The most common of these is the Lucas-Lehmer primality test. Exactly how these meth-ods work are beyond the scope of this paper.1112 While these methods are extraordinarilypowerful to test Mersenne primes, and arelargely why so many Mersenne primes havebeen found, they do not work for primes ofany other form. Recently, in 2002, the Agrawal-Kayal-Saxena primality test (AKP primalitytest) was developed. This text provided a wayto test any number for primality, no matterthe form. Today, there is a prize of $150,000funded by the Electronic Frontier Foundationto find the first prime greater than 100,000,000digits long, and GIMPS is still actively search-ing for it. In addition to trial factoring, anothermethod of factoring called "P-1 factorizations"is used to determine if a Mersenne numberis prime. It many cases P-1 factoring is fasterthan other dedicated primality tests such asthe Lucas-Lehmer test, which will be explainedlater. It was first used by John Pollard in 1974,and based off of Fermat’s little theorem, whichstates that for any prime number p, then ap � ais a multiple of p. Alternatively this meansthat ap ⌘ a(modp). The "mod" or modulo op-eration calculates the remainder upon divisionof the value on the left of the "mod" by thevalue on the right. Furthermore, two integersa and b are congruent modulo c (which must

be a real number), or "a ⌘ b (mod c)" if c di-vides a � b. In addition, "a ⌘ b(modc)" alsoimplies that when divided by c, a and b havethe same remainder. By dividing both sides bya, the statement ap ⌘ a(modp) is equivalent toap � 1 ⌘ 1(modp) or ap � 1 � 1 is a multiple ofp. An adapted version of a proof for Fermat’slittle theorem (which was not derived duringclass, but rather gathered from the prime pageslist of proofs is paraphrased below):1314

Theorem Let p be a prime which does notdivide the integer a, then ap�1 ⌘ 1(modp).

Proof. Create a list of the first p � 1 positivemultiples of a (where a is a positive integer):

a, 2a, 3a, ...(p � 1)a Suppose r and s are co-efficients of a in the list above so that ra ⌘sa(modp). This also means that r ⌘ s(modp),by dividing a on both sides. Since r and s arebetween 1 and p � 1, and therefore they areless than p, r = s. Therefore, when modp istaken for each item in the above list, each en-try will provide a unique and nonzero (sincep cannot divide any number between 1 andp � 1) list with all of the values between 1 andp � 1. Since the values in the list 1, 2, 3, p � 1are fully reduced with respect to p,

a ⇤ 2a ⇤ 3a ⇤ ...(p � 1)a ⌘ 1 ⇤ 2 ⇤ 3 ⇤ ...p �1(modp).

This can be simplified to:ap�1(p � 1)! ⌘ (p � 1)!(modp).The value (p � 1)! Can be canceled from

both sides to achieveap�1 ⌘ 1(modp).

The "P - 1" factoring methods begins bychoosing a bound B1, for which p � 1 is as-sumed to be less than B1. Next, an E is com-puted for which E = 2E2 ⇤ 3E3 ⇤ 5E5 ⇤ B, whereeach En corresponds to each of the consecutiveprime numbers less than N, and where 2E2is about the value of B1. The same shouldbe true for each En. The number B is thelargest prime number under N. The valueaE(modN) is then computed where a is an in-teger. The integer E should be a multiple ofp � 1 since E is a multiple of all of the primefactors of p � 1. However, if this is not thecase, the test will fail, and another method will



have to be used. Since E should be a multipleof p � 1, then by FermatâĂŹs little theorem,aE ⌘ 1(modp). Since aE � 1 is a multiple ofp � 1, but in general not a multiple of N, thanthe gcd(aE � 1, N) should provide a factor ofN. Since the factors of Mersenne numbers havethe form q = 2kp + 1. There is an additional en-hancement to this method simply called Stage2 that uses a B2 in addition to a B1, but requiresextra memory and is beyond the scope of thispaper. This method of P � 1 factoring is moreefficient in particular cases than the standardtrial factorization used first be Great InternetMersenne Prime Search (GIMPS). Therefore,GIMPS tries to calculate the probability that afactor will be found, as well as the computingtime required in order to decide which factor-ing method to employ when.1516

VI. The Lucas-Lehmer PrimalityTest

Trial factoring and P - 1 factoring is only usefulwhen the energy needed to test the factors isless than what would be required to run a trueprimality test. If it is unlikely that factors willbe found using these two methods, then theLucas-Lehmer primality test is used. This is atest created in 1856 by ÃL’douard Lucas, butadapted later by Derrick Henry Lehmer. Thismethod begins by defining a Mersenne num-ber, 2p � 1, with p > 2. Then a sequence is per-formed starting with S0 = 4. Each additionalSn is defined as Sn = (S2n�1 � 2)mod(2p � 1)(the modulo operation returns the remainderupon division of the number on the left bythe number on the right). This sequence isrepeated until Sp�2 is reached. As an exam-ple, let p = 5 to create the Mersenne number25 � 1 = 31. To prove that 25 � 1 is prime,begin with S0 = 4:

S0 = 4S1 = (42 � 2)mod31 = 14S2 = (142 � 2)mod31 = 8S3 = Sp�2 = (82 � 2)mod31 = 0Therefore, 25 � 1 is prime. The proof

for this theorem follows in the usual patternfor proving biconditional statements, or state-

ments in the "if, and only if" format. The proofbegins with a sufficiency argument, showingthat if Sp�2 ⌘ 0(mod2p � 1), then 2p � 1 mustbe prime. Then a necessity argument is usedto show the inverse of the statement, or thatif 2p � 1 is prime, then Sp�2 ⌘ 0(mod2p � 1).However, the proof in its entirety is very com-plex, and will not be included. While comput-ers are relatively efficient in performing mod-ulo operations, this algorithm leads to a bot-tleneck when squaring numbers. Originallyin computing, the method used to calculatesquares was to take the number, and break itup into an array. First, an operation called aFast Fourier Transform is applied, and thena particular squaring operation is conducted,followed by an inverse Fast Fourier Transform.However, in recent years, physicists and com-puter scientists Richard Crandall and BarryFagin improved this method by using an irra-tional base Fast Fourier Transform to reducecomputing time by more than half.1718

VII. Ensuring Accuracy

The final important aspect for findingMersenne primes is to double check results.The first method employed by GIMPS was tosimply run the test a second, and then a thirdtest. To check if the final value in the Lucas-Lehmer primality, the end of the value (thelast 64 bits) of Sp�2 is uploaded it to the cen-tral PrimeNet servers. If those values matched,then the test was declared a success. If not,the check would be repeated again. This checktakes about two additional years, since it issent out to slower computers to work out. Fur-thermore, when double-checking results, theoriginal S0 value is shifted randomly from 4.This ensures that there are no errors in the FastFourier Transform squaring operations sincedifferent numbers are being squared.19

VIII. Conclusions

Mersenne primes are so sought after becauseof how well algorithms such as the Lucas-Lehmer primality test work with them, and



because their factors are so specific. Despitethe great advancements in efficiency, Mersenneprimes still provide a challenge to test the lim-its of computing and efficacy. They are nothowever the only primes searched for. Oneof the most interesting distributed comput-ing searches for primes comes from Primecoin(http://primecoin.io/). Primecoin is a digitalcurrency, similar to the popular alternative Bit-coin, which uses algorithms that searches forparticular prime number chains as a way toverify digital transactions. In addition, thereis working being done on creating new algo-rithms like the Lucas-Lehmer primality test,that are more generally applicable, such as theMiller-Rabin primality test and the AKS primal-ity test. There is also innovation taking placewithin GIMPS itself to not only optimize run-ning time and efficiency, but to develop waysto run tests on graphics cards in addition toCPUs, in order to maximize parallel processing.Furthermore, anybody is able to get involvedin GIMPS today, by downloading the software,which automatically takes unused CPU cycles(minimizing effect on battery), and uses it tosearch for Mersenne primes.

Notes1"Free Mersenne Prime Search Software." GIMPS

- Free Prime95 Software Downloads - PrimeNet.January 1, 2015. Accessed April 24, 2015.http://www.mersenne.org/download/.

2"Mersenne Twister Home Page." A Random Num-ber Generator (since 1997/10). Accessed April24, 2015. http://www.math.sci.hiroshima-u.ac.jp/ m-mat/MT/emt.html.

3"Why Do People Find These Big Primes?" Prime FAQ.Accessed April 24, 2015. http://primes.utm.edu/notes/faq/why.html.

4Corbit, Dann. "Practical Use of MersennePrimes - Mersenneforum.org." Practical Use ofMersenne Primes - Mersenneforum.org. Au-gust 15, 2012. Accessed April 24, 2015.http://www.mersenneforum.org/showthread.php?t=17068.

5"Free Mersenne Prime Search Software." GIMPS- Free Prime95 Software Downloads - PrimeNet.January 1, 2015. Accessed April 24, 2015.http://www.mersenne.org/download/.

6Ibid.7Hammack, Richard H. Book of Proof. Ed. 2.1 ed. Rich-

mond, Va.: Virginia Commonwealth University, 2013. 110,261-262.

8Caldwell, Chris K. "Heuristics: Deriving the WagstaffMersenne Conjecture." Prime Pages. Accessed May 1, 2015.https://primes.utm.edu/mersenne/heuristic.html.

9Caldwell, Chris K. "The Prime Glossary." Law ofSmall Numbers. January 1, 2015. Accessed April 24, 2015.http://primes.utm.edu/glossary/page.php?sort=LawOfSmall.

10"Mathematics and Research Strategy." GIMPS - TheMath - PrimeNet. January 1, 2015. Accessed April 24, 2015.http://www.mersenne.org/various/math.php.

11Weisstein, Eric W. "Lucas-Lehmer Test."From MathWorld–A Wolfram Web Resource.http://mathworld.wolfram.com/Lucas-LehmerTest.html

12"LucasâĂŞLehmer Primality Test." Wikipedia. Ac-cessed April 24, 2015. http : //en.wikipedia.org/wiki/LucasLehmerprimalitytest.

13Caldwell, Chris K. "Proof of Fermat’s Little The-orem." The Prime Pages. Accessed April 24, 2015.https://primes.utm.edu/notes/proofs/FermatsLittleTheorem.html.

14"Fermat’s Little Theorem." Wikipedia. Accessed April24, 2015. http : //en.wikipedia.org/wiki/Fermat0sl ittletheorem.

15"P-1 Factorization Method." Mersennewiki. AccessedApril 24, 2015. http://www.mersennewiki.org/index.php/P-1.


17Weisstein, Eric W. "Lucas-Lehmer Test."From MathWorld–A Wolfram Web Resource.http://mathworld.wolfram.com/Lucas-LehmerTest.html

18"LucasâĂŞLehmer Primality Test." Wikipedia. Ac-cessed April 24, 2015. http : //en.wikipedia.org/wiki/LucasLehmerprimalitytest.




Gödel’s Incompleteness TheoremsCaleb Zakarin

Wildwood Institute for STEM Research and [email protected]

I. The Origin of Set Theory

Set theory is the study of well-ordered collec-tions of mathematical objects. Set theory is oneof the most significant and powerful ideas in allof mathematics. Countless rules and equationshave been derived from set theory. Though thestudy of sets of mathematical objects has ex-isted in some form or another for thousands ofyears, it was not until 1873 that it truly becameits own mathematical field of study. Georg Can-tor, a brilliant German mathematician, createdset theory while examining properties of realnumbers. Cantor discovered that the real num-bers and the natural numbers are not bijective(they do not have one-to-one correspondence).The natural numbers are countable, thus realnumbers are uncountable. The number of el-ements of these two sets were shown to bedifferent. The cardinality of the set of the natu-ral numbers is denoted as @0. While both setsare infinite, the set of the natural numbers isthe smallest infinity. Cantor pioneered a wayto analytically examine the number sets. Hismethods of proving the uncountable nature ofthe real numbers will be looked at in greaterdepth, as well as various other proofs that usedthe diagonal argument.

II. Cantor’s Diagonal Argument

The method of proof that Cantor employed toprove that the real numbers are uncountablewas contradiction.

Initial Claim: The set of real numbers is un-countable.

Proof. "Assume the set of real numbers andthe set of natural numbers are bijective. N iscountable, R is countable.

Suppose N and the decimals between (0,1)

are bijective. For every n 2 N there are corre-sponding decimals.

So, 1 corresponds with .a1a2a3...ax; 2corresponds with .b1b2b3...bx; 3 correspondswith .c1c2c3...cx; and f corresponds with.f1f2f3...fx, where a, b, c, and f are 2 N andx 2 N ! •.

Choose n, where n corresponds to the deci-mal .n1n2n3... nx where n1 6= a1, 0, 9; n2 6= b2,0, 9; and n3 6= c3, 0, 9 ... nx 6= zx.

As a result, n differs from every single num-ber between (0,1). It differs diagonally, not corre-sponding with the decimal place that matchesits own number. Thus n is not included within(0,1). So, this result contradicts the initial claimand it is proven that the set of real numbers isuncountable."

III. Developments from Cantor’sDiagonal Argument

The diagonalisation argument showed manyfundamental aspects of mathematics. As statedbefore, the natural numbers have the smallestcardinality of infinity: @0. By showing thatthe set of the natural numbers and the set ofthe real numbers do not have one-to-one cor-respondence, it is gathered that the real num-bers are a greater infinity. While this discoverywas extremely important, it is the diagonalmethod that is most significant. Cantor usedthis method once again to prove Cantor’s The-orem. Cantor elaborated upon this idea of dif-ferent sized infinities. The proof deals with thepower sets of sets: the amount of subsets in aset. By showing that the number of the subsetsof a set and the set itself did not have one-to-one correspondence through diagonalisation,Cantor proved that the cardinality of the powerset is always larger than the set from which it

14– Gödel’s Incompleteness Theorems – Caleb Zakarin


comes from. In 1901, the British mathemati-cian, Bertrand Russell, took the diagonalisationmethod to its fullest extent. Russell had got-ten word in 1901 of Cantor’s Theorem, eventhough it had yet to be published. Instead oflooking at the diagonal argument in terms ofits applications to number sets, Russell decidedto look at sets in general. While Russell’s para-dox is often described in language as: a setthat includes every set except itself, it can alsobe read as: a set which is different from ev-ery set, as well as itself. If X is a set that fitsthe previous description, X 2 X. This, how-ever, leads to a contradiction. If X /2 X, then acontradiction is also met. Thus we find a para-dox. While the diagonalisation method wasnot used in the same way as Cantor’s DiagonalArgument, the same concepts were employed.Russell’s Paradox led to an entire reworkingof set theory, exposing its inconsistencies. Thefoundations of mathematics were strengthenedwith the Zermelo-Fraenkel axioms, or ZFC ax-ioms, that soon followed the paradox, and KurtGodel, with insights from the diagonal argu-ment, proved his legendary IncompletenessTheorems.

IV. Principia Mathematica

Since the creation of set theory, the notionof constructing a set of axioms from whichall mathematics can be derived, has enticedcountless mathematicians. In 1931, The Aus-trian mathematician Kurt Godel put an endto the fierce search with his IncompletenessTheorems. Frege’s initial attempts, in the 19thcentury, to formalize all of mathematics weremet with Russell’s paradox. Russell showedthat naive set theory was unworkable. If X isthe set of sets that do not contain themselves,then X is not an element of X. However thismeans that X is an element of X. Thus a para-dox arises.

"Let X=(x : x /2 x)Thus X /2 X and X 2 X"After his paradox was published, Russell

and the British mathematician Alfred NorthWhitehead attempted to create a system in

which all mathematical truths could be derivedand proven from its axioms. After 17 years ofintense work, the third and final volume of thePrincipia Mathematica was published in 1927.These three volumes included extensive workon the foundations of mathematics. The mainsubjects were number theory, the real numbers,and cardinality. Later volumes were planned,but Russell and Whitehead never completedthem. With the publication of the third vol-ume though, Russell and Whitehead believedthey had laid out enough on the foundationsof mathematics to provide a framework fromwhich all mathematical truth could be derived.At first, the Principia Mathematica did not re-ceive much attention from the mathematicscommunity. People were more intrigued by thephilosophical aspects of the book. Nonetheless,there were still several mathematicians whoread the several thousand page tome. LudwigWittgenstein was one of the first to provide acritique of the Principia Mathematica. In hisTractatus Logico � Philosophicus. Wittgensteincritique was in reference to the ’axiom of re-ducibility.’ In Russell’s Theory of Types, heposited that all propositional functions havea relation at every mathematical level. Thismeans that induction is a valid method of proofbecause a function can be reduced to the firstlevel. This is absolutely necessary in under-standing the foundations of mathematics, aswithout it, functions could not be related acrosslevels. Wittgenstein believed that Russell, inhis proofs, was guilty of not recognizing thathis system of logic was dependent on an ’un-mentioned’ higher set of logic.

"6.123

’It is clear that the laws of logic can-not themselves obey further logicallaws. (There is not, as Russell sup-posed, for every ’type’ a special lawof contradiction; but one is suffi-cient, since it is not applied to it-self)’

6.1232

’Logical general validity, we couldcall essential as opposed to ac-



cidental general validity, e.g., ofthe proposition "all men are mor-tal". Propositions like Russell’s "ax-iom of reducibility" are not logicalpropositions, and this explains ourfeeling that, if true, they can onlybe true by a happy chance.’6.1233’We can imagine a world in whichthe axiom of reducibility is notvalid. But it is clear that logic hasnothing to do with the questionof whether our world is really ofthis kind or not’" (Tractatus Logico-Philosophicus).

The most important aspect of Wittgen-stein’s critique is the idea that the logic Rus-sell and Whitehead present in the PrincipiaMathematica is based on something other thanlogic. They have pushed logic as far as it cango, but logic itself cannot show that logic islogical.

V. Godel and Incompleteness

Picking up from where Ludwig Wittgensteinleft off, Kurt Godel was able to turn set the-ory on its head. He used logical self-referenceto prove that for any system capable of arith-metic, it is impossible to prove that the systemis complete.

Essential to understanding the Incomplete-ness Theorems are the concepts: completeness,consistency, and f ormal system. A f ormalsystem is a set of finite axioms from which the-orems can be derived. Completeness is wherea formal system can prove every single possi-ble truth and falsehood. Consistency is when aformal system does not prove a statement thatis both true and false (Stanford Encyclopediaof Philosophy).

VI. First and SecondIncompleteness Theorems

The first theorem states that in any complexmathematical system, there are statements in

the formal system X which cannot be proventrue or false.

Godel’s second Incompleteness Theoremwas more revolutionary: For any complex for-mal system X that has consistency, it is impos-sible to prove this consistency from the axiomsof X.

VII. Diagonal Argument andSelf-Reference

The brilliance of Godel was the way in whichhe forced formal systems to ’talk’ about ’them-selves.’ This is encapsulated in his utilizationof the Diagonal Lemma. This Lemma was firstused by Georg Cantor to prove that the realnumbers and the natural numbers do not haveone-to-one correspondence. If Cantor coulduse this Lemma to show that the reals are un-countable, theoretically, it could also be shownthat formal systems capable of arithmetic couldalso run into this ’uncountability’ problem. Thecrux of the lemma is that when a statementwith a free variable is derived from the systemX, a sentence can be made where a statementequivalent to the system can be chosen as thevariable. If X is a formal system where thestatement A(b) where b is a variable can bederived, then the sentence Y can be made. Es-sentially Y, is talking about itself here.

X ` Y $ A(Y)When the negation of this statement is

taken, there is an error because the statementcannot say whether or not the system is consis-tent or not. Thus endlessly checking every pos-sible statement is unnecessary because the sys-tem has just made clear in one step that it can-not make a decision. To prove X, Y must firstbe proved and then, of course for Y anothersystem must be employed. Using inductionthrough diagonalisation, Godel’s Theorems areproven to be true.

VIII. Bibliography

1. Raatikainen, Panu. "Godel’s Incom-pleteness Theorems." Stanford University.



Stanford University, 11 Nov. 2013. Web.08 Dec. 2014.

2. "Kurt Godel." Godel Biography. Univer-sity of St. Andrews, n.d. Web. 08 Dec.2014.

3. Whitehead, A. N. and Russell, B. Prin-cipia Mathematica. New York: Cam-bridge University Press, 1927.

4. Hofstadter, D. R. Godel, Escher, Bach: AnEternal Golden Braid. New York: VintageBooks, p. 17, 1989.

5. "Godel’s Incompleteness Theorem – fromWolfram MathWorld." Godel’s Incom-pleteness Theorem – from Wolfram Math-World. N.p., n.d. Web. 08 Dec. 2014.

6. GORDON, CHARLES K., and Prof.William A. R. Weiss. AN INTRODUC-TION TO SET THEORY. N.p.: n.p., 2008.University of Toronto, 2 Oct. 2008. Web.

7. "Russell’s Antinomy." – from WolframMathWorld. N.p., n.d. Web. 12 Dec.2014.

8. "Wittgenstein’s Logical Atomism (Stan-ford Encyclopedia of Philosophy)".Plato.stanford.edu. Retrieved 2011-12-10.

9. "Tractatus Logico-Philosophicus." Tracta-tus Logico-Philosophicus. N.p., n.d. Web.12 Dec. 2014.

10. Wahl, Russell, 2011, ?The Axiom of Re-ducibility,? in Nicholas Griffin, BernardLinsky and Kenneth Blackwell, PrincipiaMathematica at 100, in Russell (SpecialIssue), 31(1): 45-62.

11. Whitehead, Alfred North and Russell,Bertrand (1910?1913, 2nd edition 1927,reprinted 1962 edition), Principia Mathe-matica to *56, Cambridge at the Univer-sity Press, London UK, no ISBN or UScard catalogue number.

12. "Axiom of Reducibility." Axiom of Re-ducibility. Oxford, n.d. Web. 12 Dec.2014.

13. Alonzo Church, A formulation of the sim-ple theory of types, The Journal of Sym-bolic Logic 5(2):56?68 (1940)

14. 1931, Uber formal unentscheidbare Satzeder Principia Mathematica und ver-wandter Systeme, I. and On formally un-decidable propositions of Principia Math-ematica and related systems I in SolomonFeferman, ed., 1986. Kurt Godel Col-lected works, Vol. I. Oxford UniversityPress: 144-195.

15. Irvine, Andrew David. "Principia Math-ematica." Stanford University. StanfordUniversity, 21 May 1996. Web. 12 Dec.2014.

16. Bagaria, Joan. ’Set Theory.’ Stanford Uni-versity. Stanford University, 08 Oct. 2014.Web. 20 Nov. 2014.

17. Cantor, Georg. ’Diagonal Argument:Proof that real numbers are larger thanthe natural numbers.’ 1874.

18. Stoll, Robert R.’Set Theory.’ EncyclopediaBritannica Online. Encyclopedia Britan-nica, n.d. Web. 21 Nov. 2014.

19. Warner, Evan. SPLASH TALK: THEFOUNDATIONAL CRISIS OF MATHE-MATICS (n.d.): n. pag. Stanford Univer-sity. Web.

20. GORDON, CHARLES K., and Prof.William A. R. Weiss. AN INTRODUC-TION TO SET THEORY. N.p.: n.p., 2008.University of Toronto, 2 Oct. 2008. Web.

21. E.W. Beth, The Foundations of Mathemat-ics, North-Holland Amsterdam, 2nd ed.,1968.

22. Joseph Warren Dauben, Georg Cantor:His Mathematics and Philosophy of theInfinite, Harvard University Press, 1979.



23. Abraham A. Fraenkel and Yehoshua Bar-Hillel, Foundations of Set Theory, NorthHolland, Amsterdam, 1958.

24. Douglas Hofstadter, Godel, Escher, Bach,

Basic Books, 20th Anv. Edition, 1999.

25. Bertrand Russell, Principles of Mathemat-ics, Norton, New York, 1937 (first pub-lished in 1903).


� � ( )

� ( )

( )� ( )

�

R : ! R

R

19– Differential Calculus of the Can Module – Noah Goldman

: R ! R

: ! R �

( ) � ( ) = ( � ) + ( )( � )

�( ) = �

� ( ) � : R ! R �

( )

( )

( )

( ) =

�

��

( )

( )

( )

( )

( )

�

��

� ( )

( )

( ) = ( ) + � ( )( � ) + ( )( � )


( )( � ) �

�( ) =

�

( ) =�( + ) + ( + )

�

( ) =

��

�( ) �

( ) � (� ) <


= ��

( ) + ( )�

= ��

( ) � ( )�

= ��

( ) ( )�

= ��

( )

( )

�

= ��

( ) � ( )�

� �( � )

�

( ) = ( ) + ( )

( ) = ( ) � ( )

( ) = ( ) ( )

( ) =( )

( )

= (R )

�


( ) = ( )

( ) = ( )

�( )

�= ( ) ( )

= ( )�

( ) ( )�

=�

( ) ( )�

( )

= ( ) ( )

=�( )

�

�( )

�=

�( )

�

( )

( ) �! ( ) ( ( ) ( ) ( ))

( ) ( ( ) ( ) ( )) �! ( )( ( ) + ( ) � ( ))


( )( ( )+ ( )� ( )) �! ��

( )( ( )+ ( )� ( ))�

( ) = ��

( )( ( ) + ( ) � ( ))�

�. . .

�= �

�(+ � ⇤ ( ) ( ) . . .)

�

= �( ) = �( )

= =

� �

� �

= [ ] �

= [ ] �

�

( ) = ( ) . . . ( )


= [ ] = [ ]

� �

( ) =

� = ( R � )

��

�

�

: !

( ) = ( ) ( )


: !

: !

: ! �

( ) ( ) = ( ) ( ) ( )

�( ) = � �

� ( ) � : ! �

: R ! R

�


� ( ) = ��

� � ( ) � ( )� ( )

�

� ( ) = � ( ) (� ( ))� � ( )

�

� ( ) = ��

�

�( ( )) � ( ( ))

( ) � ( )

��

( ( )) � ( ( ))( ) � ( ) =

( ( )) � ( ( ))�

�( ) � ( )

�

�( ( )) � ( ( ))

( ) � ( )

�=

�

�( ( )) � ( ( ))

�

�

�

��

( ) � ( )

�

=� � ( )� ( )

� ( )

� ( ) = ��

� � ( )� ( )� ( )

�


( ) = � � ( ) =

( ) = � � ( ) =

( ) = � � ( ) =

( ) = � ( ) � � ( ) =


• •

29– Bayesian Reinforcement Learning – Owen Leddy

• •

E0 = 0, Et = Et�1 +n

�i=1

AitSit

EtAit Sit

P(At+1 = 1) =

��

�

0 : Et � lEt�l1�l : l < Et < 1

1 : Et � 1

AtP(At+1 = 1)

At+1

S0 StSt�1

�t[�(n + 1), n + 1]

�t �0

An

�t+1 = �t � r

�t+1 = �t + p


• •

S0�0


• •

F = gt2max �g

�i=1

t2i

gti

i

n = 4

c

tmax

c = 1000tmax = 100000

S0 = 0.2684, r = 1.2057, p = 0.0931, l =0.9931, �0 = 0.1699

F = 2.99141E10S0 = 0.5, r = 0.5, p = 0.05, l =

0.5, �0 = 0.2


• •

tmax

S0 �0

l

F

r p �0


• •



Creating Solid PotassiumNitrate-Based Rocket Engines

David OlinWildwood Institute for STEM Research and Development (WISRD)

[email protected]

This research is one component of an on-going project attempting to create a soundingrocket that can travel up to a significant alti-tude. A sounding rocket is essentially a rocketdesigned to collect sub orbital data while inflight. Getting a rocket into orbit costs mil-lions and is definitely unfeasible, however it iscertainly possible to create a sounding rocketwhich can still reach an impressive height andcollect actual data. Most sub-orbital rocketsare actually quite simple. They are just card-board tubes packed with solid fuel and maybea nose cone on top to make them slightly moreaerodynamic. It is easy for anyone to buy thesecardboard rocket motors from companies likeEstes, and there are no shortage of kits forbuilding rocket bodies to put them in. Theease of this process is precisely why I did notchoose it; anyone can follow the instructions to"make a rocket" but they are not really makingit. They are allowing someone else to makeit and then putting a few simple componentstogether. In order to actually learn somethingabout rocketry, I needed to create things fromscratch.

With this in mind, the first component thatneeded to be created was the motor. The motoris the key to any rocket. Technically, a singlestore-bought Estes motor could still qualify asa rocket on its own. The body of the rocketand its payload only serve to give purpose tothe motor?s launch. In a motor, an oxidationreaction is induced which then expands insidea chamber and increases the pressure withinit. That high pressure is then forced to moveto an area of lower pressure through a nozzle.The force of the exhaust rapidly exiting thenozzle is what acts against the rocket and givesit thrust. The propellant is made up of two

components: the fuel and the oxidizer. Theoxidizer rapidly accepts electrons from the fuel,causing a massive increase in heat and light,leading to expansion and increase in pressure.

Though this is the basic premise off whichmotors run, there is significant variation in thetype of motor. The two main kinds of motorsare motors that use liquid and solid propellant.Liquid propellants are, as the name would sug-gest, propellants using liquid fuel and oxidizer.Large rockets such as those that make it intoorbit tend to use liquid propellants as a mainbooster, combining super-cooled liquid oxygenas an oxidizer and liquid hydrogen as a fuel.But in an entire spacecraft launch there are of-ten many different types of propellants servingspecific functions. The other type of propellant– solid – is less efficient, but is far more man-ageable. It is composed of a relatively stablemixture of fuel and oxidizer, which is ignitedand then burns through all of its propellant.Though less effective, they are still used asboosters for some larger launch systems, suchas those of the space shuttle back when it wasin operation. In addition, their ease of use andpracticality make them the propellant of choicefor low-altitude rocketry.

Solid propellant rockets have the same basecomponents as liquid ones – a fuel and anoxidizer – but the motor design is much sim-pler. Instead of the complex cooling and intakesystems used in liquid rockets, all that is neces-sary is for the propellant to be tightly packedinto a casing. In order for it to function well, asolid nozzle needs to be placed at the bottomof the motor to direct the flow of the exhaust.In addition, it is common to give the propel-lant an open core. This increases the insidesurface area, allowing it to burn faster, giving

35– Creating Solid Potassium Nitrate-Based Rocket Engines – David Olin


it a higher exhaust velocity. Slower-burningpropellant can also be placed on top of thefaster-burning one as a time delay. When thetime delay runs out, it can ignite more fuel ontop, blasting off an end cap, which can causea parachute to deploy or send flames into thenext stage of the rocket, igniting more propel-lant. In addition, there are often other chemicalsubstances used in order to improve its effec-tiveness. In addition, binders are necessary tosolidify and liquefy the mixture respectively,in order to allow it to be poured into the motor

casing and form a solid structure.These basic parameters have been the basis

for all attempts at making rocket motors up tothis point. The first and most crucial step ofthe motor building process was preparing thepropellant. In order to create this a simple buteffective mixture of charcoal, potassium nitrate,red gum, sulfur, and acetone, was used. 76% ofthe mixture was the oxidizer, potassium nitrate,and another 18% was charcoal, the fuel. Potas-sium nitrate is a very common oxidizer andis useful because it will not react easily due

Figure 1: Burned propellant after powder was tested in a stainless steel crucible.

Figure 2: Crucible filled with tightly backed propellant shavings mid-burn.



to pressure, such as ramming the propellantinside the motor, but if sparked, say by a fuse,it will oxidize very effectively. Charcoal wasused as the fuel because it is easily obtainableand came in the form of a very fine powder. 4%sulfur was then added, sulfur is hygroscopic,meaning it absorbs water, so any moisture inthe motor would be absorbed by the sulfur in-stead of the charcoal. 4% red gum was addedbecause it is a binder. This means that thepowdered propellant mixture can more easilyform a solid pellet when rammed into a motorcasing.

These ingredients were combined to createa total 100-gram mixture, and then placed intoa rock tumbler with eight brass pellets. Therock tumbler milled the powder thoroughly,and the brass pellets increased the amountthe powder was mixed but would not sparkif they collided, ensuring the mixture wouldnot accidentally react. After milling for twelvehours, they were removed and stored.

First, a clay mixture was created to form thenozzle mold of the rocket motor. This mixturewas composed of 66% clay powder and 34%sand, it did not require any milling time. Itspurpose was simply to form a nonreactive com-pound that would become solid when rammedinto the based of the rocket motor.

Second, this propellant mixture was testedby pouring a line over the flat of a counterand igniting it with the flame of a Bunsenburner, but the line ignited quite poorly. Itignited somewhat better when placed inside astainless steel crucible, but still it burned quiteslowly. This combustion is described by thefollowing chemical equation, omitting acetoneand red gum: 2KNO3(s) + 3C(s) + S(s) ! N2(g)+ 3CO2(g) + K2S(s).

It was decided to go ahead and create arocket motor using that propellant anyway tosee how it would function. A 7.5 inch long,0.75 inch diameter piece of PVC pipe was usedas a motor casing. A mold to create a nozzleand core was created digitally and 3D printedand then placed inside the PVC pipe. Afterthat, 2 grams of the clay mixture were addedand rammed down with a piece of wood. Theclay successfully formed the hardened nozzlemold after several minutes hammering downthe piece wood. After that it was time to addthe propellant, which would total 30 grams.However, it was first necessary to add 2.25mlof Acetone in order to allow the mixture tosolidify. After the acetone was stirred in withthe rest of the propellant, it was poured in andrammed down bit by bit in the same manneras the clay mixture. The end product was

Figure 3: Test motor post-burn. Note the burnt fuel at the bottom of the ring stand.



an engine with no time delay. However, thepropellant had stuck to well not only to thesides of the casing but also to the core spindle,and when the core spindle was removed partof it broke inside of the motor. This wouldimpede exhaust flow and seal the engine shutwith melting plastic, it would also prevent thefuse from even being placed inside the motor,so it needed to be removed. Unfortunatelythe rest of the spindle was finally removed asolid pellet of fuel was removed with it. Themotor casing would need to be prepared again.However, it was decided to make the best ofthe problem and try and test the removed fuelpellet. A chisel was used to separate the fuelfrom the core spindle, and the shavings wereplaced inside of a stainless steel crucible. Theseshavings were lit with a Bunsen burner flameas well, and the results were far more success-ful than previous tests. It burned rapidly andrequired a lesser spark. It would appear thatsolidifying the mixture allowed for an easierburn as all of the propellant was more tightlypacked.

After that, a very simple motor was pre-pared. This motor was prepared the same wayas the one mentioned above except the corespindle was lubricated so it was successfullyremoved. This motor proved to be success-ful, completing a burn in 3.4 seconds with 30grams of fuel. During this burn the nozzleretained its shape.

Though a full motor with a time delay isnot quite ready, initial prospects for the suc-cess of the potassium nitrate based motor aregood. The propellant mixture is very effectiveas is the clay nozzle mold, and a basic motorwithout a time delay proved successful as aproof of concept. With some more preparationthese initial successes will yield a fully com-pleted motor, given more time this motor willdoubtlessly be able to carry a payload to a highaltitude.

References

[1] "Propellants." John F. Kennedy Space Cen-ter - KSC Fact Sheets and InformationSummaries. John F. Kennedy Space Cen-ter, Oct. 1991. Web. 21 Nov. 2014.

[2] "How Do Conventional Rockets Work?"Virtual Solar System Project. QualitativeReasoning Group, Northwestern Univer-sity, n.d. Web. 21 Nov. 2014.

[3] Braeunig, Robert A. "Basics of SpaceFlight: Rocket Propellants." Rocket andSpace Technology. N.p., 2008. Web. 19 Nov.2014.

[4] Sleeter, David. Amateur Rocket MotorConstruction. Moreno Valley, CA: Teleflite,2004. Print.



Underwater Navigations as a Platformfor STEM Curriculum

Dylan VecchioneWildwood Institute for STEM Research and Development (WISRD)

[email protected]

Navigation while diving can pose its manydifficulties. There are two types of underwaternavigation, natural navigations and compass(calculated) navigation. Students will learnwhy natural navigation is beneficial in certainenvironments, but inadequate in other homo-geneous environments, such as sandy oceanenvironments or low visibility. Students willdiscover this by attempting to navigate (natu-rally) in a grassy field, blindfolded (resemblingmurky environments). From that, students willdiscover how to calculate transect lengths usingtrigonometric functions and the PythagoreanTheorem. Students will also learn how to readmany graphs, deducing information aboutdive time, depth, and using their observationsto determine the dive limits. A sample graph(Figure 1: Dive Log) is provided below. Onecan observe that the diver increases in depthuntil reaching his maximum depth at roughly30 minutes. The diver then stays at this divesite for five minutes, before returning to thesurface (an ascent that takes approximately 15minutes). The diver then takes a 10-minutesurface interval before embarking on a seconddive. The diver descends to a maximum depth

of 25 feet for 5 minutes, before ascending backto the surface.

Once fully understanding the graph, stu-dents will analyze the two dives using actualdiving protocols (Figure 2- Recreation DivePlanner).

Using the Recreational Dive Planner (Fig-ure 2), students will be able to model theabove scenario, demonstrated in the below dia-gram, determining pressure groups, residualnitrogen time, and surface intervals, all actualcomponents of diving.

Diving provides a natural integration ofmath, science, and critical thinking, whichmolds easily into an applied curriculum that isunique, challenging, and contextualized, pro-viding a real-world application for the class.The Wildwood Institute for STEM Researchand Development and a WISRD collabora-tor, ReefQuest, are designing this curriculum,which, upon completion, will be available on-line for teacher download, as well as integratedinto preliminary classes in the form of a Geom-etry/Algebra II level project.

39– Underwater Navigations as a Platform for STEM Curriculum – Dylan Vecchione


Figure 1: Dive log

Figure 2: Recreational dive planner.



Figure 3: Recreational dive planner chart.


WISRD Journal CoverWISRD Journal _1Proofs and SummariesThe Brachistochrone Problem – Miana SmithMersenne Primes – Max CaplowGödel's Incompleteness Theorems – Caleb ZakarinOriginal ResearchDifferential Calculus of the Can Module – Noah GoldmanBayesian Reinforcement Learning – Owen LeddyCreating Solid Potassium Nitrate-Based Rocket Engines – David OlinCurriculum DevelopmentUnderwater Navigations as a Platform for STEM Curriculum – Dylan Vecchione

WISRDwisrd.org/wp-content/uploads/2016/02/WISRD_Journal_1_1-.pdf · 2017. 4. 18. · The...

Documents

Transcript of WISRDwisrd.org/wp-content/uploads/2016/02/WISRD_Journal_1_1-.pdf · 2017. 4. 18. · The...