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Plotting Coulomb’s law or the law of gravity – not quite hyperbolas
Here’s a graph of force versus distance using an inverse square law.
This is Coulomb’s law, showing the magnitude of the force between two electrically charged particles.
It looks hyperbolic – but does this actually qualify as a hyperbola?
What is a hyperbola?
There are many different yet equivalent definitions for hyperbolas, see those definitions here:
Hyperbola, Math Is Fun, The Hyperbola, Lumen, Graphs of Hyperbolas Centered at the Origin, CK-12
For our graph:
Force is plotted on the Y-axis.
‘r’ is the distance between two charged objects, plotted on the X-axis.
In the above example we used Coulomb’s law, but mathematically it is the same form as Newton’s law of universal gravitation:
K is just a constant. With gravity this constant is extremely small.
With electric attraction/repulsion the constant is many orders of magnitude larger.
So for any of these cases, is this curve a hyperbola?
No. Hyperbolas – by definition – are conic sections.
And by definition conic sections must be able to be put into this format:
Ax2 + Bxy + Cy2 +Dx + Ey + F = 0
The above equations – Coulomb’s law and Newton’s law – can’t be put into this format. Thus these curves cannot be hyperbolic.
Rational functions
So what kind of curve are these force vs distance curves?
They are not hyperbolas but they are rational functions: the ratio of two polynomials.
It is called “rational” because one is divided by the other, like a ratio.
Notice that rational functions have horizontal and vertical asymptotes, and inverse relationships, so they visually approximate hyperbolas.
One might even say that they share some properties of hyperbolas without fulfilling all the criteria of actually being one.
A special case of rational functions
Although not applicable for Coulomb’s law, one may note that rational functions of the form (ax+b)/(cx+d) are hyperbolas
As long as determinant, ad-bc, and c, are non-zero.
So hyperbolas are special cases of rational functions.
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#hyperbolas #conic #rationalequations
The Eötvös effect
The Eötvös effect is the change in perceived gravitational force caused by the change in centrifugal acceleration resulting from eastbound or westbound velocity.
The measured effect is caused by the motion of the object traveling with, or against, the rotation of the Earth.
When moving eastbound, the object’s angular velocity is increased (in addition to Earth’s rotation)
thus the centrifugal force also increases, causing a perceived reduction in gravitational force.
When moving westbound, the object’s angular velocity is decreased,
thus the centrifugal force decreases, causing a perceived increase in gravitational force.
From flatearth.ws, debunking flat earth misconceptions
In the early 1900s (decade), a German team from the Institute of Geodesy in Potsdam carried out gravity measurements on moving ships in the Atlantic, Indian, and Pacific oceans.
While studying their results, the Hungarian nobleman and physicist Baron Roland von Eötvös (Loránd Eötvös) noticed that the readings were lower when the boat moved eastwards, higher when it moved westward. He identified this as primarily a consequence of Earth’s rotation.
In 1908, new measurements were made in the Black Sea on two ships, one moving eastward and one westward. The results substantiated Eötvös’ understanding.
Relationship between eötvös effect and Coriolis effect
Some people say that the Eötvös effect is the vertical component of the Coriolis effect. Max on Physics StackExchange explains to us
In many science disciplines, casual versus formal usages become intermixed, and this is certainly one area.
Eötvös is not the vertical component of Coriolis.
The earth is both (a) spherical and (b) spinning. This produces a number of phenomena that affect bodies in motion on or near the surface of the Earth.
In casual usage these phenomena tend to be lumped together into all being called “Coriolis,” but they are actually discrete physical properties that are not related, except for the fact that they are artifacts of (a), (b), or both.
Coriolis is a conservation of angular momentum consideration when objects move north/south across a spinning sphere.
As you move away from the equator latitudinally, the same angular rate of rotation around the Earth’s C/G results in a different velocity in the east/west component, and the effects of this difference is the Coriolis Effect.
Were the Earth a cylinder instead of a sphere, there’d be no Coriolis Force. (*)
Eötvös on the other hand is a centrifugal force/orbital mechanics problem. Eötvös would still occur on a cylinder, where Coriolis would not.
There is an angular momentum force that acts east/west based on the height of an object’s trajectory or orbit, and thus would affect the vertical component of a projectile’s trajectory at long distances involving high trajectories.
But this isn’t Eötvös at all. If I shoot a projectile perfectly vertically a few miles into the air, conservation of angular momentum dictates the projectile will not land back on me, it will land several feet west of me, opposite the direction of the Earth’s spin. It may be more correct to think of this motion as the vertical component of Coriolis.
(*) This gets addressed later on this page. There would be some force, but it would different from what we see on a spherical Earth.
Cultural and historical origin of trigonometry
Cultural and historical origin of trigonometry
Hipparchus of Nicaea 190 – 120 BCE was a Hellenic (“Greek”) astronomer, geographer, and mathematician, considered the founder of trigonometry. He was born in Nicaea, Bithynia – now İznik, Turkey. Ethnically he was a Bithynian.
Hipparchus effectively developed the predecessor of Sine and Cosine through his study of chords, straight line segments whose endpoints both lie on a circular arc. He developed this for a practical reason – he was computing the eccentricity of the orbits of the Moon and Sun.
This work was continued by Claudius Ptolemy (Κλαύδιος Πτολεμαῖος) of Roman Egypt (90–168 CE). He was a Macedonian Hellenist mathematician, astronomer, and geographer.
He was a Macedonian.
Hypatia ~360-415 CE. She was a Hellenistic Neoplatonist philosopher, astronomer, and mathematician, who lived in Alexandria, Egypt, then part of the Eastern Roman Empire.
“Hypatia was known more for the work she did in mathematics than in astronomy, primarily for her work on the ideas of conic sections introduced by Apollonius. She edited the work On the Conics of Apollonius, which divided cones into different parts by a plane. This concept developed the ideas of hyperbolas, parabolas, and ellipses.
With Hypatia’s work on this important book, she made the concepts easier to understand, thus making the work survive through many centuries. Hypatia was the first woman to have such a profound impact on the survival of early thought in mathematics.”
Hypatia, Biographies of women mathematicians
There was much trade and scholarly contact between ancient Greece and India after the Indian campaign of Alexander the Great.
We soon see the continuation of trigonometry in India.
Wikimedia, Roman trade in the subcontinent according to the Periplus Maris Erythraei 1st century CE
Some classic, 4th to 5th century CE Siddhantas defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine, and inverse sine.
They used the words jya for sine, kojya for cosine, utkrama-jya for versine, and otkram jya for inverse sine.
The words jya and kojya eventually became sine and cosine respectively after a mistranslation described above.
The classic Sanskrit astronomy text, the Sūrya Siddhānta, thought to have been composed around 800 CE. It provides table of sines function which parallel the Hipparchian table of chords, though the Indian calculations are more accurate.
This Indian trigonometry was copied and translated by Arabic scholars, where it soon had a great effect in stimulating the Arabic sciences.
In the early 9th century AD, Muhammad ibn Mūsā al-Khwārizmī produced accurate sine and cosine tables, and the first table of tangents. He was also a pioneer in spherical trigonometry.
Muhammad ibn Jābir al-Harrānī al-Battānī (known as Al-Battani, or Albatenius) (853-929 AD) discovered the reciprocal functions of secant and cosecant, and produced the first table of cosecants.
By the 10th century AD, in the work of Abū al-Wafā’ al-Būzjānī, Muslim mathematicians were using all six trigonometric functions.
In 1342, the Jewish scholar Levi ben Gershon, known as Gersonides, wrote “On Sines, Chords and Arcs,” proving the sine law for plane triangles and giving five-figure sine tables.
Gersonides also created the astronomical, surveying, and navigational device known as the Jacob’s Staff.
from Peter Apian’s Cosmographia, 1524 CE
#mathematic #history
Radians
Learning goals: Students will understand
why we traditionally measure angles in degrees
that the usually stated number of degrees in a circle is a social/cultural convention, not a scientific or mathematical fact
that there exists a purely natural measure of angles (radians)
that natural units are not social/cultural conventions: rather they are relationships that exist independently of human choice.
We usually measure angles in degrees. But today we’re going to learn about another way to measure angles, called radians.
First let’s think about everyday life: Outside of school, in any construction project, we measure angles in degrees.
That just seems like the only and obvious way.
Plumbing and Heating (46.0503) T-Chart
We’re told that there are 360 degrees in a circle.
But once we get to high school trigonometry we start using a different way of measuring angles.
Asking “Why change?” might be wrong question.
A better question would be “Why did we start measuring angles in degrees to begin with?”
Because 2500 years ago ancient Sumerians discovered that there were about 360 days in a year, that’s why.
This knowledge was picked up by the ancient Babylonians.
That’s not exactly correct – we now know that there are about 365.25 days in a year.
Geography connections: Where was ancient Babylon?
Because of this, the ancient Sumerians and Babylonians used 360 and 60 as building blocks for their counting systems.
The Babylonians knew, of course, that the perimeter of a hexagon is exactly equal to six times the radius of the circumscribed circle, in fact that was evidently the reason why they chose to divide the circle into 360 degrees (and we are still burdened with that figure to this day).
A History of Pi, Petr Beckmann
That system then passed down to later Greek and Roman civilizations, Islamic civilizations, and then to the European and Asian civilizations.
We’re so used to using it that dividing a circle into 360 parts seems natural. But there’s nothing necessary or natural about this. We didn’t discover this – we defined this. It’s just a convention.
The degree is an arbitrary unit; basically any division of a circle would work as a system of measurement. The degree has the advantage that 360 divides evenly by 2, 3, 4, 5, 6, 8, 9 & 10 making it easy to mentally calculate an angle; indeed this is the major advantage of all old imperial units.
Ray Gallagher, Belfast, Northern Ireland
We could have chosen any other number. During the French Revolution there was an attempt to make a metric system version of angle measurement.
Metric right angles were defined as having 100 “degrees”
(obviously, these degrees were smaller than regular degrees.)
A metic circle would had 400 metric degrees instead of 360.
To avoid confusion these slightly smaller degrees were called gradians, or the gon grad.
360 degrees = 400 gradians
This system was used by some surveyors and engineers in Europe for some time. It is not common anymore. Here we see a French compass from 1922, divided into 400 degrees.
compassmuseum.com/diverstext/divisions.htm
From Compassipedia, part 2, the division systems.
In the end this new metric degree system didn’t catch on.
Using 360 is just easy for people to use. It is an abundant number, i.e. there are many factors. People find it easy to mentally divide the circle into 2, 3, 4, 5, 6, 8, 9, 10, 12, etc.
But there is one way to measure angles that is natural:
Radians
A radian is a special, natural angle.
There are a few equivalent ways of defining it:
the angle defined when the radius is wrapped round the circle’s perimeter
the angle defined when the arc length = radius
the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle. (this is the more complete, most accurate phrasing.)
Construct an angle of one radian
Know these abbreviations:
θ = subtended angle (in radians)
s = arc length
r = radius
We can measure any size angle, small or large, in radians.
The size of an angle θ, in radians = the ratio of the arc length to the radius of the circle
θ = s/r
Conversely, the length of the intercepted arc = radius multiplied by the magnitude of the angle
s = rθ
How does this relate to π?
The magnitude (in radians) of one complete revolution = 360 degrees
The magnitude (in radians) of one complete revolution = length of the entire circumference divided by the radius
1 revolution = (2πr / r) radians
1 revolution = 2π
Thus 2π radians = 360 degrees
Thus 1 radian = 180/π ≈ 57.295779513082320876 degrees.
What is π?
When a circle’s diameter is 1 then its circumference is π.
Distance halfway around a circle will be 3.14159265… Pi
When a circle’s radius is 1 – a unit circle – then its circumference is 2π.
Here we see a circle rolling out to 2π.
In calculus and physics, radians are usually more useful and natural to use than degrees.
Further lessons
Intuitive Guide to Angles, Degrees and Radians
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Learning Standards
Math Common Core
Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
CCSS.MATH.CONTENT.HSF.TF.A.2
Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.
Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
Science of sci-fi movies and TV shows
Time to look at the Good, the Bad, and the Ugly, of science in movies and TV shows! 🙂
Science of sci-fi movies and TV shows
Ant-Man, Fantastic Voyage & Honey I Shrunk The Kids: Miniaturization
2001 A Space Odyssey & Babylon 5 – Artificial gravity in a space station
Batman The Dark Knight
Doctor Who
Godzilla vs the scaling laws of physics
Jurassic Park
Megalodon: The Monster Shark Lives
Mermaids: The Body Found
Star Trek: The physics of warp drive
Superman Returns (2006) – Calculating his strength
The Core (Earth Science & Physics ideas)
The Expanse, TV and books, Leviathan Wakes
Tremors (featuring Graboids)
Up – Buoyancy of balloons
Books
Insultingly Stupid Movie Physics
Hollywood’s Best Mistakes, Goofs and Flat-Out Destructions of the Basic Laws of the Universe
by Tom Rogers
Would the bus in Speed really have made that jump? -Could a Star Wars ship actually explode in space? -What really would have happened if you said “Honey, I shrunk the kids”? The companion book to the hit website (www.intuitor.com/moviephysics), which boasts more than 1 million visitors per year, Insultingly Stupid Movie Physics is a hilarious guide to the biggest mistakes, most outrageous assumptions, and the outright lunacy at work in Hollywood films that play with the rules of science.
Don’t Try This At Home!: The Physics of Hollywood Movies\
by Adam Weiner
A fresh look at the basics of physics through the filmmaker’s lens. It will deconstruct, demystify, and debunk popular Hollywood films through the scientific explanations of the action genre’s most dynamic and unforgettable scenes.
Websites
Intuitor: Insultingly Stupid Movie Physics
PHYSICS IN FILMS by Costas J. Efthimiou
Thanks for reading. While you’re here see our other articles on astronomy, biology, chemistry, Earth science, mathematics, physics, the scientific method, and making science connections through books, TV and movies.
Teaching ionic formulas with Legos as manipulatives
We can use giant Legos as manipulatives to teach students about ionic bonding formulas.
We can explore cation-to-anion ratios.
The blocks may represent trivalent, divalent, and monovalent cations and anions.
from “An Interlocking Building Block Activity
in Writing Formulas of Ionic Compounds”
You might be able to get them at low cost from a community yard sale group. Just label them yourself.
Ruddick and Parrill write
LEGO blocks provide excellent representations of ions, particularly because the blocks are color coded, and the valency of the ion can be represented by the number of “dots”, or raised knobs, on a brick. For example, a blue 1 × 3 brick (1 dot wide and 3 dots long) can represent cationic Al3+. The oxide ion, O2−, can be represented by a red 1 × 2 brick (1 dot wide and 2 dots long) (Figure 1). These two types of bricks can then be assembled to make a product that helps students determine the cation-to-anion ratio in aluminum oxide and write the chemical formula.
Materials
This is just one example. Mega Bloks.
References
JCE Classroom Activity #113: An Interlocking Building Block Activity in Writing Formulas of Ionic Compounds, Kristie R. Ruddick and Abby L. Parrill, J. Chem. Educ. 2012, 89, 11, 1436–1438, 9/19/2012
https://doi.org/10.1021/ed200513y
Related
Teaching chemistry with LEGO bricks
Ryo Horikoshi, Chemistry Teacher International, 12/21/2020
“Since LEGO bricks possess varieties of shapes and colors, they can be employed to design various teaching aids, including periodic tables, molecular models, polymer structure models, and frameworks for handmade measuring instruments. The polymeric structure models are generally difficult to build with typical ball-and-stick type molecular models; however, they can be easily built, employing LEGO bricks.”
DOI: https://doi.org/10.1515/cti-2020-0017
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What is genetic material?
What is genetic material? What are genes for?
Look closely inside any living creature – plants, animals, even fungi.
If you use a microscope you can see individual cells.
Looking even more closely you can see that cells have a nucleus
Look even more closely inside the nucleus and you’ll see chromosomes.
(Here the nucleus has been punctured, and chromosomes are spilling out.)
Now if you look with some super sophisticated techniques then you’ll see that these chromosome aren’t solid.
They seem to be made of a kind of thread, something really thin that is wrapped up to make a shape.
Looking even close, this thread is made up of molecules bonded together into a kind of helix (spiral shape.)
Not just that, but two spiral shapes wrapped around each other – a double helix
This beautiful molecule here is DNA.
The DNA wraps itself up to make those chromosomes.
Why is it here? This contains the instructions of life itself.
DNA contains all of the information necessary to build a living thing.
Each separate piece of DNA – each gene – builds a different part of the organism.
So we call this genetic material.
There’s another important type of genetic material, RNA
In most forms of life DNA is the “original copy”, the master blueprint.
Cells work by making a working copy of the blueprint.
This copy is made of mRNA (messenger RNA) molecules.
That RNA then leaves the nucleus (see above animation) and gets used by the cell to build things.
So both DNA and RNA are types of genetic information.
Genetic info in viruses
Although viruses are not alive in the same way that cells are, they also have genetic material.
Some viruses have DNA, others have RNA.
The basic idea is still the same. The DNA, or RNA, in a virus is the instruction set to build more virus particles.
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How does the Schrodinger equation create orbitals?
How does the Schrödinger equation (from quantum mechanics) create atomic orbitals and molecular orbitals that?
Where do these beautiful three dimensional shapes come from?
Why do they have the shapes that they do?
This lesson assumes that you have already learned about –
waves and superposition
constructive and destructive interference
the classical models of the atom
Neils Bohr and his semi-quantum mechanical model of the atom
the Schrödinger model of the atom
Physicists use the Schrödinger equation to model an e- around a nucleus.
Hence, e- = electron
Let’s first review the idea of superposition: Notice what happens here when two waves pass through the same space, at the same time.
The waves are not bouncing off of each other – they pass through each other.
Waves can add together – constructive interference. That’s what we see here.
Below we see two waves that add together to create a region where they cancel out. This is destructive interference.
Can we make waves appear to stand still? Consider what happens when two waves come at each other, at just the right speed and height:
A standing wave is produced!
We see this in music all the time. Pluck a string on a guitar or violin.
Musicians call the first standing wave the fundamental, or first harmonic.
Higher frequency standing waves are called overtones.
This violin string isn’t quite showing a standing wave, but it gets close.
We can get standing wave on any physical object, like on a cymbal, or on any flat round disk.
Your loudspeakers, on a stereo system or in earbuds, move like this.
Well, in the Schrödinger equation:
Constructive interference leads to regions where we’re more likely to find the e-.
Destructive interference leads to regions where we’re less likely to find the e-.
Here we model an electron as a standing wave around the nucleus of an atom. Compare the left and right side.
On the left a standing wave is produced, when the wavelength is of such a length that it creates constructive interference. On the right the wavelength has a different length, and no constructive behavior develops.
Allowed Not allowed
This is what happens in atoms – e- aren’t solid objects.
Instead, e- are understood to be quantum phenomenon that follow a wave equation!
When the wavelength of the e- allows for constructive interference, that is where it has an effect.
Of course, that model for e- around an atom is too simple. It is flat.
So let’s show a standing wave in three dimensions.
How do these equations lead to the spherical “s” orbitals?
This next image is from 6.3 Development of quantum theory
Here scientists use the Schrödinger equation to model an e- as standing waves in three dimensions.
from Opentextbc.ca, Chapter 6. Electronic Structure and Periodic Properties of Elements
What would it look like inside these spheres?
Below is a “traveling wave resulting from a point source emitting symmetrically in 3 dimensions producing spherical wavefronts.”
One of the upper quadrants of the region has been removed in order to expose the internal structure of the wave”
from intothecontinuum.tumblr.com
Just like the 2D case we can also create a ‘”standing wave resulting from an outward and inward traveling wave”
from intothecontinuum.tumblr.com
Now that we have made it here, here is a chapter on electron orbitals from ChemPRIME (Moore et al.)
This chapter is great for honors or AP physics or chemistry students.
Creating d and f orbitals
The same general idea is true for how d orbitals and f orbitals appear.
There are many possibilities for the wave function of an e- with a higher amount of energy.
We add all of them together – this is superposition.
Most of those waves cancel each other out (destructive interference.)
Whatever remains is there and assumes a certain shape (constructive interference.)
Here is a simplified version – different wave possibilities adding together to create a d orbital.
Molecular orbitals
Hybrid orbitals, what we see in molecules, are more complex examples of the same idea:
We see wave functions interfering with each other.
When two orbitals overlap we get more complex forms of interference, leading to new shapes.
Here, a hydrogen atom 1s orbital bonds with another Hydrogen atom 1s orbital.
This creates a H2 molecule with a sigma orbital (σ).
https://www.grandinetti.org/molecular-orbital-theory
This is a representation of two oxygen atoms merging into an O2 molecule in the singlet state.
This shows the most accurate representation for the actual shape of the molecule.
The original 2s and 2p atomic orbitals can be seen merging to create Sigma and Pi orbitals, which bind the atoms together.
The 1s orbitals do not combine and still show the individual atoms.
This GIF is from Wikimedia Commons, O2 MolecularOrbitals Anim.gif
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.
Teaching quantum mechanics in high school
I would love to hear how science teachers are teaching quantum mechanics in high school. Many of us plan to have a week long sequence in Honors Physics, or AP Physics, or perhaps in a science elective.
This could even be done in regular college-prep level physics classes if the students are so motivated. Having discussions like these – even done totally qualitatively, without math, are hallmarks of an inspiring science classroom.
I’m putting together here my resources on teaching various aspects of quantum mechanics, in what seems to be a reasonable order. Please understand that this is not an online textbook. These articles were each developed separately, so there is a lot of overlap.
Also, there is no mathematics, calculus, or differential equations required.
The nature of reality itself! The allegory of the cave
The wave nature of matter
Why did we have to develop modern physics? What is wrong with just keeping classical physics?
Early quantum theory
Schrodinger model of the atom
Schrodingers cat
Articles about the consequences of quantum mechanics
What are covalent bonds between atoms?
The normal force: So, objects really never touch?
How time and space could be a quantum error correcting code
The origins of space and time
Quantum teleportation (and it isn’t what you think)
Time’s arrow may be traced to quantum source
The quantum thermodynamics revolution
The four possible types of multiverses
Parallel universes in quantum mechanics
Karpman’s drama triangle
Practical psychology – Thoughts on Karpman’s drama triangle, narcissistic personality disorder, and finding an alternative – the empowerment triangle.
This phenomenon can occur in dysfunctional family or community dynamics; it is especially prevalent in social media arguments. This model explains much of the psychology behind virtue signaling and moral grandstanding.
Karpman’s drama triangle is a tool used in psychotherapy, specifically transactional analysis. Once a person’s role in this dynamic is realized, people can develop skills that create a healthier and more productive dynamic, e.g. the empowerment triangle.
Let’s look at the drama triangle in action:
This next section excerpted from TED: The Empowerment Dynamic
The Rescuer – looks for victims, businesses or causes to save and are quick to jump-in and save the day, even when others are responsible. Rescuers believe they will be appreciated and valued for their good deeds. They feed off of crisis’ so they can be needed and valued for their help.
The Persecutor – is controlling, blaming, critical, oppressive, authoritarian, rigid… Persecutors believe they must win and convince others that they are right. They have little compassion for another’s perspective or way of doing things.
The Victim – feels powerless and at the mercy of life’s events. Avoids taking responsibility for their actions, finding it easier to blame others or their circumstances. Suffering is a perpetual state for victims. Self-pity. Interestingly, these feelings then create an odd state of entitlement and specialness.
Lynne Forest writes:
“Each person has a primary or most familiar role – what I call their “starting gate” position. This is the place from which we generally enter, or “get hooked” onto, the triangle. We first learn our starting gate position in our family of origin.
Although we each have a role with which we most identify, once we’re on the triangle, we automatically rotate through all the positions, going completely around the triangle, sometimes in a matter of minutes, or even seconds, many times every day.”
Why do people fall into the drama triangle?
Behaviors in the drama triangle, while dysfunctional, are highly seductive.
The persecutor gets to feel powerful.
The victim gets pity, and a pass from responsibility.
The rescuer feels like a saviour; they perceive themselves to have moral superiority. They feel that only they can rescue the victim. Ironically, rescuers themselves can become persecutors of those they address if others do not accept the rescuer’s demands to be seen as saviour.
Ironically the rescuer does not really help the victim. In fact they often can keep the victim in place by promoting the victimhood narrative.
Victims only break out of this dynamic by rejecting powerless. One needs to reclaim personal agency and accountability. One needs to develop allies of equals, not rescuers who manipulate. By doing so then the former victim takes power away from the persecutor – they no longer needs the rescuer.
Image created by Robert Kaiser
What the Drama Triangle has in common with narcissistic personality disorder
Sarah Davies writes
This drama triangle is a dynamic often seen with narcissists and is what relentlessly plays out in relationships of narcissistic abuse and other toxic relationships.
The ‘victim’ position is the “poor me” stance. The person in this position sees themselves as being victimized, bullied, being hard done by, helpless, hopeless, persecuted or oppressed.
The Persecutor: The persecutor is often the bully narcissist on the attack. This is the position of blaming, shaming, controlling, being aggressive, oppressive, judgmental or authoritative, threatening and/or arrogant.
The Rescuer: The rescuer is often the classic codependent, echoist, enabler, fixer and/or helper. The rescuer tends to respond to the real or portrayed ‘helplessness’ of the victim. The person in the rescue position will assume responsibility on the ‘victims’ behalf….
The rescuer will take on responsibility for situations or issues that are not theirs…. the narcissist shifts quite skillfully between any one of these positions… they can use the position of rescuer to control and manipulate.
Moving to the empowerment triangle
Ideally people will want to transition from the drama triangle model to an empowerment triangle model. This happens when people recognize that their previous dynamic was not working, and they become willing to learning new ways to talk with others. When we do so, we develop new skills, as shown in this table:
Let’s see what this change looks like:
Image created by Robert Kaiser
KaiserScience 
