The Pythagorean Theorem: Engineering Applications

August 15, 2017 (15/08/2017) was Pythagorean Theorem Day. I was not even aware that there was such a day until this year, but it brought to my attention some really interesting engineering applications of which I was not previously aware. Let’s take a look at a few interesting ones.

Antenna analysis1

A probe designed for elliptically polarized field measurements is employed for a system with n identical dipoles which all intersect at the center of an xy plane in a Cartesian coordinate system. The angles between those dipoles are all identically equal to π/n. The first dipole crosses the y axis at an angle γ (Figure 1).


Figure 1 An n dipole system diagram

In this example, the probe is illuminated by a monochromatic plane wave with angular frequency Ω which propagates in the z plane direction. Its electrical components are as follows:


in which the amplitudes are A and B.

Next, the author makes the assumption that the sizes of the system are small as compared with the wavelength, so the mutual coupling influences between the antennas are negligible. The author next makes the assumption that all of the antennas are loaded with square-law detectors, so with these conclusions we can describe their directional patterns to be sinusoidal in nature.

Now, the square of the output voltage of the ith antenna (ν) is as follows:


where 1i is the vector of the ith dipole.

Now, we can calculate the output voltage from n dipoles by the use of the Pythagorean Theorem as follows from Reference 1:

If, in a circle are inscribed n secants which intersect themselves in the center of that circle, the angle between two adjacent secants is π/n, and the first of them crosses the axis of symmetry of the circle at angle γ.

Then the sum of the directional sines and cosines of the secants is then constant and equals n/2.

We can write this result as follows:


The formulas in Equation 3 are valid for n > 2. The notation used in the formulas is identical with that of Figure 1. When n = 2, the formulas are in the form of the Pythagorean Theorem.

By taking the sum of Equation 2 and using Equation 3 for n detectors, we get the square of the output voltage, V, of the system. For elliptical polarization, this is as follows:


Now, for circular polarization, A = B and


And for linear polarization, such as, B = 0, and we get


Here we can see that for the formulas in Equation 3, for n being even, it is a simple sum of n/2 “standard” Pythagorean Theorems.

The author is puzzled regarding the result when n is odd, so how can it be shown in this case, the last part of the sum equals ½, or its angle is a multiple of π/4?

Can anyone solve this?

Reducing energy consumption in wireless sensor networks2

The Generalized Pythagorean Theorem also has an application in creating a wireless sensor networks (WSN) topology that can minimize energy consumption.

National Instruments defines a WSN as follows:

A wireless sensor network (WSN) is a wireless network consisting of spatially distributed autonomous devices using sensors to monitor physical or environmental conditions. A WSN system incorporates a gateway that provides wireless connectivity back to the wired world and distributed nodes (see Figure 2). The wireless protocol you select depends on your application requirements. Some of the available standards include 2.4 GHz radios based on either IEEE 802.15.4 or IEEE 802.11 (Wi-Fi) standards or proprietary radios, which are usually 900 MHz.


Figure 2 A WSN consists of a wireless gateway and distributed wireless nodes (Image courtesy of National Instruments)

Simply put, the autonomous sensor nodes access data from the environment or other such physical conditions of interest, then process/condition it usually with an on-board microcontroller, and finally forward that data via wireless transceiver for processing via a star network, cluster tree or multi-hop wireless mesh network and ultimately via a wired system like Ethernet to a system such as a programmable automation controller (PAC).

Power consumption of the sensor nodes is critical since their power source is usually some sort of battery system or energy harvesting method. The network topology is a strong determining factor on the power source of the node to enable adequate energy for that node to continuously do its job. Reference 2 shows a way to optimize the power consumption of a node by using the Generalized Pythagorean Theorem to optimize the network topology choice for best power consumption.

The paper discusses two possibilities for the sensor nodes to transfer their data from node to node; that is, direct or indirect transmission via another node. The distance between node x and y is d(x,y). Let’s look at three nodes a,b, and c where the distance d(b,c) between nodes b and c differ according to angle γ. Figures 3 and 4 show the difference.


Figure 3 Three nodes in space with 0 < γ <  π/2 (Image courtesy of Reference 2)


Figure 4 Three nodes in space with 0 < γ < π/2 (Image courtesy of Reference 2)

The Generalized Pythagorean Theorem for this example is:


The author of Reference 2 goes on to do an analysis of WSN nodes by first looking at the nodes sending their position to the gateway and performing an energy mode and analysis from References 3 and 4. We look at a radio energy dissipation model as shown in Figure 5.


Figure 5 Radio energy dissipation model (Image courtesy of Reference 5)

Now we can see the energy spent by the radio transmitter in Equation 2 and the energy spent by the receiver in Equation 3:


Efs and Emp are the amplifier energies respectively in a free space model (with d2 power loss) and in a multipath fading (with d4 power loss) channel models. D is the distance between the transmitter and the receiver and was very much a critical parameter in the energy equation. The multipath model was used, but if the distance was less than d0, a threshold distance, then the free space model would be used. d0 was defined in Reference 6 as follows in Equation 4:


Next, Reference 2, goes on and takes Equation 1, Figures 3 and 4 to get the following:


By the use of Equations 2, 5, and 6 we can state that if 0 < γ < π/2, the node “b” transmits directly to node “c”, however, if π/2 < γ < π, then node “b” transmits its information first to “a”, a transitional relay, which in turn transmits it to node “c”. This takes more system power for a particular transition of data than the first on which is only a single transition.

Here is where Reference 2 proposes a unique approach to optimize energy consumption in a WSN. If a node needs to forward its data to a gateway that is not in a distance that makes it a first transmission choice, the data then needs to be send via multi-hop mode. The source node sends out a request message which each neighbouring node receives and stores the source ID and its (x,y) as response and then waits for an acknowledgement. After the node has a list of neighbouring nodes it will execute an algorithm to choose the optimum hop. Reference 2 goes further and adds conditions in case one node has gone off line and further evaluates the effectiveness of this proposed approach and its performance based upon simulations. The effectiveness of the approach is then detailed.

Bad data detection7

Reference 7 discusses a geometric interpretation of minimum variance estimation in terms of the Pythagorean Theorem.  We analyze bad data detection among redundant measurements and simplify this process by the use of the Dual Pythagorean Theorem, outlined in Reference 7, which is implied in the solutions of Linear, Quadratic, and Gaussian (LQG) estimation and control problems (Figure 6).


Figure 6 (a) The Pythagorean Theorem and (b) The Dual Pythagorean Theorem (Image courtesy of Reference 7)

The Dual Pythagorean Theorem clarifies the most important features of the bad data detection problem in power system state estimation.

I will not go into the ‘gory’ details here, please see Reference 7 for that, but Figure 7 shows the result of some equation manipulations and the addition of a second measurement to a given set of measurements in Reference 7 to get the regions of low and high likelihood of detection of bad data from considering error variance and minimum variance estimates as well as obtaining the normalized residue whose value is compared to a limit to either accept or reject the ith measurement. See Figure 7 for a graphic representation of the minimum error after adding a second measurement to a previous set of measurements.


Figure 7 (a) shows low likelihood of error detection and (b) shows the high likelihood of error detection (Image courtesy of Reference 7)

A Pythagorean treelet-shaped fractal antenna10,11

Here’s a useful design for an antenna in the 0 to 6 GHz region that can be used for Wi-Max, WLAN, some IEEE C band frequencies, and NATO G bands. The fractal, or broken, approach allows a larger set of parameters for controlling the antenna characteristics. Different iterations and etching is shown that will increase the resonance frequencies. Based on simulation results, proposed antenna has good miniaturization and light weight.

The IED3 simulator (version 14.10) was used in this effort.

The Pythagorean Fractal treelet is a planar antenna constructed by small patches shaving a square shape. The name Pythagorean treelet fractal antenna is used because it obeys the Pythagorean Theorem. In this case, a right triangle is formed when three squares are touching each other (Figure 8).


Figure 8 The proposed antenna geometry with finite ground plane using microstrip feeding (Image courtesy of Reference 10)

The antenna in this design is printed over an RT/Duroid substrate.

Reference 11 shows that by using a fractal geometry (i.e., Pythagoras tree), the design results in a multi-frequency and ultra-wide bandwidth operation antenna without making any further modifications, such as incorporation of U or L slots and stacking techniques which offer an added advantage. However, a supplementary use of such modifications will certainly help in antenna size reduction with a further improved performance.

Other Pythagorean ideas

Some other interesting Pythagorean Theorem relationships are outlined in Reference 8 which shows the Pythagorean Theorem’s relationship to the derivative function and Reference 9 which shows an educational game based on the Pythagorean Theorem for game-based learning.


  1. Letter to the Editor: A Puzzle: How to Prove the Pythagorean Theorem in Antenna Analysis, Tomasz Dlugosz, IEEE Antennas and Propagation Magazine, Vol. 55, No. 3, June 2013
  2. Reduction of Energy Consumption In WSN using the Generalized Pythagorean Theorem, S. Bouarafa, R. Saadane, D. Abouttajdine, IEEE, 2015
  3. Chopard and M. Droz, “Cellular Automata Modeling of Physical Systems,” Cambridge, 1998.
  4. Ilachinski, “Cellular Automata: A Discrete Universe,” World Scientific.
  5. W.B. Heinzelman, A.P. Chandrakasan, H. Balakrishnan, “Application specific protocol architecture for wireless microsensor networks”, IEEE Transactions on Wireless Networking (2002).
  6. Wolfram, “A New Kind of Science,” Wolfram Media, Inc., Champaign, 2002.
  7. The Dual Pythagorean Theorem and Bad Data Detection, Paul H. Haley, Proceedings of the IEEE, Vol. 71, No. 4, April 1983.
  8. From the Pythagorean Theorem to the definition of the derivative function, Jacek Stańdo, Gertruda Gwóźdź-Łukawska, Jan Guncaga, IEEE 2012.
  9. Educational Game Design on Pythagorean Theorem For Game Based Learning Using 6i’s Component, Haritz Cahya Nugraha, Pranoto Hidaya Rusmin, 2015 4th International Conference on Interactive Digital Media (ICIDM), December 1-5, 2015 Bandung – Indonesia.
  10. Pythagorean Tree Shaped Antenna for Multi-Band Application, Renu Sharma, Dipa Nitin Kokane, International Conference on Electrical, Electronics, and Optimization Techniques (ICEEOT) – 2016
  11. Pythagoras Tree: A Fractal Patch Antenna For Multi-Frequency And Ultra-Wide Bandwidth Operations, A. Aggarwal and M. V. Kartikeyan, Progress In Electromagnetics Research C, Vol. 16, 25–35, 2010

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