Phanthom Computing

It be amazing how humans be! The typical person will accept anything that he is told by a trusted authority without questioning it. Especially when, that message itches one's ears. This is a common phenomenon. Which occurs with radio, television, and the World Wide Web.

This afternoon we were enjoying our twice per week course on computer organization. Which also touches on computer architecture. In a sense, they are different sides of the same coin, independent but related. And, we spoke briefly of the major types of machines, CISC and RISC, and their generations.

Somehow or another, we started talking about mainframe computers. In that, life experiences permit recalling newscast and periodical reports about the Data General computer. Who the United States government was using in the Vietnam War effort. It was called a mainframe. And, it was claimed that it was as big as a warehouse.

Question? Where be it?

When roughly twelve years old, a family vacation was had that visited Fredericksburg Naval Museum. It was in Virgina. It is believed. At least, it was among the Southeastern states. It had a display of the United States military power, naval ships and artillery. One interesting fact from that trip was this. At roughly seventy-three inches in height and one hundred and eighty-five pounds, the role of a gunner could not be had. Those gentleman, who manned the naval cannons, were much smaller.

Yet, proof positive existed of these American artifacts of warfare from the Second World War.

Living among America's Heartland included a fieldtrip during second grade. Which visited the Strategic Air Force Museum. Which is near Interstate 80 West and Highway 370. It has a number of well-preserved aircraft. Those who played a vital role in past war efforts and in protecting the country during peacetime.

Where is that Data General? Where are those National Museums of military might. Which showcase that mainframe computer. Which is as large as a warehouse.

One of the past sentences is not missing a question mark. It will all sink in, in a moment. 

And, America has its Museum of National History and the Smithsonian.

It even has a National Museum that details some of the work done by George Washington Carver near Neosha and Diamond, Missouri. Which is amazing! Because, some Americans still tell individuals like him that they were released from slavery roughly a hundred and fifty years ago and they truly have not been set free. They simply are humored. So, practicing genocide is easier.

Long story short, if a Negro who is best-known for inventing peanut butter has a museum. Where is that Data General or a similar museum. Which would be the mainframe. That is filled with arrays of vacuum tubes and large ferromagnetic cores?

Might one fathom that these computers are propaganda phantoms?

Could anyone ever Phanthom this? Most likely not, or, it would have been done decades ago. Well-before, secondary school was exited!

This simply is an example of American Propaganda. That mythical lesson has been repeated so many times. That everyone accepts it as common knowledge. Early mainframe computers were the size of a warehouse. Oh! Everyone knows that! Really?

This is how a number of misconceptions become accepted as fact.

However, it is clearly recallable from readings of Judeo-Christian Scripture from the 70s, 80s, and even the 90s that some red-letter words said, "A lying tongue is only for a moment." It is believed that that might have been recorded in HIS famous Sermon among the passages in Matthew. And, seeing that, the New Testament Teachings are based upon the Old Testament. That statement spoken was a paraphrase of HIS Law. Matthew 5:13-20 (KJV). Which stated in older editions of the Bible that the penalty for lying was the removal of one's tongue. And, it was recalled from early moments of reading Scripture during second grade that the penalty for a first theft was the removal of a right hand. And, the second penalty was the removal of the left. That is the pre-millennial truth.

But, as the second President George Bush is famous for saying, "What is truth?" And, he also was credited with fighting the Hunafa. Who it believed are those who seek the "original truth". Well, seeing that, earlier readings of Revelations said that the beast would come a deceive many feigning that he was the MESSIAH, when he was actually the anti-Christ, suggests that siding with the Hunafa is not that bad in GOD's Eyes. Be that El-Roi?

Are the modern renditions of Scripture part of that deception?

Because, certain key passages are missing.

It often was wondered. Is it truly time for a Hebrew resettlement of Isreal. It was said, during the 70s, in Revelations, and as read in Bibles from the late-1880s; that the anti-Christ would reestablish Isreal first. As, the Eschaton began. Who led Isreal's resettlement after the Second World War? Written textbooks from the 50s and 60s will share a telling account; however, he was not a Hebrew!

So, who be he? It be a question for the ages...

But, the constant repetition of a whopper(prevarication) makes it a holy grail truth. And, therein lies the "beef". Wendy and her burger really was not a personal favorite. Some be down with the KING. And, the highly permissive gospel who is preached from modern pulpits that speaks that one is "free in Christ" when he simply "believes and does not doubt" such that he might freely commit any manner of sin, willful or not, and be immediately cleansed from such by Christ's Blood, Death, Burial, and Resurrection. Such simply be not so; however, those ears that itch for such a life's bondage in the name of liberty, like the modern message of sexual liberty [Roman 1, Galatians 5:19, Ephesians 5:6, Jude 7(KJV)], who has men becoming women and women becoming men via the means of medicinal processes or simply that typical adolescent sexual behavior who extends throughout adulthood for many, brings forth a damnation that be meet. And if, "one believes and does not doubt", he will accept Christ's Teaching during HIS Mountaintop Sermon [Matthew 5:13-20(KJV)]. Do not ever lose your savor. Because when, it be gone. It be gone permanently [2 Timothy 2:15(KJV)].

But, on a more mellow note, have a wonderful day! Because, Salvation via HIS Grace was, is, and will never be free! Saying so is a misinterpretation of the truth, that tickles the ears of those with a selfish and spoilt heart. Because, it costs much pain and suffering on a hill called Golgotha. Someone perished. Someone spent three days in the grave. Someone fought with the devil. So, he might snatch the keys called "death" and "hell" from satan's hands. So, we might be free! Someone fought a arduous battle. So, we might have a chance at Eternal Life! So, why cannot modern Christians show HIM enough love and respect that they obey HIS Word. Because, as Scripture states obedience is better than sacrifice.

And, that is the untold Truth of Salvation and the Gospel. It will cost you something! You must obey HIM! For, it one does not! That person is basically saying that HE is not WHO HE says HE is! Which is what? And, what are the costs of such! 

Every modern Protestant denomination and the Catholicism, those who profess a faith in Christ, all have their traditions. The Hebrew community experienced the same phenomena over the millennia of her existence. What did GOD say concerning such traditions.

[Mar 7:8-9, 13 KJV] 8 For laying aside the commandment of God, ye hold the tradition of men, [as] the washing of pots and cups: and many other such like things ye do. 9 And he said unto them, Full well ye reject the commandment of God, that ye may keep your own tradition. ... 13 Making the word of God of none effect through your tradition, which ye have delivered: and many such like things do ye.

[Col 2:8 KJV] 8 Beware lest any man spoil you through philosophy and vain deceit, after the tradition of men, after the rudiments of the world, and not after Christ.

[1Pe 1:18 KJV] 18 Forasmuch as ye know that ye were not redeemed with corruptible things, [as] silver and gold, from your vain conversation [received] by tradition from your fathers;

Although, life be rather pleasant with modern conveniences and such. And, we have what we feel are some great advances. Plus, we feel that we have made some great accomplishments. Do not be deceived. These are wicked and perilous times in the developed world. Especially, the western world. It might be the case that in some developing nations these vices and blasphemes do not exist; however, such placed be never seen personally.

[Psa 23:4 KJV] 4 Yea, though I walk through the valley of the shadow of death, I will fear no evil: for thou [art] with me; thy rod and thy [distaff] they comfort me.

[Isa 40:31 KJV] 31 But they that wait upon the LORD shall renew [their] strength; they shall mount up with wings as eagles; they shall run, and not be weary; [and] they shall walk, and not faint.

Mathematics := [ Consistent | Complete ]

While studying computer science, during a undergraduate and a graduate career, a course was taken in formal methods. We spoke of algebraic specifications and other formal systems used when describing an application. Means existed for verifying that the specifications were correct. It is the case that such philosophical systems, like mathematics, might be consistent or complete but not both. If, it is consistent. One might prove something as conclusively true or false and be certain of the result. But, one cannot prove somethings. If, it is complete. One can prove any and all things; however, the system will not be consistent. So, some provable theorems would arise that would be contradictory. 

Question? Is the set of prime numbers infinite? What is your answer? Who proved it so? Was it you, Clyde? Or, was it Euclid? And, with how many based did Euclid work? Some communities during his era counted by ten. One famous one counted by twenties. And, those digits also served as letters. Plus, a very powerful seafaring community counted by sixties.

Let us consider the number line. It has a pair of extremes, a positive and negative infinity, plus a zero. How many total positions are on that number line? [infinity] + [infinity] + 1. That is for both of the extremes and the locus of zero. So, that total number of loci is the [effective infinity] for that number line.

So, let us pause and consider what [infinity] is using the tenth base. If, we assume that that [infinity] holds the greatest possible digit in base ten, or 9. What would [infinity] be? Most probably, it would be a unending sequence of 9s. Yet, that is the [infinity] at the terminus and not the [effective infinity]. Who could not be represent in the tenth base. It would require a greater one.

And, the [infinity] in that greater base, say hexadecimal, would represent a much larger quantity than that in the lower base. That might be proven based upon the underlying principles of the positional numbering system. And, as a result, the [effective infinity] in that larger base would be much greater than the one in the smaller base and larger than its [infinity].

What would the [infinity] be in hexadecimal? Some might say that it would be an extension of the semester end grade reports of all of the modern mathematicians in history. If, their proof-work was regraded based upon the results which could be derived from this construction.

Answer: It would be an endless and unbounded sequence of Fs.

One common construction seen while studying mathematics was this. One could place any infinite set in a injective correspondence with the positive integers. This includes the evens and the primes; however, these sets grow at different rates. So if, we fixed a location like ninety-nine. One hundred integers exist between zero and ninety-nine, inclusive of the endpoints. How many are even numbers? Fewer than one hundred, correct? What about primes? Fewer still, right-o?

Let us pause and review. What does this tell us about these sets for the fixed locus of a positive [infinity] in any base? What about its associated [effective infinity]?

So, one might and must conclude that the set of primes at [infinity] or the [effective infinity] has a much smaller cardinality than the set of integers. And, that would be for any base by construction, including an infinite one. What does this mean? Is you wrong, Clyde? Or, be that Euclid's error? Well, that be you for you!

Seeing that, a sermon could be given that is consistent with Judeo-Christian and Islamic teachings concerning the folly in idolizing any man. It is best that this passage be abbreviated here. Why? Knowledge puffeth up and divides; while, charity edifieth. And, that word charity is properly translated as love. So, please show other enough love that you do not idolize them. For, HE said, "I will have not any other gods before ME." And, idols must be thrown down. So, HE receives the praise and adoration that HE deserves.

So, mathematicians and any other professionals, especially computer scientists, who are the new kids on the block, take the accomplishments of your field and its great names all with a grain of salt. 

And, add some Light while you are at it![Matthew 5:13-20 (KJV)]

Goldbach's Conjecture and Cantor's Argument

 A note was posted yesterday April 27, 26 that described why Cantor's diagonalization argument only describes exceeding the upper boundary set by the largest value that one might represent in base ten; however, that value is only finite and does not represent a true infinity.

Yet, that was only opinion supported with handwaving proof-work. And if, it does not stand the scrutiny of accomplished mathematicians. And, the diagonalization argument concerning infinity holds. Then, one might say that the true "countable" infinity represented by the number line is the cardinality of the integers found on it between (-999...) and (+999...). 

Each of these extreme values have an absolute value of (+999...). And when, they are added. They equal twice (+999...). Which is only representable in a base larger than ten. And, that effective infinity based upon the count of the integers on the number line is even.

Goldbach's Conjecture supposes that every even number is the sum of a pair of primes. Consider, this effective infinity for the base ten number line. It is the sum of composites only (999...) and (999...). Which are each the product of 9 and (111...). And, the largest possible prime smaller than (999...) is (999...) minus 2. And if, this value is prime. The largest even number which is possibly a sum of primes is (999...) minus 2 plus (999...) minus 2. Which is twice (999...) minus 4. So, twice (999...), the effective infinity of the number line, and twice (999...) minus 2, the last even number before this "infinity", are not a composition of primes. These are the minimum pair of even quantities between zero and the effective infinity which are contradictions for Goldbach's Conjecture.

One might argue the contrary that this is the very reason why the infinities identified by Cantor are fallacies. Seeing that, a long-forgotten derivation supporting Goldbach's Conjecture shared that it holds if one can find a singular prime between n and its square root. If so, a sufficient number of primes exists for yielding all of the even numbers less than n. It is thought that such is the case. Although, the nature and details of the proof-work have been long lost over the decades. That was a derivation seen during secondary school calculus and an enrichment module taught by the instructor.

As learnt during Philosophy 100 Logic at a small college in the rural Heartland, one might prove anything in the affirmative if one of his premises is false. Which is a further argument that Cantor's diagonalization argument has its limitations and has been misinterpreted among those who practice the Theory of Computation.

Shy a Doctoral Degree

The very reason why it was decided that earning a doctoral degree in computer science was impractical is the immaturity of the field. It seemed that as learning advanced certain basic aptitudes and understandings were lost.

One of the most glaring was the notion that infinity is not reachable.

This is contradicted by G. Cantor’s diagonalization argument. Which is fine for reasoning about the comparative growth of sets; however, it only truly shows that our positional numbering system has limitations. Which are not found among other numbering systems. Like, those based upon the grouping of quantity symbols. Such as, the system of Roman numerals.

For, it is personally believed that Cantor did not find nor fix infinity at a point. Consider, Cantor’s argument that the largest possible base ten number (999…) is along a diagonal in the table he created. Which is necessarily not square. As, it is often represented. When, this argument is “proven”. Since, the cardinality of a set of numbers in base ten outpaces its length significantly. Intuitively, how many numbers might be held is a set of numbers who are nine digits in length? One billion or ten with an exponent of nine.

Yet, adding any number in the set of all possible numbers representable by base ten and this largest number (999…) will yield a value smaller than the largest possible number found in base thirty-six (ZZZ…). With the alphabet serving as digits for numerals greater than ten.

So, Cantor did not locate a “true” infinity. His argument proves that quantities exist beyond the largest possible base ten value; however, that scalar is simply finite. Which is intuitive since ten is also. So, our base ten positional number system has limits.

This represents a limit of computation for the base but not computability in general.

It is highly possible that such a numbering system supporting a infinite base does not. Seeing that, infinity plus infinity is a new infinity. Yet, for mankind it is often most comfortable discussing computation in terms of base ten.

The popularity of this argument, Cantor’s diagonalization, and others that were seemingly as nonsensical are the very reason why doctoral studies were left. It was hoped that certain fundamental skills and aptitudes would never be lost by practicing very popular and community mandated dysfunctions as a doctoral-level education and career were sought. When, one might show that subtraction is not computable based upon Cantor’s argument. One has reached a sufficient and sound contradiction in his proof-work showing that Cantor’s argument is a fallacy; however, it seems that among modern computer scientists this is only fuel and motivation for more baffling proof-work establishing the hardness and intractability of some problems who were only meant as curriculum-based heuristics.

The words of a very well-respected and widely published professor in the Texas System will never be forgotten. He said, “If, you want your PhD. You will do it this way [and not question it]!” Then, he suggested a book by G. Polya on proper proof-work.

These ideas were shared with Co-Pilot from Microsoft and the system tanked. Seemingly, it feels that integers cannot have a infinite length; yet, if they cannot, then one cannot construct the diagonalization argument of Cantor.

And although, it has been many years since his name was first seen along with his diagonalization proof. It was thought that his first name was Gregor and not Georg. Yet, those are memories arising from the early-80s.

PHDP - Personal Hypertext Document Processor

Within the last three months, a general-purpose controller was produced along the lines of the CABOOSE project mentioned earlier in this web history. It was written in JAVA and might be combined with a JEE Servlet or a Helidon server using object composition.

As with CABOOSE, it places a partition between the static and dynamic view concerns. So, these might be created in parallel by software team members. This approach greatly simplifies application development.

Simply put, creating a PHDP application requires completing a storyboard of static concerns. One who has placeholders for any dynamic content. The placeholders for the dynamic "hotspots" are replaced in the view is rendered by the controller.

It is deemed general-purpose; because, one might build a vast array of different applications with the same controller mechanism. Which only requires parameterization. In fact, as a simple composable and inheritable object, it might be used among client-side applications.

This application was built using Microsoft Co-Pilot. Who did a wonderful job at suggesting various coding approaches. And, it was built in fewer than forty person-hours with less than a full KLOC during lulls in demand while working as a mathematics tutor at a local community college.

The history behind its development is a very long story. Who actually starts during the early-70s. When, a parent worked at the local medical center and was one of the first research technicians hired who was a person of color. 

She offered a suggestion one day during a research seminar, and she was immediately chastised by a doctoral level researcher. Who claimed that research ideas do not come easily. And that, they take a great deal of time developing. She claimed that her son, her only child at the time, could come up with a worthwhile research idea. 

At which, the researcher claimed that, if this author produced a fundable research idea before graduating secondary school, he would receive a full scholarship for a college education at the local state university. Which the author did during the Fall of '88. Albeit, he sought an education elsewhere for better or worse.

That confrontation in that weekly seminar, likely on a Friday, during the early-70s set the stage for a single person, this author, having a great influence on modern science and industry. Much, if not all, of the work developing those "suggestions" was done by others, who leaned upon those insights given, were credit them as the "originators" of those ideas. Which was not actually true. They incubated and hatched the products which arose from those ideas. Some of which came from a elementary school pupil.

The idea for a general-purpose controller arose while studying Pascal at Vanderbilt University during the Spring of '89. Seeing that, a starting document was sought for every program that one might write as an undergraduate. One sunny weekend afternoon was spent daydreaming and brainstorming in a dorm-room that was part of the freshman Quadrangle. Having personal contacts among the ranks of the international research community and in senior and executive leadership at SUN Microsystems, many of those ideas were developed. For one, the graduate assistant who taught that Pascal course asked if she could Zerox the notes drawn on pages of the author's research journal. After, some of those ideas were presented during classroom discussions. And, they proliferated through the ranks of computer science and software engineering professors among Vanderbilts departments at that time.

The author and his notes had been the subject of prying eyes. While, he attended secondary school. So, he had learnt that he should keep the best material hidden and non-obvious. Until, he found a serious collaborator. Which was the case for these notes.  Which had notions for the feature set of object-orientation, a series of related languages, including JAVA, the rudiments of frameworks and patterns, general-purpose controllers and other novel concepts who have had quite an impact since '90 listed in obvious terms.

In fact, modern computing, including the ACM, have worked diligently at covering over the actual origins of these innovations. For one, they claim that the first OOPSLA conference was in '86. Truth be told, it was during the early-90s. It was the renaming of a conference on software, languages, and programming which focused on structured development that had been occurring since the '70s or '80s. Objects, as abstract data types, did not land upon the computing scene until the Spring of '89. The construct called an object in Simula 67 was simply a dynamic type, like those found in Python 1.0. Which was not a object-oriented language. Objects as abstract data types came in a later version of Python. They have the same name but are entirely different things.  

And, many of the "best ideas" had that weekend afternoon were non-obvious concepts hidden behind the surface of the notes recorded. As such, IBM developed a "Domino" server. Who was not built upon a "domino" data structure. Which was seen as a configuration of interacting objects who relied upon dynamic invocation and started with the "initial 'tip' of the first one" yielding a cascade of computational activity. Which would settle upon a result.

Long story short, PHDP was a long-standing idea for a general-purpose controller. Who was a concept called PHP, Cold Fusion, and other "obvious" names in a personal research and idea journal kept during the '90s. These ideas were shared while looking for a collaborator; however, others took what "little information" that they were given and developed what they thought was intended. Which always was much more than needed. As, the source for PHDP and CABOOSE do show.

The source of PHDP might be found here: 

complete-web/phdp: HTML Document Preprocessor and General-Purpose Controller for Web Applications

Irrationals - Once Again

This account was temporarily lost. The password was forgotten, and the account could not be recovered via the traditional means. This was saddening; because, the readership had reach nearly 25K and it was gaining momentum. And, with the commotion that teaching college courses be, a concerted effort at recovering the account could not be achieved. By chance, Google sent a e-mail message which asked if the account could be validated and it used a smartphone number for recovery. So, the account was available for usage again.

Hopefully, that readership will resume. Although, it has been nearly a decade since the account was lost.

This forum was used for sharing some ideas which seemed valuable; however, they were not yet commonplace knowledge or were controversial. All of the work is "original" emanating from this sole author.

One topic and lesson that shall be taught again is that concerning "irrational" numbers. Which after this short mathematical development will seem quite possibly like a "irrational" concept.

During fourth grade, our class learnt the art of long division along with other much more advanced concepts often met in secondary school and college. By chance, one of the more advanced topics taught by our instructor, Mrs. Kaiser, was the development of the number hierarchy with its various sets: natural numbers, integers, rational numbers, irrational numbers, reals, and etcetera. It is believed that she also mentioned those who be complex, transcendental, and such; however, she said that we would learn more about those later. She was just introducing a few of the fundamentals.

Seeing that, we had just learnt about long division. We had discovered the power of the "vinculum". It is that bar who rest above the repeating part of a decimal number when one is determining the quotient of the division problem with its decimal expansion.

It dawned upon the author that the usage of the vinculum might yield a means for writing irrational numbers as ratios of integers. Which would make them rational and not irrational.

How might this be done? One asks!

The observation was made when the instructor described "pi" as a number with an infinite and never-ending decimal expansion. And, a comment was shared by the author. At which, the instructor exclaimed, "Do you want a Field Medal?"

That comment was this. [If, one places that bar used for repeating decimals above the zero in a ten. He would produce a "big honking ten" (it was fourth-grade), and he could multiply pi by a special form of one, the "big honking ten" divided by the "big honking ten" and produce a ratio of integers who each have an infinite length.]

So, seeing that, this editor will not produce a vinculum, we will describe our ten with a infinite number of zeros after it as 1(0)' . The ' signifies that the number or numbers in the parentheses repeats indefinitely. 

And,

[pi * 1(0)'] / 1(0)' = [pi * 1000...] / [1000...]

Which represents the ratio of an integer with a infinite number of digits divided by the same. So, although it might never be rendered by hand or mechanically, it is a ratio of integers. Unless, the number system does not allow for integers of infinite length. Which is does. 

At the age of nine, the author knew very little about Field Medals. And, life never permitted pursuing mathematics as a career field. Such that, a thesis, a dissertation, a research publication could be prepared that merited a Field Medal. If, such an idea is potentially sufficient for one.

So, this notion is shared once more; because, it seems that the concept of the irrational number is simply that, irrational. From, a simple-minded perspective that uses discrete mathematics concepts. An unending number of integers exist. When, these are composed with themselves once. The count of the total possible compositions is the square of infinity. Which is also infinity. So, the number of ratios must cover and map with all possible numbers. This is by the pigeon-hole principle.

Albeit, one might "prove" otherwise using a "diagonalization" argument which fixes the locus of infinity. Which by simple reasoning is unreachable and not fixable.

This is why Poincare said that Cantor [should cease with his insanity and arguing that infinity was reachable via an argument of diagonalization].

Arguments of diagonalization versus a very large fixed, finite value are great for comparing and contrasting the growth rate of mathematical objects like functions and sets; however, they place a hard-fixed and artificial cap on mathematical reasoning about the infinite. When, one claims that it is a fixed point with nothing beyond it. Such permits the "irrational" among mathematical reasoning.

Simple Heuristic for High-School Algebra Students

Team. It seems that an unfortunate situation is occurring in modern history. Old books are slowly being destroyed and recycled.

And, in that language is in constant flux with slang terms from the urban dictionary like "throke" filling the airways of modern conversation.

Having a reference handy is quite wise.

For one, a term in modern computing which has been misused is "heuristic". It simply means a challenging problem in mathematics which might require that one recall some facts from early subjects which they should have learnt well and mastered. That lesson was learnt while scanning an old dictionary that was the property of the author's grandmother while he was in seventh or eighth grade.

At, which time he constructed a few "different" very simple ones.

The modern connotation for this term in computing is a "impossible" problem. Which might require an inordinate amount of computation resources in its resolution. And, it is often proposed before classrooms that an efficient solution in time and space resources is not feasible.

And, on the topic of "intractable" problems:

The following "simple" problem might be classed as a heuristic. It is a "simple" challenge question for algebra students who have just learnt the art of simple polynomial multiplication .

"How might one produce a pocket square from a 'bow-tie' or an 'handkerchief'?" provided the following binomials and trinomial. The diagrams are at the "base" of this www-history. And, one of the solutions might be somewhat shocking.

Actually, this is drawn from a classic lesson in algebra that can enter a segue that describes a complementary lesson in graph theory.

Although, this first question might be a simple one for many "highschool-aged" algebra students.

It would be a "splendid" conceptual basis for an "heuristic" that mixed principles from numerous subfields such as topology, abstract algebra, optimization, set theory, or others. When, first hidden under layers of postulates, theorems, corollaries, relationships, and axioms. This is all that mathematicians ever do. They often meld a challenging question around a simple mathematical construct.

One might spend an entire career building substantial and meaningful problems around a few such simple relational and logical building blocks.

Such is the case with ECT - elliptical curve technology - it was drafted up by the author. And, he was rather surprised that it reached the "realm of use" by "professional-grade" mathematicians apparently for ciphering information. The phrase near the original construction that was drawn in that notebook while lollygagging and daydreaming during a trigonometry and pre-calculus class be "could this be used for hiding information". This was an hint for the student who sat beside the author in class who was working on modules in upper division subjects, like undergraduate topology, while in high-school. He would often peer over and look at the recreational mathematics notes sketched by this loon.

Finally, a note for those taught otherwise. The mnemonic used for describing this procedure is often called based upon [ First, Outer, Inner, Last ] when addressing this question of finding the product of a couple of binomials such as ( A + B ) * ( C + D ) or our bow-tie. However, this is "best" remembered using the diagram below. It is more descriptive than the four-letter mathematical term drawn from the first capital letter of each word. The phrase is the equivalent of calculated profanity that permanently fixes a learning barrier in the mind of many students. One might decimate this barrier if he recognizes the relationships that occur between each term when forming the product. These fully-realized understanding makes the associated acronym as tenuous as the tin extracted from the hills of Tennessee drawn thin and spun around a cylinder of cardboard..