Category: Mathematics

Mathematics uses many concepts in threes. The first structure mathematically is a triangle. There are acute, right, and obtuse angles. Trigonometry is the study of the relationship of the sides of a triangle. Have your heard of Pascal's Triangle?

Posted on by

Pythagorean triple

A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. Such a triple is commonly written (a, b, c), and a well-known example is (3, 4, 5). If (a, b, c) is a Pythagorean triple, then so is (ka, kb, kc) for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are coprime (that is, they have no common divisor larger than 1). For example, (3, 4, 5) is a primitive Pythagorean triple whereas (6, 8, 10) is not. A triangle whose sides form a Pythagorean triple is called a Pythagorean triangle, and is necessarily a right triangle.

32 + 42 = 52

The name is derived from the Pythagorean theorem, stating that every right triangle has side lengths satisfying the formula a2 + b2 = c2; thus, Pythagorean triples describe the three integer side lengths of a right triangle. However, right triangles with non-integer sides do not form Pythagorean triples. For instance, the triangle with sides a=b=1 and c=2 is a right triangle, but (1,1,2) is not a Pythagorean triple because 2 is not an integer. Moreover, 1and 2 do not have an integer common multiple because 2 is irrational.

Pythagorean triples have been known since ancient times. The oldest known record comes from Plimpton 322, a Babylonian clay tablet from about 1800 BC, written in a sexagesimal number system. It was discovered by Edgar James Banks shortly after 1900, and sold to George Arthur Plimpton in 1922, for $10.

Plimpton 322, a Babylonian clay tablet from about 1800 BC
Plimpton 322, a Babylonian clay tablet from about 1800 BC

One example of a Pythagorean triple is a=3, b=4, and c=5: Ancient Egyptians used this group of Pythagorean triples to measure out right angles. They would tie knots in a piece of rope to create 3, 4, and 5 equal spaces. Three people would then hold each corner of the rope and form a right triangle!

3-4-5 triangle using rope in Egypt
3-4-5 triangle using rope in Egypt

right triangles during the construction process to help determine the slope of the pyramid. The Pythagorean Theorem states that given a right triangle with sides of length a and b respectively and a hypothenuse of length c, the lengths satisfy the equation a2 + b2 = c2.

Pyramids at Giza
All three of Giza’s famed pyramids and their elaborate burial complexes were built during a frenetic period of construction, from roughly 2550 to 2490 B.C.

When searching for integer solutions, the equation a2 + b2 = c2 is a Diophantine equation. Thus Pythagorean triples are among the oldest known solutions of a nonlinear Diophantine equation. The simplest linear Diophantine equation takes the form ax + by = c, where a, b and c are given integers. The solutions are described by the following theorem: This Diophantine equation has a solution (where x and y are integers) if and only if c is a multiple of the greatest common divisor of a and b.

Posted on by — Leave a comment

The Three Laws of Recursion

Like the robots of Asimov, all recursive algorithms must obey three important laws:

  • A recursive algorithm must have a base case.
  • A recursive algorithm must change its state and move toward the base case.
  • A recursive algorithm must call itself, recursively.

Recursion is the process of defining a problem (or the solution to a problem) in terms of (a simpler version of) itself. For example, we can define the operation “find your way home” as: If you are at home, stop moving. Take one step toward home.

Let’s begin our discussion of recursion by examining the first appearance of fractals in modern mathematics. In 1883, German mathematician George Cantor developed simple rules to generate an infinite set:

Cantor’s rule for an infinite set

There is a feedback loop at work here. Take a single line and break it into two. Then return to those two lines and apply the same rule, breaking each line into two, and now we’re left with four. Then return to those four lines and apply the rule. Now you’ve got eight. This process is known as recursion: the repeated application of a rule to successive results. Cantor was interested in what happens when you apply these rules an infinite number of times.

George Cantor

Dichotomy paradox – Zeno’s

“That which is in locomotion must arrive at the half-way stage before it arrives at the goal.”

— as recounted by Aristotle, Physics VI:9, 239b10

Suppose Atalanta wishes to walk to the end of a path. Before she can get there, she must get halfway there. Before she can get halfway there, she must get a quarter of the way there. Before traveling a quarter, she must travel one-eighth; before an eighth, one-sixteenth; and so on.

Zeno’s paradox was recursive by cutting the distance in half each time to the infinitesimal. This is also how the Tortoise beat the Hair by questioning time over distance.

Recursive Function Calls

The tortoise and the Hair – the paradox of time
int factorial(int n) 
{ if (n == 1) { return 1; } 
  else { return n * factorial(n-1); } }

A function that does call others is called a nonleaf function. … The factorial function can be rewritten recursively as factorial(n) = n × factorial(n – 1). The factorial of 1 is simply 1. The image shows an object trace of the factorial function written as a recursive function. Each call goes in the run time stack until the base case is reached, and the the stack is popped as the result is passed to each function on the stack.

Five Factorial (5!) in recursion

What Is a Fractal?

The term fractal (from the Latin fractus, meaning “broken”) was coined by the mathematician Benoit Mandelbrot in 1975. In his seminal work “The Fractal Geometry of Nature,” he defines a fractal as “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole.”

Recursion in Nature

Looking closely at a given section of the tree, we find that the shape of this branch resembles the tree itself. This is known as self-similarity; as Mandelbrot stated, each part is a “reduced-size copy of the whole.”

The Three Laws of Robotics

Isaac Asimov was an American writer and professor of biochemistry at Boston University. During his lifetime, Asimov was considered one of the “Big Three” science fiction writers, along with Robert A. Heinlein and Arthur C. Clarke. A prolific writer, he wrote or edited more than 500 books.

  • A robot may not injure a human being or, through inaction, allow a human being to come to harm
  • A robot must obey the orders given it by human beings except where such orders would conflict with the First Law
  • A robot must protect its own existence as long as such protection does not conflict with the First or Second Laws
Partial sources: https://natureofcode.com/book/chapter-8-fractals/, Wikipedia, Google
Posted on by

Noise by Daniel Kahneman | 3 Distinctions

The Michael Shermer Show with Daniel Kahneman – Noise: A Flaw in Human Judgment

DESCRIPTION

Imagine that two doctors in the same city give
different diagnoses to identical patients. Now
imagine that the same doctor making a different
decision depending on whether it is morning or
afternoon, or Monday rather than Wednesday.
This is an example of noise: variability in
judgments that should be identical.

Shermer speaks with Nobel Prize winning
psychologist and economist Daniel Kahneman
about the detrimental effects of noise and what
we can do to reduce both noise and bias, and
make better decisions in: medicine, law, economic
forecasting, forensic science, bail, child
protection, strategy, performance reviews, and
personnel selection.

Video clip – 3 minutes

Noise by Daniel Kahneman | 3 Distinctions

SUPPORT THE PODCAST
If you enjoy the podcast, please show your
support by making a donation. Your patronage will
ensure that sound scientific viewpoints are heard
around the world.
https://www.skeptic.com/donate/
SPONSOR
Wondrium
https://wondrium.com/shermer
SPONSOR
Brilliant
https://brilliant.org/MichaelShermer/

Listen to The Michael Shermer Show via Apple
Podcasts, Spotify, Amazon Music, Google
Podcasts, Stitcher, iHeartRadio, and Tuneln.
https://www.skeptic.com/michael-sherm..

Full Video – Noise by Daniel Kahneman

Full video at https://youtu.be/5CFjERpwFys

Posted on by

Michio Kaku: 3 mind-blowing predictions about the future

What lies in store for humanity? Theoretical physicist Michio Kaku explains how different life will be for your descendants—and maybe your future self, if the timing works out.
▸
15 min
—
with
Michio Kaku

Michio Kaku: 3 mind-blowing predictions about the future
  1. We will become a space-faring species
  2. We will expand the brain’s capabilities
  3. We will defeat cancer

About

Michio Kaku (Japanese: カク ミチオ, 加来 道雄, born January 24, 1947) is an American theoretical physicist, futurist, and popularizer of science (science communicator). He is a professor of theoretical physics in the City College of New York and CUNY Graduate Center. Kaku is the author of several books about physics and related topics and has made frequent appearances on radio, television, and film. He is also a regular contributor to his own blog, as well as other popular media outlets. For his efforts to bridge science and science fiction, he is a 2021 Sir Arthur Clarke Lifetime Achievement Awardee.

Sources
Posted on by

The Metre – the repeating circle & triangulation

The Metre (meaning measure) was one ten-millionth of the distance from the North Pole to the Equator! France embarked on a first large scale measurement. It took 7 years to measure the distance from Dunkirk to Barsalona. They used triangulation with an instrument called the Repeating Circle along with trigonometry.

The standardization of measurement: the Metre

Creating the Metre – a universal standard

By the 16th century, there we over 250,000 weights and measures in Europe. This effected trade, navigation, building plans, etc. Fire hoses would not connect from town to town. France chose to create a standard by measuring something unchangeable. They chose the Earth. Before this standardization, the human body (the Ruler of the land) would make new measurements upon gaining power.

The Repeating Circle

Repeating Circle
Repeating Circle

DESCRIPTION

This is one of two double repeating circles that Ferdinand Rudolph Hassler, the first superintendent of the U. S. Coast Survey, ordered from Edward Troughton in London in 1812, and that was shipped in 1815. The large circle may be angled from vertical to horizontal to the opposite vertical position. It is graduated to 10 minutes, and read by four verniers and two magnifiers to single minutes.

A repeating circle is a geodetic instrument with two telescopes that is designed to reduce errors by repeated observations taken on all parts of the circumference of a circle. The form was developed by the Chevalier de Borda, first executed by Etienne Lenoir in Paris around 1789, and popular for about 50 years.

Ref: F. R. Hassler, “Papers on Various Subjects Connected with the Survey of the Coast of the United States,” Transactions of the American Philosophical Society 2 (1825): 232-420, on 315-320 and pl. VII. “The Repeating Circle Without Reflection, as made by Troughton,” in The Cyclopaedia: or, Universal Dictionary of Arts, Sciences, and Literature, edited by Abraham Rees (London, 1819), Vol. VII, Art “Circle.”

Image credit:

NAME: repeating circle MAKER: Troughton and Simms PLACE MADE: United Kingdom: England, London MEASUREMENTS: overall: 32 1/8 in x 26 3/4 in x 17 in; 81.6356 cm x 67.945 cm x 43.18 cm upper circle: 17 1/2 in; 44.45 cm circle at base: 13 1/2 in; 34.29 cm telescope: 24 in; 60.96 cm overall; base: 16 3/4 in x 15 1/4 in x 16 in; 42.545 cm x 38.735 cm x 40.64 cm overall; horizontal circle: 13 in x 23 in x 20 in; 33.02 cm x 58.42 cm x 50.8 cm ID NUMBER PH.314640 CATALOG NUMBER 314640 ACCESSION NUMBER 208213

Continue reading The Metre – the repeating circle & triangulation
Posted on by

Three types of hypotheses

* Simple hypothesis

* Complex hypothesis

* Null hypothesis

What is a hypothesis

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more variables. An independent variable is something the researcher changes or controls. A dependent variable is something the researcher observes and measures.

Daily apple consumption leads to fewer doctor’s visits.

In this example, the independent variable is apple consumption — the assumed cause. The dependent variable is the frequency of doctor’s visits — the assumed effect.

Three steps to hypothesis testing

  • State the null hypothesis and alternative hypothesis
  • Decide on test static and critical value
  • Compute p-value. If P-value is less than the critical value reject the null hypothesis and accept the alternative hypothesis

Simple hypothesis

Simple hypotheses are ones which give probabilities to potential observations. The contrast here is with complex hypotheses, also known as models, which are sets of simple hypotheses such that knowing that some member of the set is true (but not which) is insufficient to specify probabilities of data points.

Normal curve​
Simple hypothesis – Bell Curve

Complex hypothesis

Complex hypothesis is that one in which there are multiple dependent as well as independent variables. Example: Global warming causes icebergs to melt which in turn causes major changes in weather patterns.

Complex hypothesis
Complex hypothesis

Null hypothesis

A null hypothesis is a hypothesis that says there is no statistical significance between the two variables in the hypothesis. … For example, a null hypothesis would be something like this: There is no statistically significant relationship between the type of water I feed the flowers and growth of the flowers.

null hypothesis
Null hypothesis

Create a Null Hypothesis

Depending on your study, you may need to perform some statistical analysis on the data you collect. When forming your hypothesis statement using the scientific method, it’s important to know the difference between a null hypothesis vs. the alternative hypothesis, and how to create a null hypothesis. 

  • A null hypothesis, often denoted as H0, posits that there is no apparent difference or that there is no evidence to support a difference. Using the motivation example above, the null hypothesis would be that sleep hours have no effect on motivation.
  • An alternative hypothesis, often denoted as H1, states that there is a statistically significant difference, or there is evidence to support such a difference. Going back to the same carrot example, the alternative hypothesis is that a person getting six hours of sleep has less motivation than someone getting eight hours of sleep.
Posted on by

Cantor’s threefold division of infinity

Excerpts from Infinity and the Mind

Rudy Rucker
Infinity and the mind by Rudy Rucker
Infinity and the Mind

… This threefold division [of infinity] is due to Cantor, who, in the following passage, distinguishes between the:

  • Absolute Infinite
  • Physical infinities
  • Mathematical infinities

The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully indepen dent other-worldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number, or order type. I wish to make a sharp contrast between the Absolute and what I call the Transfinite, that is, the actual infinities of the last two sorts, which are clearly limited, subject to further increase, and thus related to the finite,

PHYSICAL INFINITIES

There are three ways in which our world appears to be un bounded and thus, perhaps, infinite. It seems that time cannot end. It seems that space cannot end. And it seems that any interval of space or time can be divided and subdivided endlessly. We will consider these three apparent physical infinities in three subsections.

Temporal Infinities

One of the chief consequences of Einstein’s Special Theory of Relativity is that it is space-time that is fundamental, not isolated space which evolves as time passes. I will not argue this point in detail here, but let me repeat that on the basis of modern physical theory we have every reason to think of the passage of time as an illusion. Past, present, and future all exist together in space-time.

So the question of the infinitude of time is not one that is to be dodged by denying that time can be treated as a fixed dimension such as space. The question still remains: is time infinite? If we take the entire space-time of our universe, is the time dimension infinitely extended or not?

Möbius strip ​
Möbius strip

Spatial Infinities

Whether or not our space is actually infinite is a question that could conceivably be resolved in the next few decades. Assuming that Einstein’s theory of gravitation is correct, there are basically two types of universe: i) a hyperspherical (closed and
unbounded) space that expands, and then contracts back to a point; ii) an infinite space that expands forever. It is my guess that case i) will come to be most widely accepted, if only because the notion of an actually infinite space extending out in every direction is so unsettling.

The fate of the universe in case i) is certainly more interesting, since such a universe collapses back to an infinitely dense space-time singularity that may serve as the seed for a whole new universe. In case il), on the other hand, we simply have cooling and dying suns drifting further and further apart in an utterly empty black immensity … and in the end there are only ashes and cinders in an absolute and eternal night.

The question we are concerned with here is whether or not space is infinitely large. There seem to be three options: i) There is some level n for which -dimensional space is real and infinitely extended. The situation where our three-dimensional space is infinitely large falls under this case. ii). There is some n such that there is only one n-dimensional space. This space is to be finite and unbounded, and there is to be no reality to n + 1 dimensional space.

The situation where our three-dimensional space is finite and unbounded, and the reality of four-dimensional space denied, falls under this case. iii). There are real spaces of every dimension, and each of these spaces is finite and unbounded. In this case we either have an infinite number of universes, duoverses, triverses, etc., or we reach a level after which there is only one n-verse for each n.

Read more in his book Infinity and the Mind

Posted on by — Leave a comment

Three geometries – Euclidean, Hyperbolic, Elliptical

Three geometries - Euclidean, Hyperbolic, Elliptical
Three geometries – Euclidean, Hyperbolic, Elliptical

There are precisely three different classes of three-dimensional constant-curvature geometry: Euclidean, hyperbolic and elliptic geometry. The three geometries are all built on the same first four axioms, but each has a unique version of the fifth axiom, also known as the parallel postulate. The 1868 Essay on an Interpretation of Non-Euclidean Geometry by Eugenio Beltrami (1835 – 1900) proved the logical consistency of the two Non-Euclidean geometries, hyperbolic and elliptical.

The parallel postulate is as follows for the corresponding geometries.

Euclidean geometry: Playfair’s version: “Given a line l and a point P not on l, there exists a unique line m through P that is parallel to l.” Euclid’s version: “Suppose that a line l meets two other lines m and nso that the sum of the interior angles on one side of lis less than 180°. Then m and n intersect in a point on that side of l.” These two versions are equivalent; though Playfair’s may be easier to conceive, Euclid’s is often useful for proofs.

Hyperbolic geometry: Given an arbitrary infinite line l and any point P not on l, there exist two or more distinct lines which pass through P and are parallel to l.

Elliptic geometry: Given an arbitrary infinite line land any point P not on l, there does not exist a line which passes through P and is parallel to l.

Source and more info: https://en.m.wikibooks.org/wiki/Geometry/Hyperbolic_and_Elliptic_Geometry

Posted on by — Leave a comment

Physics has a law that explains everything. And it’s brought to you by the number three

Jamie Seidel: News Corp Australia Network

The rule of three has become something akin to a social law of gravity — as if the number is behind everything.

Three groups of experimentalists have independently observed a strange state of matter that forms from three particles of any type and at any scale, from practically infinitesimal to infinite.

Forget pairs. They’re old pat. And 42? We still don’t know the question.

Three rings to rule them all, and in the darkness bind them ... apologies Lord Of The Rings purists.
Three rings to rule them all, and in the darkness bind them … apologies to Lord Of The Rings purists.

Comedians insist three is the best pattern to exploit perceptions and deliver punchlines; three features prominently in titles, such as The Three Little Pigs, Three Musketeers, Goldilocks and the Three Bears; even the Romans believed three was the ultimate number: “Omne trium perfectum” was their mantra — everything that comes in threes is perfect.

Now, it seems Mother Nature may also think in threes. Especially at the very edge of physics — quantum mechanics.

A Soviet nuclear physicist first proposed the idea back in the 1970s — and was met with derision.

For 45 years number-crunchers around the world have been attempting to topple Vitaly Efimov’s idea and prove his equations wrong.

Shih-Kuang Tung of the University of Chicago
Shih-Kuang Tung of the University of Chicago holding Borromean rings, which have often been used as a symbol of unity. If one ring is removed, all three fall apart.

They’ve failed; and his “outlandish” theory is now on the point of being proven.

Most importantly, Efimov felt that sets of three particles could arrange themselves in an infinite, layered pattern. What form these layers take helps determine the makeup of matter itself.

Jump forward four decades, and technological advances now allow his groups of three quantum particles to be studied and manipulated.

The quantum condition — now known as Efimov’s state — is visible only under supremely cold conditions. Matter, when chilled to a few billionths of a degree above Absolute Zero, does strange things …

If you want the technical details, read Quanta Magazine’s article which examines the recent research papers.

Continue reading Physics has a law that explains everything. And it’s brought to you by the number three
Posted on by

Big Oh(O) Big Theta(Θ) Big Omega(Ω)

big O Notation

Analysis of Algorithms

Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. It is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation.

Big Oh Running Time Complexity
Big Oh Running Time Complexity

Two simple concepts separate properties of an algorithm itself from properties of a particular computer, operating system, programming language, and compiler used for its implementation. The concepts, briefly outlined earlier, are as follows:

• The input data size, or the number n of individual data items in a single data instance to be processed when solving a given problem. Obviously, how to measure the data size depends on the problem: n means the number of items to sort (in sorting applications), number of nodes (vertices) or arcs (edges) in graph algorithms, number of picture elements (pixels) in image processing, length of a character string in text processing, and so on.

• The number of elementary operations taken by a particular algorithm, or its running time. We assume it is a function f(n) of the input data size n. The function depends on the elementary operations chosen to build the algorithm.

Algorithms are analyzed under the following assumption: if the running time of an algorithm as a function of n differs only by a constant factor from the running time for another algorithm, then the two algorithms have essentially the same time complexity. Functions that measure running time, T(n), have nonnegative values
because time is nonnegative, T(n) ≥ 0. The integer argument n (data size) is also nonnegative.

Definition 1 (Big Oh)

Big Oh Rank Meaning
Big Oh Rank Meaning

Let f(n) and g(n) be nonnegative-valued functions defined on nonnegative integers n. Then g(n)is O(f(n)) (read “g(n)is Big Oh of f(n)”) iff there exists a positive real constant c and a positive integer n0 such that g(n) ≤ c f(n) for all n > n0.

Note. We use the notation “iff ” as an abbreviation of “if and only if”.

Continue reading Big Oh(O) Big Theta(Θ) Big Omega(Ω)
Posted on by — Leave a comment

Pascal’s Triangle

Relation between-Pascals triangle and fibonacci series

Pascal, Blaise (1623-1662)
Pascal, Blaise (1623-1662)

Pascal’s Triangle was originally developed by the ancient Chinese, but Blaise Pascal was the first person to discover special patterns contained inside the triangle. They teach his ideas in various schools online in math courses. You probably also heard of this guy from your high school math teacher.

Triangular numbers appear in Pascal’s Triangle. In fact, the 3rd diagonal of Pascal’s Triangle gives all triangular numbers as shown below:

 

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1
1 8 28 56 70 56 28 8 1
1 9 36 84 126 126 84 36 9 1

Continue reading Pascal’s Triangle

Posted on by

Three ways to see an atom

Atom

A remarkable photo of a single atom trapped by electric fields has just been awarded the top prize in a well-known science photography competition. The photo is titled “Single Atom in an Ion Trap” and was shot by David Nadlinger of the University of Oxford
A remarkable photo of a single atom trapped by electric fields has just been awarded the top prize in a well-known science photography competition. The photo is titled “Single Atom in an Ion Trap” and was shot by David Nadlinger of the University of Oxford.

If there is no way in the world to see an atom, then how do we know that the atom is made of protons, electrons, neutrons, the nucleus and the electron cloud?

There are three ways that scientists have proved that these sub-atomic particles exist. They are direct observation, indirect observation or inferred presence and predictions from theory or conjecture.

Atomic model
Atomic model

Scientists in the 1800’s were able to infer a lot about the sub-atomic world from The Periodic Table of Elements by Mendeleyev gave scientists two very important things. The regularity of the table and the observed combinations of chemical compounds prompted some scientists to infer that atoms had regular repeating properties and that maybe they had similar structures.

 

J.J. Thompson
J.J. Thompson

Other scientists studying the discharge effects of electricity in gasses made some direct discoveries. J.J. Thompson was the first to observe and understand the small particles called electrons. These were called cathode rays because they came from the cathode, or negative electrode, of these discharge tubes. It was quickly learned that electrons could be formed into beams and manipulated into images that would ultimately become television. Electrons could also produce something else. Roentgen discovered X-rays in 1895. His discovery was a byproduct of studying electrons. Protons could also be observed directly as well as ions as “anode” rays. These positive particles made up the other half of the atomic world that the chemists had already worked out. The chemists had measured the mass or weight of the elements. The periodic chart and chemical properties proved that there was an atomic number also. This atomic number was eventually identified as the charge of the nucleus or the number of electrons surrounding an atom which is almost always found in a neutral, or balanced, state.

Ernest Rutherford
Ernest Rutherford

Rutherford proved in 1911, that there was a nucleus. He did this directly by shooting alpha particles at other atoms, like gold, and observing that sometimes they bounced back the way they came. There was no way this could be explained by the current picture of the atom which was thought to be a homogeneous mix. Rutherford proved directly by scattering experiments that there was something heavy and solid at the center. The nucleus was discovered. For about 20 years the nucleus was thought to consist of a number of protons to equal the atomic weight and some electrons to reduce the charge so the atomic number came out right. This was very unsettling to many scientists. There were predictions and conjectures that something was missing.

James Chadwick
James Chadwick

In 1932 Chadwick found that a heavy neutral particle was emitted by some radioactive atoms. This particle was about the same mass as a proton, but it had a no electric charge. This was the “missing piece” (famous last words). The nucleus could now be much better explained by using neutrons and protons to make up the atomic weight and atomic number. This made much better sense of the atomic world. There were now electrons equal to the atomic number surrounding the nucleus made up of neutrons and protons.

Wilhelm Conrad Röntgen raggi X
Wilhelm Conrad Röntgen raggi X

Mr. Roentgen’s x-rays allowed scientists to measure the size of the atom. The x-rays were small enough to discern the atomic clouds. This was done by scattering x-rays from atoms and measuring their size just as Rutherford had done earlier by hitting atoms with other nuclei starting with alpha particles.

Cyclotron - 1930 particle accelerator
Cyclotron – 1930 particle accelerator

The 1930’s were also the time when the first practical particle accelerators were invented and used. These early machines made beams of protons. These beams could be used to measure the size of the atomic nucleus. And the search goes on today. Scientists are still filling in the missing pieces in the elementary particle world. Where will it end? Around about 1890, scientists were lamenting the death of physics and pondering a life reduced to measuring the next decimal point! Discoveries made in the 1890’s proved that the surface had only been scratched.

Carbon Atomic Model
Carbon Atomic Model

Each decade of the 1900’s has seen the frontier pushed to smaller and smaller objects. The explosion of knowledge has not slowed down and as each threshold has been passed the amount of new science seems to be greater even as we probe to smaller dimensions. Current theories (if correct) imply that there is even more below the next horizon awaiting discovery

Text Author: Paul Brindza, Experimental Hall A Design Leader

Source: https://education.jlab.org/qa/history_04.html