This project originated with an idea to create a deployable relief shelter that would expand and contract for deployment. The project focused on the design and the structure itself, the mathematics behind the structure, and then used these to develop a physical prototype in the form of a scale model.
Stage 1: Compact
Stage 2: Semi Expanded
Stage 3: Fully Expanded
Chuck Hoberman is best known for his principle creation: The Hoberman Sphere, a toy which was popularized in the 1990’s. However creating a toy was not what inspired Hoberman to envision such a structure, instead he envisioned the structure to serve an architectural purpose to create transformative designs. He wanted to explore how art, engineering, and architecture can be fused together to form something new. The goal of doing such was to create architectural spaces that are not bound by the means of their structure but are instead defined by the transformation of its surrounding structure. The sphere is comprised of interconnected rings that are assembled from individual segments that create a scissor mechanism.
This scissor mechanism is one of the basic components used in several different types of transformative structures, which makes it an ideal starting point to create a transformative structure of one’s own. Exploring this topic is personally engaging because it pertains to my personal interest in architecture and design. I have an interest in developing a project that incorporates a transformative structure, but in order to do so it is necessary to understand the mathematic relationships that drive the mechanics of the scissor component, and thus of the Hoberman Sphere.
Therefore, the aim of this exploration is to leverage geometric relationships between the components and the form they compose when linked together to create a series of equations that model the characteristics of a specific Hoberman Sphere given specific parameters. Lastly, a visual representation of the mechanism will be created so designers can visualize the structure. This will be personally beneficial as someone who wants to create an expanding and contracting shelter by using the mechanics set forth in this exploration.
Fundamental Design of the Joist
It is most logical to begin this exploration by examining the individual components of the Hoberman Sphere. In doing so, their relationships to each other will become apparent, which will form the foundation for the geometric properties used to create relationships between them.
The core component of the Hoberman Sphere is a joist with an angled bend at its center point, designated by a. The distance between the center point and each side’s endpoint is the same, this length is designated by L. Additionally, although not strictly necessary to be modeled, the joist has a width which is designated by c. The design of the joist is shown in Figure 1.1. The joist is then attached to other joists, at their center points to form a segment. A circular ring is then formed by connecting the endpoints of each segment to the corresponding endpoints on another segment. Finally the sphere is comprised by arranging multiple rings in three dimensional space and affixing them using connectors. The connectors themselves are typically square so that two rings can be attached to each other at 90 degree angles. These intersecting rings form the overall sphere shape of the structure.
Geometrical Relationships Between Joists
Now that an understanding of how each component interacts with one another, it is now possible to begin to analyze their relationships geometrically.
The first thing that needs to be understood is how the angle in the joist affects the rest of the mechanism. The angle present in the joist is the same as the angle present in the segment, and thus determines the number of segments that are needed to make a ring of segments. This makes logical sense because each joist is bent at an angle. When the joists are overlapped the angle present on each side of the segment is now half of what it is in the joist itself. However, since two joists form the segment the overall angle between the endpoints is the same as the angle found in the original joist’s design. This discovery led to an important thought: the angle between its endpoints remain constant even when the distance between them varies, as shown in Figure 2.1. This angle, between endpoints, is equal to the half of the angle present in the component. This is critical because it allows multiple segments to be arrayed together to form a regular polygon. As the angle between the segments is changed the only thing that changes is the distance between the endpoints.
Based on that understanding, the number of segments needed to create each of the circular rings can be calculated, this is shown in Formula 2.1. The logic behind this calculation is based on the properties of polygons and the property aforementioned. Since all polygons must have an exterior angle of 360° the number of segments needed to comprise a ring can be calculated by diving 360° by our desired angle. Adversely, the angle of the joist can be calculated by dividing 360° by the number of desired segments. For instance, if we wanted to have an angle of 45 degrees in each joist then we know that 8 segments are necessary to enclose the polygon. Adversely if we know we wanted 8 segments, we can calculate that each joist would need to be designed with a 45 degree angle in it. Table 2.1 shows examples of the relationship between the joist’s angle and the number of segments needed to enclose a polygon.
Figure 2.1, Equation 2.1, Table 2.1
Geometrical Relationships Between Segments
By examining how multiple segments are joined together the next property of the Hoberman Sphere is revealed. When two segments are combined a rhombus is formed between them, and this shape is what dictates the size of the Hoberman sphere. As the distance between the endpoints is decreased the rhombus’ width increases. As the width between the endpoints is increased the rhombus’ width decreases. This arrangement is visible in Figure 3.1.
Therefore, the dimensions of the sphere can be calculated by using the dimension of the rhombus. When each rhombus is formed by connecting one of the center points with an endpoint, then connecting that endpoint to the other center point, connecting that center point to the other endpoint, and lastly connecting that endpoint back to the original center point. The geometry of the rhombus can be divided into four triangles, as shown in Figure 3.2. This is helpful to understand since to calculate the width and height of the rhombus the Pythagorean Theorem of triangles will have to be used. In order to calculate the relationship between the rhombus formed by the adjoining segments it is necessary to understand the geometry that informs the shape of the rhombus. The rhombus inherits its side lengths from the length specified in the design of the component defined as L. Therefore, since all of the rhombus’s sides are equal, it can be divided into four right triangles. These each will have a hypotenuse of length L since that is the side length of the rhombus. However since the rhombus is not always a perfect square it cannot be assumed that the two side lengths are equal. Therefore the side length can not be calculated unless we know one of the lengths already.
A visualization of the shape overall is necessary for this task; imagine the assembly of the segments to form the ring. If we connect all of the center points of the segments we can see that they form a polygon. It is known that the rhombus’s width is a variable that determines the overall size of the polygon. Therefore we can express the width between segments as the variable s. The manipulation of this parameter by the end user is what will cause the sphere to expand and contract. In layman’s terms when you pull on the side of the Hoberman Sphere you are effectively changing the width of the rhombus that are created between the segments.
Given that the rhombus was divided into four right triangles, it is evident that the width of each triangle is half of the width of the overall rhombus. Thus the triangles with is given by 1/2 × s. With this knowledge, Pythagorean’s Theorem can be used to find the height of each triangle. This process is show in Computation 3.1. Therefore since the triangle’s height is equal to half of the rhombus’ height, the height of the rhombus is equal to twice that of the triangle.
Area and Volume of the Enclosed Polygon
With this established our findings about the properties of the rhombus, present between segments, can be synthesized to find the properties of the rings that make up the Hoberman Sphere. Now that this relationship is know it is possible to model other aspects of the Hoberman Sphere by leveraging properties of a polygon.
The first property we need to understand is the relationship between the polygons side length, given by L, and its radius. This relationship is given in Formula 4.1. Since the side length of the polygon is known and the number of sides is known, the radius can be calculated. Formula 4.2 can then be used to calculate the area of the polygon, as shown in Computation 4.1. However, this formula uses a variable known as the apothem, which is the distance from the polygons center to the center of one of its sides. Luckily there is a formula to calculate the apothem from the radius, shown in Formula 4.3. Lastly, the formula for finding the perimeter of a polygon is shown in Formula 4.4. However, this is not a practical representation of the area inside each ring, since the polygon is formed by connecting the center points of the individual segments half of the structure of the segments are actually inside of the polygon. To account for this, half of the area of the rhombus must be subtracted from the area of the polygon. To this, it is necessary to calculate the area of a triangle using Formula 4.5. The calculations necessary to find the area of the inner half of the structure is shown in Computation 4.2 and the calculations needed to calculated the area enclosed by the ring of segments is shown in Computation 4.3.
From the area we can also find the volume of the enclosed polygon by multiplying 1/2 of the area by 2π, this is shown in Formula 4.6. This is fairly logical step to take if we imagine that the area each presents an individual slice of a sphere. If we rotate this slice around an axis, a solid of revolution will be formed from the slices which will have the volume set forth in Formula 4.6.
Formula 4.1, Formula 4.2
Formula 4.3, Formula 4.4
Formula 4.5, Formula 4.6
Modeling the Hoberman Sphere
The next logical step in the exploration is to combine all of the properties that have been discovered thus far together to get a holistic view of how a specific Hoberman Sphere would perform given the parameters of the number of segments, the length of each side of a joist, and the width at a given moment. In order to model the characteristics of the Hoberman Sphere, it must be graphed each time the parameters are adjusted. However before parameters can be adjusted, the math that drives the graphing must first be discovered.
This process of discovery begins by setting exploratory values for each of our variables. In this instance, the length of each joist’s side (L) was set to be 2, the number of segments was (n) 8, and the width between each segment (s) was set to 2. By employing the familiar equation set forth in Formula 4.1, the radius of the polygon can be found. In order to graph the polygon it must be represented by inscribing it inside of a circle so its corners touch the perimeter of the circle. In order to do this, it is necessary to find coordinates on the circles perimeter that are spread apart by the angle of each segment. This angle is known in degrees by using Formula 2.1, however it is needed in radians for this application, because we will be using sine and cosine functions to locate points on the circle’s perimeter. So instead of representing circle by 360 degrees, it is represented by 2π radians. This is reflected in Formula 5.1.
In order to determine the points on the circle, the properties of the Unit Circle needs to be recalled. Recalling that cosine(t) represents the x coordinate on the circle and sine(t). Using the equation in Formula 5.1, the angular separation between points can be calculated. In order to do this we use Formula 5.2, where P represents an integer from [1…n] and a is the angle present as a result of using Formula 5.1. What is occurring in this function is that each time the function is called it evaluates a different value for P, which has the effect of evaluating it for each angle. This is easiest to understand through an example: imagine that the radius at this instance is 2, and that four segments comprise the polygon. Using Formula 5.1, it is evident that each segment will represent 1/2 π of the overall circle. Therefore if we multiply this value by the values in the set P we will get the position of each angle. This is shown in Computation 5.2. The same process is used to find the y coordinate, however the sine equation is used instead, as shown in Formula 5.3.
Next the values from these two functions are plotted on the graph. The next step is to create the points that represent the end points of the segments. In order to do this the inner radius and outer radius of the points needs to be found. Once this is complete it is necessary to find the points on this radius where the endpoints of the segments would fall. To do this, a variation of the equation shown in Formula 5.2 and Formula 5.3 where used. The radius value was swapped for the inner and outer value of the radius respectively, which is calculated by adding or subtracting h from the radius. Additionally both functions were shifted by π/n. The purpose of this shift was to align the points with the center of the rhombus formed by the segments. The π/n was deduced from Formula 5.1, but the point only needs to be shifted to the center. Therefore it needs to be shifted by half of the angle of the entire segment. Thus if the equation in Formula 5.1 is multiplied by 1/2 it represents this angle. These new equations are shown in Formula 5.4, Formula 5.5, Formula 5.6, and Formula 5.7. Next it is necessary to collect the values generated by Formula 5.2 into a set called Uradius and Formula 5.3 into a set called Vradius, the values generated by Formula 5.4 into a set called Uradius-inner and Formula 5.5 into a set called Vradius-inner, the values generated by Formula 5.6 into a set called Uradius-outer and Formula 5.7 into a set called Vradius-outer. These points are graphed by simply inputting (Uradius, Vradius), (Uradius-inner, Vradius-inner), (Uradius-outer, Vradius-outer) into the graphing application.
The last component of graphing the Hoberman Sphere is to connect these points with each other to outline the shapes formed by the joists in the segments. To do so, the equation in Formula 5.8 is used. The formula was devised by using the simple properties of graphing a line on a graph. The first step was to find the slope of the line, which is simple since we know two sets of points. To do this use Formula 5.9, which calculates the slope of a line. Next the lines needed to be shifted horizontally so they start at the right point and not the origin. To do this, the U value is subtracted from the x value. Next the function needs to be shifted vertically, for this we just add the V value to the function. Lastly we restrict the domain of the function to be between the two points. That way the lines only go between the points and not past them. However because there were so many points to connect, multiple versions of the equation had to be created. There are two differences with these other equations, they either: are the inverse of the originals in order to reflect the top portion of the graph over the y-axis or they are just connecting different coordinates. Regardless of what they are connecting, they all work the same. These variations are show in Formulas 5.10. The resulting graph is shown in Graph 5.1.
The last part of this investigation references back to the property uncovered about the rhombus enclosed by when two segments are joined together. I wondered about the rate of change of the rhombus’ height, represented by 2h, in relationship to the change of the side length of the polygon, represented by s. To solve this, a hypothetical situation was set up as follows: A Hoberman Sphere’s joist has a side length of 8, the current length of the polygon’s edge is also 8, and the polygon’s width is increasing at a rate of 2 units per minute. With this information the rate at which h is decreasing. This is done by substituting these known values into Pythagorean’s Theorem and then deriving the equation. This will produce two rates of change, one if which we know, leaving one to be solved for. This is shown in Computation 6.1.
Reflection and Conclusions
In the process of exploring this topic, I learned more about geometry than I have in the traditional classroom setting, since I have never taken a math class that focuses on strictly geometry. I found it enjoyable to ponder the relationships between the basic geometric ideas and the more advance ones: connecting Pythagorean’s Theorem with sine and cosine functions, and eventually incorporating some calculus to find how the geometric values change in response to each other. Also as a visual learner, the creation of the graph was fundamental to working through the exploration. By comparing the graph to an actual ring of sections that enclose a polygon I could check my work along the way. Additionally through the process of creating it, I effectively made a tool to visualize the ring of segments, which could be used by others in the future to aid in the creation of their own transformative structure. Lastly I learned more about a field I am interested, how transformative design can be used in architecture. Now with my deep understanding of how the structure functions, I can create models of it for use in my own architectural models.
In addition to being personally rewarding, the mathematical results of this exploration were also successful. The exploration began by analyzing the geometry of the individual components to find their relationship to the overall geometry of the sphere. These relationships were represented mathematically by linking the radius to the inputed parameters. Then the radius was used to calculate other properties, such as the apothem, which later set the foundation for finding the area and volume of the Hoberman Sphere. Lastly these properties were all linked together using the equations of lines in a graph, which offered a simple representation of the mathematics that drive the mechanics of the Hoberman Sphere. With this my original goal for the exploration, to create a simple model of the Hoberman Sphere, was achieved.
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