Gothic v. Roman Arches – Trigonometry Lab

How Did This Lab Begin? 

In Trigonometry Class, we started to learn in this class was on Quadratic Functions and Graphs. In learning this topic, we have learned the topic in conjunction with the arch-building techniques of the Romans and the Gothic Arch styles. One of our Expected Student Outcome states that students would be able to demonstrate their innovative and informed thinking to solve problems and communicate effectively. To follow this guideline, we, as a class, decided to build an arch of our own, one in a Roman semi-circular style and Gothic style, parabolic arch. We were divided into two groups to build two arches.

What Is Our Question? 

Between circular arch and parabolic arch, which one is stronger?   
Hypothesis
If the type of the arch is related to the strength of the arch, then the Gothic Arch will be stronger than that of the Roman Arch.
Procedure
First of all the semi-circle of Roman Arch and Gothic arch was drawn according to mathematic formulas. The two arches were built so that the inside of the arch was described by following equations.
  • Equation 1: Roman Arch (circular)
    arch height = square root of [72^2 + (horizontal dist from center of arch)^2], measured in millimeters

  • Equation 2: Gothic Arch (parabolic)
    y = -0.045x^2 + 0.001x + 230 where:
    y = height (from base) in millimeters
    x = horizontal distance from center of arch in millimeters

Then the frame was made with cardboard box. First, the carbord box was cut into 4 semi circles and 4 parabolas of same size to make the ribs of the frame. Next, the frame was covered on top of the ribs and the whole thing was attached to the base, which was also a cardboard box. Then the water and plaster was mixed to make cement. The cement was made to put it on the both frame with small rocks. The Roman Arch was done in one class period and Gothic Arch was done in two class periods. After the cement on Roman Arch and Gothic Arch was hardened,  the arches were tested by standing on top to see how strong it was. The Gothic Arch was strong enough to hold a person that is 80 kg but Roman Arch broke right after person that is 80 kg stepped on it. The Roman Arch broke even before the other foot was on the air. 
* Check out pictures of procedure: http://blog.daum.net/ekthak323/3

Result

The final result of this experiment is that the experimenters tried to assess the strength of the arch by putting a direct downward force onto the two arches. Previously, the original plan was to hang a force of weight from the arch being on top and the weight being hung on the arch to show the difference in the strength of the two arches. But, the plan was changed so that the experiment would test the strength of the arch when the downward force of human weight is applied onto the arch. It was observed that the Gothic Arch was much stronger and could withstand several trials of different human weight pushing downward on the arch. However, the Roman Arch, when stood upon, immediately broke in half and cracked into two pieces.
Some of the observations about the two arches are as follows:
1. The break that was formed along the edges of the Roman Arch was only on the cements, suggesting that there was not much amount of rocks in the middle part of the arch.
2. As previously said, the strength of the arch probably was dependent on the thickness of the arch, in which the Gothic Arch was much thicker than the Roman Arch.

Discussion

There are many factors in this experiment that could have potentially affected the results of this experiment. Indeed, as previously predicted, the thickness of the arch layer was the critical factor towards the overall strength of the arch. When the arch was being built, the structural organization of the Gothic Arch made it harder to be built, to attach the rocks and cement onto the cardboard base structure. Thus, the experiment required the experimentees to take precaution in evenly spreading the rocks and cement throughout the arch.
Meanwhile, the construction of the Roman Arch was much easier, as the overall height was not as tall as the Gothic Arch. Also, in the process of the construction, plaster was originally used to fix the rocks onto the cardboard structure, and was soon demolished as it was proven to be not as strong as using the cement. This could have potentially affected the results of the overall experiment.
Another factor was that the height difference could have affected the results as well. In fact, the Roman Arch was only one-fourth of the height compared to the Gothic Arch. Given that same pressure of the human weight’s downward force was applied, it would be indeed true that the Roman Arch would not be as strong compared to that of Gothic Arch.
Conclusion 
The hypothesis, “If the type of the arch is related to the strength of the arch, then the Gothic arch will be stronger than that of the Roman arch”, is consistent with the results of the experiment; however, there are many flaws, or sources of errors in this experiment that can make the results not fully true according to the experiment. Although the class has strived to accomplish a rather fair experiment by having the width (the gap between) similar length, the class has failed to make the thickness of the arch similar to each other, making the results obviously skewed to the “thicker” Gothic Arch than the “thinner” Roman Arch. If this experiment was conducted once again, the class would strive for the similar thickness and length of the arch. The class as a whole learned first-hand of arch building through the usage of cement and rock stacking as well as experienced a brief fresco painting, mimicking Michelangelo’s painting experience with many artworks such as the painting the ceiling of the Sistine Chapel. Through the results seen in the experiment, although flawed, we could clearly see the difference between the Gothic arch and the Roman arch – the Gothic being stronger than the Roman arch because of the Gothic arch’s strength to withhold heavy weight on top of the arch multiple times. This conclusion explains the reason why in daily life, such as the bridges found all over the world, a parabolic arch is used to hold the bridge up. Through the entire experience, we found the application of math in some of our crucial daily aspects of real life.
 
Thanks for reading this blog post. 
-BAIS Trigonometry Class of 2013-2014

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