# The Gateway Arch

## by CliffordWhite, published Aug 9, 2015

The Gateway Arch Aug 9, 2015
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# See the World!

This Challenge charged us makers to create something iconic -- a place or famous thing on this planet. The thing I decided to model for the Challenge is not just a famous place to see in the world. While it is a world-famous landmark, It is also a monument to those that went into the unknown to #SeeTheWorld. The people of the United States of America have always been among the forefront of those seeking to see new parts of this world. From our very pilgrimage to North America to Lewis and Clark's adventure, we have run full-speed into the unknown. From Alaska to the Moon, Americans have always been curious people.

The Gateway Arch in St. Louis, Missouri was built from 1963-1966 to specifically honor Lewis and Clark's expedition into the American west during the years of 1803 and 1804 and America's Westward expansion. At 630 feet, it is the tallest monument in the entire United States.

## Design

The Arch's architect, Eero Saarinen, used very strict mathematics to describe the shape of his Arch. The use of these equations made it possible to make an exact 3D model using SolidWorks.

### Centroid Curve:

The centroid curve is a line that passes through an imaginary point along the Arch's cross-section. This curve is considered a form of a catenary curve, and is governed by the following equation:
`y = 693.8597 - 68.7672cosh(0.0100333x)`
Where Y is height about the ground, and X is distance from the center of the arch. Because SolidWorks does not support the hyperbolic cosine function, I converted that equation into a function of `e^(x)` using the definition of the hyperbolic cosine. The new equation became:
`y = 693.8597 - 34.3836(e^(0.0100333x)+e^(-0.0100333x))`
Graphing this from `-299.2239` to `+299.2239` gave an accurate model of the centroid curve.

### Cross-Section

Because the Arch's cross-section is not constant throughout, another function is used to calculate the area of any particular cross-section. (A cross-section being defined as a plane perpendicular to the centroid curve at any point):
`Q = 125.1406cosh(0.010333x)`
Where Q is cross-sectional area. Because all cross-sections are equilateral triangles, I then derived the equation for an equilateral triangle's base length as a function of its area:
`B = (2*3^(3/4)*sqrt(Q)) / 3`
Where B is the base length. I then combined these two equations to create an equation for the base length of a cross section as a function of its distance from the center of the arch:
`B = (2*3^(3/4)*sqrt(125.1406cosh(0.010333x))) / 3`
(Yeah, I know that's a mess, but it works).

### Modeling

I then added a number of points, at measured distances from the center, on the centroid curve that I had already graphed in SolidWorks. I created planes perpendicular to the centroid that contained these points. I then made sketches on these planes of each cross-section, using the dimensions calculated from the equations above. After creating about a dozen and a half of these cross-sections, I used a loft feature to create the solid shape of the arch.

### 3D Print-ability

In order to make the arch 3D printable, I had to split it into several separate sections, interconnected with pins. I used AutoDesk MeshMixer to scale and rotate the sections so that a flat side would always be facing the print bed. I also designed a base so it would be able to stand on its own.

## The Middle Two Sections

Because I wanted to model the Arch in a high level of detail, but also wanted to be able to fit it in a sane amount of space, I had to come to a compromise -- I would model the solid shape of the entire arch at 1:1200 scale, then model the middle two sections at 1:120 scale. I also created a base for these two sections so that they make a classy display on their own.

## Conclusion

The Gateway Arch stands as a masterpiece of mathematical architecture, beautiful as both art and math. It was a technologically unprecedented project that was engineered and built flawlessly. Today it stands as a monument to those who dared to #SeeTheWorld.

### Credits

All the information I used in designing the Arch is available from various sources online, including but not limited to; The National Park Service, The University of Houston, The Mathematical Sciences Research Institute, and Wikipedia. Send me a personal message if you want any of the specific URLs I consulted.

# Instructions

## Main Arch

Print two each of the files `bottom.stl`, `top.stl`, and `ArchBase.stl`. The arch sections require a raft, but otherwise print without support material, although a small amount may help the quality of `top.stl`. Print one copy of `pins.stl`. I printed these parts at a layer height of 0.1mm. Glue of your choice is recommended when assembling the pieces.

## Middle Sections

The middle sections print pretty easily. Just print them upright without support material. The base is equally straight-forward. I again used 0.1mm layer height.

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