Tipping points & feedback loops

Consider:

the nature of a tipping point,
tipping points as initiating self-amplifying feedback cycles and a change of state, and
the relevance of this to climate change.

Tipping points

Anyone who has ever tilted too far back on a chair understands the concept of tipping points. Once your legs push your chair past the tipping point, there’s the shock of toppling backwards. Stopping your legs from that unwise pushing now has no impact; your only hope is that a friend might grab you to stop you from crashing to the floor.

Normally, when sitting in a chair, gravity is your friend. It keeps you safely seated. However, once you lean back past the tipping point, the change of state is that gravity suddenly becomes dangerous. Gravity now tips you further backwards, which increases the portion of your weight causing the tipping and tips you more strongly backwards. This is a self-amplifying feedback cycle that sets in just past the tipping point.

Examining the tipping point of a building brick offers insights into:

the toppling of a person in a chair,
the toppling of a glass of wine,
the concept of climate tipping points, and
climate change

On this page:

Conclusions

Conclusion: Amplifying feedback starts when the tilt passes the tipping point

(1) The tilt past the tipping point increases		(2) The weight acting to topple the block increases

(4) The angular velocity		(3) The angular acceleration increases

The diagram shows a feedback cycle. While this feedback loop is dominant:

an increase in the tilt past the tipping point causes
an increase in the block’s weight that acts to topple the block by exerting a twisting force (torque), which causes
an increase in the angular acceleration of the block, which causes
an increase in the angular velocity of the block, which closes the loop, causing
an increase in the tilt past the tipping point.

(The study of physics often presents situations where the acceleration is constant. This is not true for a toppling block where the block’s acceleration increases as it topples.)

After you push the block past its tipping point, the more it tilts, the more it will tilt. The feedback cycle becomes dominant and amplifies the tilting. (See my page on self-amplifying feedback loops.)

Many climate feedback cycles pose a significant threat to life as we know it. One of these is the amplifying feedback loop that escalates both “global heating” and the “release of methane from melting tundra”. (See my page on the rising levels of methane.)

Conclusion: Definition of a tipping point

The tipping point of a block is the critical angle of tilt at which the weight of the block begins to cause it to topple, triggering a self-amplifying feedback loop that escalates the tilt.

The above definition suggests a more general definition: The tipping point of a system is when a property of the system exceeds a critical value, and this excess triggers a self-amplifying feedback loop that increases the excess of this property.

Conclusion: We are shoving the planet past tipping points

Unfortunately, humans continue to increase the level of carbon dioxide in the atmosphere, the very factor that drives global heating. This means that we are pushing our planet towards climate tipping points at an alarming rate. The planet is gaining 74 times as much energy each year as the United States uses in a year. (See my page on how fast the Earth is heating up.) It appears that we will not gently nudge our climate past a tipping point. We have our foot hard on the accelerator, rocketing towards or past tipping points.

Conclusion: We risk irreversible global heating

As we push the Earth past climate tipping points, we will set in motion feedback processes, like the methane release feedback mentioned above. These feedback cycles would then contribute to the ongoing global heating caused by high levels of carbon dioxide. The risk is that we are setting in motion powerful climate dynamics and, once these feedbacks become dominant, humans will not be able to reduce the heating. It’s like the loss of control when you lean too far back in a chair, and it starts toppling backwards.

The greatest risk is that after pushing past one tipping point, we would then be powerless to limit the heating, and the climate would push past the tipping points of other feedback loops. We would face a cascade of tipping points and vicious feedback cycles that could destroy the climate that has nurtured us.

A discussion informed by the maths of a toppling block

Consider standing a brick (or a block of wood or a wine glass) on a table and tilting the block by pushing the top of the block with your finger. It will rotate on the edge that stays on the table. It will lean more and more, like the leaning tower of Pisa, until it reaches its “tipping point”. After you push the block just beyond its tipping point, the weight of the block makes it unstable. The block will topple until it hits the table, rotating around the edge that stays on the table.

We can understand the motion of a toppling block in mathematical detail. This maths informs this discussion about “climate tipping points” and the triggered “climate self-amplifying feedback loops”. Many authors call these loops “positive feedback”.

I include a lengthy maths section that you can skip if you prefer. I would have written about a “wine glass tipping on a table”, but the mathematics and physics of a “toppling brick” are simpler.

The movement of the block

State 1: The block standing on its base

Initially, the block is standing tall on a table. When you push it a short distance with your finger and then remove your finger, the block will return to its standing position. Gravity causes this stability as it holds the block against your finger when you withdraw your finger, allowing the block to move back to its standing position with your finger.

Small disturbances will not destabilise this block, so this is a state of stable equilibrium.

For the block, the “pushing finger” initiates the tilt of the block and is “the initial disturbance” of the system. For our climate, the “pushing finger” is the human burning of fossil fuels and forests. This burning has greatly elevated the levels of CO2 in the atmosphere and caused the build-up of heat on our planet, known as global heating.

State 2: The tilted block

As your finger pushes the block, tilting it, you are supporting some of the block’s weight with your pushing finger. It takes less and less pushing to move the block as it increasingly supports its own weight on the edge that stays in contact with the table.

State 3: The tipping point

At the tipping point, the block comes into balance, and you no longer have to push the block at all to keep it tilted. In theory, you could remove your finger, and the block would remain in place, balanced all by itself.

However, the block is in a state of delicate balance, and the slightest vibration or wind will destabilise it. It is in equilibrium, but this is an unstable equilibrium.

State 4: The toppling block

Once your finger pushes the block past the tipping point, it starts toppling, rotating on one edge.

State 5: The toppled block

The block stops rotating when it hits the table. The fall can be destructive as the block can break if it is fragile, like a wine glass. If the block is heavy enough, the table can break.

Diagram of the block as it topples

The diagrams show a cross-section of the block, halfway through the depth of the block.

Skip the maths

Click here to jump over the maths section.

Variables, Notation & Terms

You may find it helpful to refer to the diagram below as you read this. Try opening this page twice. Then, place the diagram of the block on the right of your screen, and the maths on the left.

The diagram shows these variables.

H = Block Height = H metre = 4 metre
W = Block Width = W metre = 1 metre
Set the block depth = block width = W metres
M: Block mass = M kilogram = 10 kilogram
g: Acceleration due to gravity = g = 9.8 metre per second, per second
T or Tilt degrees: The pushing finger on the top-left edge of the block at E pushes to the right, so the block tilts, rotating clockwise around the edge on the table at A. The angle of tilt away from vertical of the block = Tilt = T degrees. Initially, the tilt is zero, and after the block has fallen, the tilt is 90 degrees.
Line AD = The “landing side” of the block on which it finally lands.
Line ED = The “top of the block”, the uppermost side of the block when it is standing upright.
Line AE = The “diagonal” of the block.
Block edge A = The “axis of rotation”, the edge of the block that remains on the table and about which the block rotates.
Point C = The “centre of mass” of the block.
Line AC = The weight of the block acts through the centre of mass at point C. It twists the block around the axis of rotation at A. It’s like a plumber’s wrench with the plumber exerting force through point C to twist the block.
F = The force of the block’s weight acting at the centre of mass C
I = The moment of inertia of the block
Q = The torque or twisting force on the block, exerted by the weight force
u degrees = the constant “tipping point” tilt of the block.
s degrees = The “strength angle” is the “tilt past the tipping point” in degrees. As this angle increases, the weight force increases the strength of its twist on the block.
The Y-axis = the vertical axis.
The X-axis = the horizontal axis.

Mathematical notation:

The * in the equations indicates multiplication
Square root of 2 = Sqrt(2)

The centre of mass

The centre of mass (COM) of the block is located at its centre. It is at the point (C), where the two diagonals meet, halfway along the diagonal EA. (This assumes that the weight of the block is uniformly distributed throughout the block.)

As the block rotates, the centre of mass travels in a part-circle around the axis A, with the radius being the length AC = R metres

Pythagoras gives the length of the diagonal of the block:
Diagonal = Sqrt( W * W + H * H ) = 4.12 metre

The weight force acts at a length = half the diagonal length away from the axis of rotation
Radius = R = 0.5 * Sqrt( W * W + H * H ) = 2.06 metre

A diagram of the block standing on the table

the block standing tall showing the tipping point angle

The tilt at the tipping point

From the above diagram, notice that as the block rotates clockwise through u degrees, the initially vertical “landing side” rotates through u degrees. And the “diagonal through A” also rotates until this diagonal becomes vertical. At this point, both the diagonal and the “centre of mass” are directly over the supporting edge of the block at A. This is the tipping point.
Tipping point tilt = u degrees

When you set the width and the height, you also set the angle u. Consider the right-angle triangle AED:
Tan(u) = opposite side / adjacent side = W / H = ¼ = 0.25
So u = 14.0 degrees

The clockwise twisting force: Torque (Q)

The block’s weight exerts a force on the block (F)
F = M * g = 10 * 9.8 = 98 Newton

The angle between the line AC and the vertical weight force is s degrees. Angle s is also the “tilt beyond the tipping point” as
s = T – u.

The block rotates about the edge A, with a part of the weight force causing the rotation = F * Sin(s).

The clockwise twisting force or torque (Q) is equal to the length AC = R multiplied by the component of the weight force acting at right angles to the wrench.
Torque = Q = R * F * Sin(s)
So, Q = R * M * g * Sin(T – u)

Clockwise Torque Q = 202.03 * Sin( Tilt – 14)

How Torque changes with tilt.

Graph of torque versus tilt for the toppling block

The graph shows that:

The torque is zero at a tilt of 14 degrees. This is the tipping point, and it makes sense, as there is no twisting force when the block is supporting its own weight at this tipping point.
The torque continues to increase from zero as the Tilt increases from the tipping point at 14 degrees to when the block has fallen at Tilt = 90 degrees.
The clockwise torque is negative, i.e. anticlockwise, from when the block is standing on its base at Tilt = 0 until the block reaches the tipping point at Tilt = 14 degrees. This makes sense as, for these tilts, the weight force exerts an anticlockwise twisting force on the block and the pushing finger works against this anticlockwise torque as it tilts the block.

Moment of Inertia

As an object’s mass increases, the force required to move it increases. Similarly, as an object’s “moment of inertia” increases, the twisting force required to rotate it increases.

From above, we have the torque applied to the block (Q). By working out the “moment of inertia” of the block (I), we can work out the angular acceleration (A) because:
Q = I * A

The “moment of inertia of the block for rotation around the edge A” equals “the moment of inertia for rotation around the centre of mass” plus the “mass of the block” multiplied by the square of “the distance of the centre of mass from the edge A”.

Moment of inertia for rotating the block around the centre of mass
= M * (W * W + H * H) / 12
(See Wikipedia)

The moment of inertia of the block for rotation around the edge A is I:
I = M * (W * W + H * H) / 12 + M * R * R
So, I = 1.25 * M * R * R = 56.67

Acceleration of the tilting

Acceleration of the tilt: A = Torque / Moment of Inertia
So, A = Q / I

Q = R * M * g * Sin(T – u)
I = 1.25 * M * R * R = 56.67

A = 0.75 * g * Sin( T – u) / R

A = 3.57 * Sin ( T – 14) degrees per second per second

Acceleration of the tilt equals the torque divided by the moment of inertia (I). As the moment of inertia is a constant = 56.67, the acceleration follows the same pattern as the torque in the above graph. As the tilt increases from 14 degrees at the tipping point, the angular acceleration also increases.

Differential equation

A = 3.56 * Sin(Tilt – 14)
Now tilt-acceleration = A = d/dt (d Tilt/dt), so
d/dt (d Tilt/dt) = 3.56 * Sin(Tilt – 14)

This is a second-order non-linear differential equation involving Tilt and time.

Differential equations similar to this could be used to model other feedback loops. Before I worked through this, I often wondered about how maths would represent feedback. Of course, being able to model the toppling block barely scratches the surface of understanding how to model the complex climate feedback loops, such as the Tundra methane self-amplifying feedback loop.

A reader has suggested a book about the general structure of tipping points: “Catastrophe Theory” by V.I.Arnold.

Solving the differential equation

I wrote a computer program to solve this equation for the initial conditions:
. Time = 0 seconds
. Tilt = 35 degrees, i.e. 21 degrees past the tipping angle, and
. Tilt Speed = 0 degrees per second, i.e. initially at rest
. Tilt acceleration: The differential equation shows that the tilt-acceleration only depends on the “tilt”, so the “initial tilt” of 35 degrees sets the “initial tilt acceleration”.

Then you can work out what happens when the block tilts 0.5 degrees further, from 35 to 35.5 degrees. The differential equation gives the exact tilt-acceleration at both 35 and 35.5 degrees. As an approximation, you can use the “average tilt acceleration” as the “constant acceleration” over the step, and calculate the tilt-speed at 35.5 degrees and the time taken for the movement to 35.5 degrees.

By repeating this stepping process until the block hits the table at a tilt of 90 degrees, you can gain a detailed understanding of how the block topples. For each time considered, you have the time, tilt, tilt-speed and tilt-acceleration. You have “solved” the differential equation for the given initial conditions.

Note that the discussion here is in degrees; however, when performing calculations, use “radians” for angles, “radians/second” for speed, and “radians per second per second” for acceleration.

The toppling time for this block, given these initial conditions, is 1.07 seconds. If you need greater accuracy, you can use smaller steps in tilt.

Rotational energy of the toppling block

You can also calculate the “rotational energy” of the block for each time considered.

If you want to stop the toppling block, say by catching it, you need to counteract the rotational energy that the block acquires as it topples.

Let w = the angular speed of the block in radians per second
Let I = the moment of inertia of the block

The rotational energy of the block = 0.5 * I * w * w

Graph of the kinetic energy of the toppling block versus time

This graph of the kinetic energy of the rotating block shows how the kinetic energy increases slowly for the first 0.4 seconds, and then increases rapidly. Now, the greater the rotational energy, the more energy you have to exert to stop the block from toppling. Therefore, if you need to prevent the toppling, it is far easier to do so early before the rotational energy becomes too large.

The impact of changing the size of the block

The angular acceleration = 0.75 * g * sin(Tilt – u) / R

Consider (a) a given tilt angle “Tilt”, and (b) a given ratio of height to width, i.e. a given tipping point angle “u”, then the angular acceleration depends only on “R”, the radius of the motion of the centre of mass. As you increase R, you decrease the angular acceleration and so increase the toppling time.

Now, R = 0.5 * Sqrt(H*H + W*W), so increasing the height or the width of the block will increase the toppling time equally.

Also, note that the mass of the block is not included in the equation and so does not influence the toppling time.

If you increase the height while keeping the width the same, you make it easier to topple the block because the “tipping point tilt” decreases, and you have to tilt the block less to reach the tipping point.

End of the maths section

This is the end of the maths section

The acceleration of the tilt increases as the block topples

There are three measures concerning the tilt to consider here:

The tilt, measured in degrees,
The tilting speed, i.e. the rate of change of tilt over time, measured in degrees per second, and
The acceleration of tilting, i.e. the rate of change of the tilting speed over time, in degrees per second per second.

As the block tilts further past its tipping point, more of its weight acts to topple it, so the twisting force increases. Physics dictates that as the force increases, the acceleration also increases. As the tilt of the toppling block increases, so does the acceleration of the tilt.

Amplifying feedback takes over after the tipping point

The more the block tilts, the more the torque or weight acting to topple the block increases, and the block accelerates into the fall, with increasing acceleration. The more it tilts, the more it is going to tilt. After the block passes its tipping point, an amplifying feedback loop becomes dominant, and we have:

(1) The tilt past the tipping point increases		(2) The weight acting to topple the block increases

(4) The angular velocity		(3) The angular acceleration increases

Diagram: The self-amplifying feedback loop that is dominant as the block topples

While the tilting feedback loop is dominant:

an increase in tilt
increases the twisting force, which
increases the acceleration of the rotation, which
increase the angular velocity, which increases the tilt and closes the loop.

For a long time, I have wondered about the relationship between tipping points and amplifying feedback loops. Here, we have a clear relationship.

After the tipping point, a new controlling factor emerges

When the block tips even slightly beyond the tipping point, the above self-amplifying feedback cycle becomes dominant. Control of the block moves from (1) the pushing finger to (2) gravity via the feedback cycle.

For our climate, the initial disturbing factor is the concentration of carbon dioxide in the atmosphere, which humans are increasing. After the tipping point, a factor like methane release could take control, via an amplifying feedback loop of “more heating causing more methane release from frozen Tundra, causing more heating”.

Regaining control after passing a tipping point

To stop the block from crashing after it passes its tipping point, you have to (1) stop the continued pushing by the finger and (2) counter the new controlling factor, gravity, by something like catching the block.

To limit global heating after passing a climate tipping point, you must eliminate the initial disturbing factor: high levels of CO2 in the atmosphere. On top of this, you have to find a way of countering a new heating force. If this new heating force were the release of vast amounts of methane from melting tundra, then we would face a difficult problem, as the Tundra covers about 20% of the Earth’s land surface.

Above, I have considered taking the block past its tipping point, and then holding it stationary to see how long it takes to hit the table, i.e. the initial angular velocity of the block is zero. I did not consider the block being pushed past the tipping point at speed because, when I tried an experiment to measure the toppling time, it was even hard to get the block stationary before release. And the experimental brick toppled so fast, in about 1 second.

What is happening to our climate is very different. There is little sign that humans are inhibiting the “controlling pushing finger” of the greenhouse gases in our atmosphere. We are not gently taking our climate to just past its tipping point. We are sweeping the climate wine glass off the table by prolonged drunken debauchery.

Cascade of feedback loops

When the toppling block hits the table, a cascade of repercussions can occur. For instance, if the block is heavy, the table may be stressed beyond its breaking point, and subsequently, the floor may also be stressed beyond its breaking point.

When considering climate, the escalation of global heating due to one feedback loop, such as methane release, could lead to the climate pushing past the tipping point of other amplifying feedback loops, resulting in a cascade of these heating dynamics.

This is the big danger of the current global heating. It could push the climate past one tipping point, and this could then lead to a cascade of other climate tipping points.

Work needed to get to the tipping point

The block is initially in a stable equilibrium, and the pushing finger must do work to elevate the block to the tipping point before it topples.

For the methane feedback loop, the work is the heating needed to melt the Tundra ice and release the methane. The current high levels of greenhouse gases are heating the planet and doing this work. We are already seeing melting Tundra and methane release.

The duration of the motion

For the block, the action occurs in seconds. Our complex climate system operates quite differently, with its effects unfolding over thousands of years. Significant heating is soon evident in increased atmospheric temperatures and increased ocean surface temperatures. However, some impacts of heating take time. For example, the melting of ice occurs slowly, as it requires a large amount of heat. This means that increased sea levels resulting from melting ice will become evident gradually. Also, the deep ocean currents move slowly, redistributing heat around our vast oceans.

The reversibility of feedback loops

The feedback loop of “more tilt leading to more tilt” is not reversible. In a shortened form, this loop is (1) an increase in tilt tends to (2) increase the twisting force, which tends to (3) increase the tilt. Reversing the first element of this loop, you have a decreased tilt, which decreases the twisting gravity force. However, gravity is still providing a clockwise twist, so the tilt will still increase, not decrease. So this amplifying loop governing the toppling block will not run in reverse.

Many of the self-amplifying feedback loops linked to climate change are also irreversible. For example, the methane loop cannot be reversed. Once warming releases Tundra methane into the atmosphere, subsequent cooling will not capture the methane, freeze it, and return it underground.

However, some climate feedbacks are reversible. For example, the “warming leading to decreased Arctic sea-ice” loop is reversible, as cooling can create more ice, which can lead to further cooling.

Identifying the tipping point

For the block, the tipping point occurs when the centre of gravity of the block moves directly above the supporting edge of the block.

While science makes it clear that our planet is heating, the climate tipping points are not clear-cut.

For Tundra methane release, the tipping point would occur when “the annual heating impact of increasing atmospheric methane” equals “the maximum annual cooling impact of reducing atmospheric CO2 levels”. This is not a particularly useful concept, as we are still increasing atmospheric CO2 levels, seemingly unable to reduce CO2 emissions, let alone lower the CO2 level in the atmosphere.

Brief Conclusions

This analysis suggests a relationship between tipping points and self-amplifying feedback loops that may be general, i.e. after passing a tipping point, a self-amplifying feedback loop gains control of the process.

Once we hit the climate tipping point associated with, for example, the release of methane, then a feedback loop amplifying the methane release could become dominant, and we could expect a cascade of other climate tipping points to be breached. We must act with the utmost urgency to reduce greenhouse gases in the atmosphere, and we hope that we have not already passed one of these tipping points.

Updated 24 March 2021 and 11 July 2025