# 1.1.3: Making Scaled Copies - Mathematics

## Lesson

Let's draw scaled copies.

Exercise (PageIndex{1}): More or Less?

For each problem, select the answer from the two choices.

1. The value of (25cdot (8.5)) is:
1. More than 205
2. Less than 205
2. The value of ((9.93)cdot (0.984)) is:
1. More than 10
2. Less than 10
3. The value of ((0.24)cdot (0.67)) is:
1. More than 0.2
2. Less than 0.2

Exercise (PageIndex{2}): Drawing Scaled Copies

1. Draw a scaled copy of either Figure A or B using a scale factor of (3).
2. Draw a scaled copy of either Figure C or D using a scale factor of (frac{1}{2}).

Exercise (PageIndex{3}): Which Operations? (Part 1)

Diego and Jada want to scale this polygon so the side that corresponds to 15 units in the original is 5 units in the scaled copy.

Diego and Jada each use a different operation to find the new side lengths. Here are their finished drawings.

1. What operation do you think Diego used to calculate the lengths for his drawing?
2. What operation do you think Jada used to calculate the lengths for her drawing?
3. Did each method produce a scaled copy of the polygon? Explain your reasoning.

Exercise (PageIndex{4}): Which Operations? (Part 2)

Andre wants to make a scaled copy of Jada's drawing so the side that corresponds to 4 units in Jada’s polygon is 8 units in his scaled copy.

1. Andre says “I wonder if I should add 4 units to the lengths of all of the segments?” What would you say in response to Andre? Explain or show your reasoning.
2. Create the scaled copy that Andre wants. If you get stuck, consider using the edge of an index card or paper to measure the lengths needed to draw the copy.

The side lengths of Triangle B are all 5 more than the side lengths of Triangle A. Can Triangle B be a scaled copy of Triangle A? Explain your reasoning.

### Summary

Creating a scaled copy involves multiplying the lengths in the original figure by a scale factor.

For example, to make a scaled copy of triangle (ABC) where the base is 8 units, we would use a scale factor of 4. This means multiplying all the side lengths by 4, so in triangle (DEF), each side is 4 times as long as the corresponding side in triangle (ABC).

### Glossary Entries

Definition: Corresponding

When part of an original figure matches up with part of a copy, we call them corresponding parts. These could be points, segments, angles, or distances.

For example, point (B) in the first triangle corresponds to point (E) in the second triangle. Segment (AC) corresponds to segment (DF).

Definition: Scale Factor

To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.

In this example, the scale factor is 1.5, because (4cdot (1.5)=6), (5cdot (1.5)=7.5), and (6cdot (1.5)=9).

Definition: Scaled Copy

A scaled copy is a copy of an figure where every length in the original figure is multiplied by the same number.

For example, triangle (DEF) is a scaled copy of triangle (ABC). Each side length on triangle (ABC) was multiplied by 1.5 to get the corresponding side length on triangle (DEF).

## Practice

Exercise (PageIndex{5})

Here are 3 polygons.

Draw a scaled copy of Polygon A using a scale factor of 2.

Draw a scaled copy of Polygon B using a scale factor of (frac{1}{2}).

Draw a scaled copy of Polygon C using a scale factor of (frac{3}{2}).

Exercise (PageIndex{6})

Quadrilateral A has side lengths 6, 9, 9, and 12. Quadrilateral B is a scaled copy of Quadrilateral A, with its shortest side of length 2. What is the perimeter of Quadrilateral B?

Exercise (PageIndex{7})

Here is a polygon on a grid.

Draw a scaled copy of this polygon that has a perimeter of 30 units. What is the scale factor? Explain how you know.

Exercise (PageIndex{8})

Priya and Tyler are discussing the figures shown below. Priya thinks that B, C, and D are scaled copies of A. Tyler says B and D are scaled copies of A. Do you agree with Priya, or do you agree with Tyler? Explain your reasoning.

(From Unit 1.1.1)

## Unit 1 Big Ideas

This week your student will learn about scaling shapes. An image is a scaled copy of the original if the shape is stretched in a way that does not distort it. For example, here is an original picture and five copies. Pictures C and D are scaled copies of the original, but pictures A, B, and E are not.

In each scaled copy, the sides are a certain number of times as long as the corresponding sides in the original. We call this number the scale factor. The size of the scale factor affects the size of the copy. A scale factor greater than 1 makes a copy that is larger than the original. A scale factor less than 1 makes a copy that is smaller.

1. For each copy, tell whether it is a scaled copy of the original triangle. If so, what is the scale factor?
2. Draw another scaled copy of the original triangle using a different scale factor.
1. Copy 1 is a scaled copy of the original triangle. The scale factor is 2, because each side in Copy 1 is twice as long as the corresponding side in the original triangle. 5 oldcdot 2 = 10 , 4 oldcdot 2 = 8 , (6.4) oldcdot 2 = 12.8
2. Copy 2 is a scaled copy of the original triangle. The scale factor is frac12 or 0.5, because each side in Copy 2 is half as long as the corresponding side in the original triangle. 5 oldcdot (0.5) = 2.5 , 4 oldcdot (0.5) = 2 , (6.4) oldcdot (0.5) = 3.2
3. Copy 3 is not a scaled copy of the original triangle. The shape has been distorted. The angles are different sizes and there is not one number we can multiply by each side length of the original triangle to get the corresponding side length in Copy 3.

When two different situations can be described by equivalent ratios, that means they are alike in some important way.

An example is a recipe. If two people make something to eat or drink, the taste will only be the same as long as the ratios of the ingredients are equivalent. For example, all of the mixtures of water and drink mix in this table taste the same, because the ratios of cups of water to scoops of drink mix are all equivalent ratios.

If a mixture were not equivalent to these, for example, if the ratio of cups of water to scoops of drink mix were 6:4 , then the mixture would taste different.

Notice that the ratios of pairs of corresponding side lengths are equivalent in figures A, B, and C. For example, the ratios of the length of the top side to the length of the left side for figures A, B, and C are equivalent ratios. Figures A, B, and C are scaled copies of each other this is the important way in which they are alike.

If a figure has corresponding sides that are not in a ratio equivalent to these, like figure D, then it’s not a scaled copy. In this unit, you will study relationships like these that can be described by a set of equivalent ratios.

## 1.2: Scaling F (10 minutes)

### Activity

This task enables students to describe more precisely the characteristics of scaled copies and to refine the meaning of the term. Students observe copies of a line drawing on a grid and notice how the lengths of line segments and the angles formed by them compare to those in the original drawing.

Students engage in MP7 in multiple ways in this task. Identifying distinguishing features of the scaled copies means finding similarities and differences in the shapes. In addition, the fact that corresponding parts increase by the same scale factor is a vital structural property of scaled copies.

For the first question, expect students to explain their choices of scaled copies in intuitive, qualitative terms. For the second question, students should begin to distinguish scaled and unscaled copies in more specific and quantifiable ways. If it does not occur to students to look at lengths of segments, suggest they do so.

As students work, monitor for students who notice the following aspects of the figures. Students are not expected to use these mathematical terms at this point, however.

• The original drawing of the letter F and its scaled copies have equivalent width-to-height ratios.
• We can use a scale factor (or a multiplier) to compare the lengths of different figures and see if they are scaled copies of the original.
• The original figure and scaled copies have corresponding angles that have the same measure.

### Launch

Keep students in the same groups. Give them 3–4 minutes of quiet work time, and then 1–2 minutes to share their responses with their partner. Tell students that how they decide whether each of the seven drawings is a scaled copy may be very different than how their partner decides. Encourage students to listen carefully to each other’s approach and to be prepared to share their strategies. Use gestures to elicit from students the words “horizontal” and “vertical” and ask groups to agree internally on common terms to refer to the parts of the F (e.g., “horizontal stems”).

Speaking: Math Language Routine 1 Stronger and Clearer Each Time. This is the first time Math Language Routine 1 is suggested as a support in this course. In this routine, students are given a thought-provoking question or prompt and asked to create a first draft response in writing. Students meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . .?” and “Can you say that another way?” Finally, students write a second draft of their response reflecting ideas from partners, and improvements on their initial ideas. The purpose of this routine is to provide a structured and interactive opportunity for students to revise and refine their ideas through verbal and written means.
Design Principle(s): Optimize output (for explanation)

How It Happens:

Use this routine to provide students a structured opportunity to refine their explanations for the first question: “Identify all the drawings that are scaled copies of the original letter F drawing. Explain how you know.” Allow students 2–3 minutes to individually create first draft responses in writing.

Invite students to meet with 2–3 other partners for feedback.

Instruct the speaker to begin by sharing their ideas without looking at their written draft, if possible. Provide the listener with these prompts for feedback that will help their partner strengthen their ideas and clarify their language: “What do you mean when you say….?”, “Can you describe that another way?”, “How do you know that _ is a scaled copy?”, “Could you justify that differently?” Be sure to have the partners switch roles. Allow 1–2 minutes to discuss.

Signal for students to move on to their next partner and repeat this structured meeting.

Close the partner conversations and invite students to revise and refine their writing in a second draft.

Provide these sentence frames to help students organize their thoughts in a clear, precise way: “Drawing _ is a scaled copy of the original, and I know this because.…”, “When I look at the lengths, I notice that.…”, and “When I look at the angles, I notice that.…”

Here is an example of a second draft:

“Drawing 7 is a scaled copy of the original, and I know this because it is enlarged evenly in both the horizontal and vertical directions. It does not seem lopsided or stretched differently in one direction. When I look at the length of the top segment, it is 3 times as large as the original one, and the other segments do the same thing. Also, when I look at the angles, I notice that they are all right angles in both the original and scaled copy.”

If time allows, have students compare their first and second drafts. If not, have the students move on by working on the following problems.

## Lesson 1

Here is a portrait of a student. Move the slider under each image, A–E, to see it change.

1. How is each one the same as or different from the original portrait of the student?
2. Some of the sliders make scaled copies of the original portrait. Which ones do you think are scaled copies? Explain your reasoning.
3. What do you think “scaled copy” means?

### 1.2: Scaling F

Here is an original drawing of the letter F and some other drawings.

Expand Image

Description: <p>An original drawing of the letter F and 7 other drawings on a grid. In the original drawing, the vertical segment is 4 units, the top horizontal segment is 2 units, and the bottom horizontal segment is one unit. In drawing 1, the vertical segment is 6 units, the top horizontal segment is 3 units, and the bottom horizontal segment is 1 and 1 half units. In drawing 2, the vertical segment is 8 units, the top horizontal segment is 4 units, and the bottom horizontal segment is 2 units. In drawing 3, the vertical segment is 4 units, the top horizontal segment is 4 units, and the bottom horizontal segment is 3 units. In drawing 4, the vertical segment slanted, from the bottom endpoint is 4 units down and 1 unit over from the top endpoint, the top horizontal segment is 2 units, and the bottom horizontal segment is 1 unit. In drawing 5, the vertical segment is 6 units, the top horizontal segment is 3 units, and the bottom horizontal segment is 2 units. In drawing 6, the vertical segment is 2 units, the top horizontal segment is 1 unit, and the bottom horizontal segment is 1 unit. In drawing 7, the vertical segment is 12 units, the top horizontal segment is 6 units, and the bottom horizontal segment is 3 units. </p>

Scaled copies of rectangles have an interesting property. Can you see what it is?

Here, the larger rectangle is a scaled copy of the smaller one (with a scale factor of frac<3> <2>). Notice how the diagonal of the large rectangle contains the diagonal of the smaller rectangle. This is the case for any two scaled copies of a rectangle if we line them up as shown. If two rectangles are not scaled copies of one another, then the diagonals do not match up. In this unit, we will investigate how to make scaled copies of a figure.

## 1.1.3: Making Scaled Copies - Mathematics

Many people have asked for a rubric to use for the project - the rubric I use is blank so the students can decide on the criteria they should be assessed on. As far as presenting the project to the students, I kind of "winged" the presentation and walked the students as a class through the first steps. The students took off with the project and didn't need much more guidance.

Nice collection,it is very good to post the some pictures in the schools,it is very attractive to the schools.IB Schools in Bangalore

I love this idea! I also teach 7th grade math and am trying to build a collection of hands on projects for next year. What a great way to have kids understand scale factor while having fun and then being able to COLOR it. oh man, they are going to love it! Thank you for the idea!

I liked this idea of having the kids scale up candy wrappers and tried this activity also. They had a fun time while learning in the process. They also got to eat the candy at the end! Thanks for sharing this idea.

## What Is a Scale in Math?

A scale in mathematics refers to the ratio of a drawing in comparison to the size of the real object. A ratio is a relative size that represents typically two values. For example, 1:3 pears and grapefruits represents that there is one pear for every three grapefruit.

In using scale, the ratio represents sizes of actual drawings or models. If the scale is 1:10, then the model or drawing is 10 times smaller than the actual object. If a die-cast car is listed as a 1:10 or 1/10 diecast, then the actual car is 10 times larger than the model car.

Scale is often used to represent items like diecast cars, maps and other items. A real horse may be 1,500 mm high, but the drawing of the horse may be 150 mm high. As with the diecast car, this scale is represented by writing the ratio 1:10.

Using scale drawings can also help with representing buildings. Architects often use scale when drawing a design, or to build models to show the design to others. A doll house is a good example of something that can be represented in scale. If a doll house modeled after a real house is 50 times smaller, then that scale can be represented by writing the ratio 1:50.

## Contents

In mathematics, one can consider the scaling properties of a function or curve f (x) under rescalings of the variable x . That is, one is interested in the shape of f (λx) for some scale factor λ , which can be taken to be a length or size rescaling. The requirement for f (x) to be invariant under all rescalings is usually taken to be

for some choice of exponent Δ , and for all dilations λ . This is equivalent to f being a homogeneous function of degree Δ .

An example of a scale-invariant curve is the logarithmic spiral, a kind of curve that often appears in nature. In polar coordinates (r, θ) , the spiral can be written as

Allowing for rotations of the curve, it is invariant under all rescalings λ that is, θ(λr) is identical to a rotated version of θ(r) .

### Projective geometry Edit

The idea of scale invariance of a monomial generalizes in higher dimensions to the idea of a homogeneous polynomial, and more generally to a homogeneous function. Homogeneous functions are the natural denizens of projective space, and homogeneous polynomials are studied as projective varieties in projective geometry. Projective geometry is a particularly rich field of mathematics in its most abstract forms, the geometry of schemes, it has connections to various topics in string theory.

### Fractals Edit

It is sometimes said that fractals are scale-invariant, although more precisely, one should say that they are self-similar. A fractal is equal to itself typically for only a discrete set of values λ , and even then a translation and rotation may have to be applied to match the fractal up to itself.

Thus, for example, the Koch curve scales with ∆ = 1 , but the scaling holds only for values of λ = 1/3 n for integer n . In addition, the Koch curve scales not only at the origin, but, in a certain sense, "everywhere": miniature copies of itself can be found all along the curve.

Some fractals may have multiple scaling factors at play at once such scaling is studied with multi-fractal analysis.

Periodic external and internal rays are invariant curves .

If P(f ) is the average, expected power at frequency f , then noise scales as

with Δ = 0 for white noise, Δ = −1 for pink noise, and Δ = −2 for Brownian noise (and more generally, Brownian motion).

More precisely, scaling in stochastic systems concerns itself with the likelihood of choosing a particular configuration out of the set of all possible random configurations. More context is needed here. Probability and entropy are definitely related to choosing a particular configuration, but it is not obvious how scale invariance is connected to this. This likelihood is given by the probability distribution.

Examples of scale-invariant distributions are the Pareto distribution and the Zipfian distribution.

### Scale invariant Tweedie distributions Edit

Tweedie distributions are a special case of exponential dispersion models, a class of statistical models used to describe error distributions for the generalized linear model and characterized by closure under additive and reproductive convolution as well as under scale transformation. [1] These include a number of common distributions: the normal distribution, Poisson distribution and gamma distribution, as well as more unusual distributions like the compound Poisson-gamma distribution, positive stable distributions, and extreme stable distributions. Consequent to their inherent scale invariance Tweedie random variables Y demonstrate a variance var(Y) to mean E(Y) power law:

where a and p are positive constants. This variance to mean power law is known in the physics literature as fluctuation scaling, [2] and in the ecology literature as Taylor's law. [3]

Random sequences, governed by the Tweedie distributions and evaluated by the method of expanding bins exhibit a biconditional relationship between the variance to mean power law and power law autocorrelations. The Wiener–Khinchin theorem further implies that for any sequence that exhibits a variance to mean power law under these conditions will also manifest 1/f noise. [4]

The Tweedie convergence theorem provides a hypothetical explanation for the wide manifestation of fluctuation scaling and 1/f noise. [5] It requires, in essence, that any exponential dispersion model that asymptotically manifests a variance to mean power law will be required express a variance function that comes within the domain of attraction of a Tweedie model. Almost all distribution functions with finite cumulant generating functions qualify as exponential dispersion models and most exponential dispersion models manifest variance functions of this form. Hence many probability distributions have variance functions that express this asymptotic behavior, and the Tweedie distributions become foci of convergence for a wide range of data types. [4]

Much as the central limit theorem requires certain kinds of random variables to have as a focus of convergence the Gaussian distribution and express white noise, the Tweedie convergence theorem requires certain non-Gaussian random variables to express 1/f noise and fluctuation scaling. [4]

### Cosmology Edit

In physical cosmology, the power spectrum of the spatial distribution of the cosmic microwave background is near to being a scale-invariant function. Although in mathematics this means that the spectrum is a power-law, in cosmology the term "scale-invariant" indicates that the amplitude, P(k) , of primordial fluctuations as a function of wave number, k , is approximately constant, i.e. a flat spectrum. This pattern is consistent with the proposal of cosmic inflation.

Classical field theory is generically described by a field, or set of fields, φ, that depend on coordinates, x. Valid field configurations are then determined by solving differential equations for φ, and these equations are known as field equations.

For a theory to be scale-invariant, its field equations should be invariant under a rescaling of the coordinates, combined with some specified rescaling of the fields,

The parameter Δ is known as the scaling dimension of the field, and its value depends on the theory under consideration. Scale invariance will typically hold provided that no fixed length scale appears in the theory. Conversely, the presence of a fixed length scale indicates that a theory is not scale-invariant.

A consequence of scale invariance is that given a solution of a scale-invariant field equation, we can automatically find other solutions by rescaling both the coordinates and the fields appropriately. In technical terms, given a solution, φ(x), one always has other solutions of the form

### Scale invariance of field configurations Edit

For a particular field configuration, φ(x), to be scale-invariant, we require that

where Δ is, again, the scaling dimension of the field.

We note that this condition is rather restrictive. In general, solutions even of scale-invariant field equations will not be scale-invariant, and in such cases the symmetry is said to be spontaneously broken.

### Classical electromagnetism Edit

An example of a scale-invariant classical field theory is electromagnetism with no charges or currents. The fields are the electric and magnetic fields, E(x,t) and B(x,t), while their field equations are Maxwell's equations.

With no charges or currents, these field equations take the form of wave equations

where c is the speed of light.

These field equations are invariant under the transformation

Moreover, given solutions of Maxwell's equations, E(x, t) and B(x, t), it holds that Ex, λt) and Bx, λt) are also solutions.

### Massless scalar field theory Edit

Another example of a scale-invariant classical field theory is the massless scalar field (note that the name scalar is unrelated to scale invariance). The scalar field, φ(x, t) is a function of a set of spatial variables, x, and a time variable, t .

Consider first the linear theory. Like the electromagnetic field equations above, the equation of motion for this theory is also a wave equation,

and is invariant under the transformation

and so it should not be surprising that massive scalar field theory is not scale-invariant.

#### Φ 4 theory Edit

The field equations in the examples above are all linear in the fields, which has meant that the scaling dimension, Δ , has not been so important. However, one usually requires that the scalar field action is dimensionless, and this fixes the scaling dimension of φ . In particular,

where D is the combined number of spatial and time dimensions.

Given this scaling dimension for φ , there are certain nonlinear modifications of massless scalar field theory which are also scale-invariant. One example is massless φ 4 theory for D =4. The field equation is

(Note that the name φ 4 derives from the form of the Lagrangian, which contains the fourth power of φ .)

When D =4 (e.g. three spatial dimensions and one time dimension), the scalar field scaling dimension is Δ =1. The field equation is then invariant under the transformation

The key point is that the parameter g must be dimensionless, otherwise one introduces a fixed length scale into the theory: For φ 4 theory, this is only the case in D =4. Note that under these transformations the argument of the function φ is unchanged.

The scale-dependence of a quantum field theory (QFT) is characterised by the way its coupling parameters depend on the energy-scale of a given physical process. This energy dependence is described by the renormalization group, and is encoded in the beta-functions of the theory.

For a QFT to be scale-invariant, its coupling parameters must be independent of the energy-scale, and this is indicated by the vanishing of the beta-functions of the theory. Such theories are also known as fixed points of the corresponding renormalization group flow. [6]

### Quantum electrodynamics Edit

A simple example of a scale-invariant QFT is the quantized electromagnetic field without charged particles. This theory actually has no coupling parameters (since photons are massless and non-interacting) and is therefore scale-invariant, much like the classical theory.

However, in nature the electromagnetic field is coupled to charged particles, such as electrons. The QFT describing the interactions of photons and charged particles is quantum electrodynamics (QED), and this theory is not scale-invariant. We can see this from the QED beta-function. This tells us that the electric charge (which is the coupling parameter in the theory) increases with increasing energy. Therefore, while the quantized electromagnetic field without charged particles is scale-invariant, QED is not scale-invariant.

### Massless scalar field theory Edit

Free, massless quantized scalar field theory has no coupling parameters. Therefore, like the classical version, it is scale-invariant. In the language of the renormalization group, this theory is known as the Gaussian fixed point.

However, even though the classical massless φ 4 theory is scale-invariant in D=4, the quantized version is not scale-invariant. We can see this from the beta-function for the coupling parameter, g.

Even though the quantized massless φ 4 is not scale-invariant, there do exist scale-invariant quantized scalar field theories other than the Gaussian fixed point. One example is the Wilson-Fisher fixed point, below.

### Conformal field theory Edit

Scale-invariant QFTs are almost always invariant under the full conformal symmetry, and the study of such QFTs is conformal field theory (CFT). Operators in a CFT have a well-defined scaling dimension, analogous to the scaling dimension, , of a classical field discussed above. However, the scaling dimensions of operators in a CFT typically differ from those of the fields in the corresponding classical theory. The additional contributions appearing in the CFT are known as anomalous scaling dimensions.

### Scale and conformal anomalies Edit

The φ 4 theory example above demonstrates that the coupling parameters of a quantum field theory can be scale-dependent even if the corresponding classical field theory is scale-invariant (or conformally invariant). If this is the case, the classical scale (or conformal) invariance is said to be anomalous. A classically scale invariant field theory, where scale invariance is broken by quantum effects, provides an explication of the nearly exponential expansion of the early universe called cosmic inflation, as long as the theory can be studied through perturbation theory. [7]

In statistical mechanics, as a system undergoes a phase transition, its fluctuations are described by a scale-invariant statistical field theory. For a system in equilibrium (i.e. time-independent) in D spatial dimensions, the corresponding statistical field theory is formally similar to a D -dimensional CFT. The scaling dimensions in such problems are usually referred to as critical exponents, and one can in principle compute these exponents in the appropriate CFT.

### The Ising model Edit

An example that links together many of the ideas in this article is the phase transition of the Ising model, a simple model of ferromagnetic substances. This is a statistical mechanics model, which also has a description in terms of conformal field theory. The system consists of an array of lattice sites, which form a D -dimensional periodic lattice. Associated with each lattice site is a magnetic moment, or spin, and this spin can take either the value +1 or −1. (These states are also called up and down, respectively.)

The key point is that the Ising model has a spin-spin interaction, making it energetically favourable for two adjacent spins to be aligned. On the other hand, thermal fluctuations typically introduce a randomness into the alignment of spins. At some critical temperature, Tc , spontaneous magnetization is said to occur. This means that below Tc the spin-spin interaction will begin to dominate, and there is some net alignment of spins in one of the two directions.

An example of the kind of physical quantities one would like to calculate at this critical temperature is the correlation between spins separated by a distance r . This has the generic behaviour:

#### CFT description Edit

The fluctuations at temperature Tc are scale-invariant, and so the Ising model at this phase transition is expected to be described by a scale-invariant statistical field theory. In fact, this theory is the Wilson-Fisher fixed point, a particular scale-invariant scalar field theory.

In this context, G(r) is understood as a correlation function of scalar fields,

Now we can fit together a number of the ideas seen already.

From the above, one sees that the critical exponent, η , for this phase transition, is also an anomalous dimension. This is because the classical dimension of the scalar field,

where D is the number of dimensions of the Ising model lattice.

So this anomalous dimension in the conformal field theory is the same as a particular critical exponent of the Ising model phase transition.

Note that for dimension D ≡ 4−ε , η can be calculated approximately, using the epsilon expansion, and one finds that

In the physically interesting case of three spatial dimensions, we have ε =1, and so this expansion is not strictly reliable. However, a semi-quantitative prediction is that η is numerically small in three dimensions.

On the other hand, in the two-dimensional case the Ising model is exactly soluble. In particular, it is equivalent to one of the minimal models, a family of well-understood CFTs, and it is possible to compute η (and the other critical exponents) exactly,

### Schramm–Loewner evolution Edit

The anomalous dimensions in certain two-dimensional CFTs can be related to the typical fractal dimensions of random walks, where the random walks are defined via Schramm–Loewner evolution (SLE). As we have seen above, CFTs describe the physics of phase transitions, and so one can relate the critical exponents of certain phase transitions to these fractal dimensions. Examples include the 2d critical Ising model and the more general 2d critical Potts model. Relating other 2d CFTs to SLE is an active area of research.

A phenomenon known as universality is seen in a large variety of physical systems. It expresses the idea that different microscopic physics can give rise to the same scaling behaviour at a phase transition. A canonical example of universality involves the following two systems:

Even though the microscopic physics of these two systems is completely different, their critical exponents turn out to be the same. Moreover, one can calculate these exponents using the same statistical field theory. The key observation is that at a phase transition or critical point, fluctuations occur at all length scales, and thus one should look for a scale-invariant statistical field theory to describe the phenomena. In a sense, universality is the observation that there are relatively few such scale-invariant theories.

The set of different microscopic theories described by the same scale-invariant theory is known as a universality class. Other examples of systems which belong to a universality class are:

in piles of sand. The likelihood of an avalanche is in power-law proportion to the size of the avalanche, and avalanches are seen to occur at all size scales.
1. The frequency of network outages on the Internet, as a function of size and duration.
2. The frequency of citations of journal articles, considered in the network of all citations amongst all papers, as a function of the number of citations in a given paper. [citation needed]
3. The formation and propagation of cracks and tears in materials ranging from steel to rock to paper. The variations of the direction of the tear, or the roughness of a fractured surface, are in power-law proportion to the size scale.
4. The electrical breakdown of dielectrics, which resemble cracks and tears.
5. The percolation of fluids through disordered media, such as petroleum through fractured rock beds, or water through filter paper, such as in chromatography. Power-law scaling connects the rate of flow to the distribution of fractures.
6. The diffusion of molecules in solution, and the phenomenon of diffusion-limited aggregation.
7. The distribution of rocks of different sizes in an aggregate mixture that is being shaken (with gravity acting on the rocks).
8. The key observation is that, for all of these different systems, the behaviour resembles a phase transition, and that the language of statistical mechanics and scale-invariant statistical field theory may be applied to describe them.

### Newtonian fluid mechanics with no applied forces Edit

In order to deduce the scale invariance of these equations we specify an equation of state, relating the fluid pressure to the fluid density. The equation of state depends on the type of fluid and the conditions to which it is subjected. For example, we consider the isothermal ideal gas, which satisfies

### Computer vision Edit

In computer vision and biological vision, scaling transformations arise because of the perspective image mapping and because of objects having different physical size in the world. In these areas, scale invariance refers to local image descriptors or visual representations of the image data that remain invariant when the local scale in the image domain is changed. [8] Detecting local maxima over scales of normalized derivative responses provides a general framework for obtaining scale invariance from image data. [9] [10] Examples of applications include blob detection, corner detection, ridge detection, and object recognition via the scale-invariant feature transform.

## Accommodations

Students requesting accommodations for this course due to a disability must provide a current Authorization for Accommodation (AFA) letter issued by the Office for Students with Disabilities (OSD) which is located in University Center 202 behind Center Hall. The AFA letter may be issued by the OSD electronically or in hard-copy in either case, please make arrangements to discuss your accommodations with me in advance. We will make every effort to arrange for whatever accommodations are stipulated by the OSD. For more information, see here.