A Mathematical Model for Space Planning to Minimize Travel Distance
Authors: Han Li 
Published: July 2019

IntroductionThe purpose of this research is to investigate how mathematical knowledge can be applied to space planning in design. To begin, it is useful to know how mathematics was utilized in the past to reach creative and logical design solutions. The design process is a multilayered and multifaceted investigation and it aims to find an optimized solution for a defined problem. Since optimization is a mathematical term, it is reasonable to assume many design problems can be solved using mathematical models. Mathematics and design were inseparable from the beginning. In ancient civilizations people build houses and temples facing north. However, without a compass, how would one know which direction is true north? “North” was a decision. Our ancestors decided to call north the direction to which the shortest shadow of the day points (Evans 1998, 28). However, it is difficult to tell where the shadow is shortest. Our ancestors used simple tools such as a stick and a string to draw an arc on the ground. They observed and marked the shadow where it first intersects with the arc and also where it last intersects with the arc. They would draw a line between the points and find the middle point. What our ancestors really cared about was their relationship with the sun and exposure to sunlight. Although it is very easy for a modern person to understand this ancient method, it is not easy to reach this solution independently if the answer has not been given. The orientation of a building is a design problem and it was solved by simple mathematical deductions by our ancestors. A good designer cannot be ignorant about the various aspects of science related to design. For example, design decisions for space planning—location of the fire exit, travel distance, orientation of the building—can be calculated using mathematical models, especially optimization problem solving. A typical question in calculus illustrates how mathematics can be applied to design problem solving (Tsishchanka 2010). A farmer has 2400 ft. of fencing and wants to fence off a rectangular field that borders a straight river. He needs no fence along the river. What are the dimensions of the field that has the largest area? The mathematical model is: Maximize: A = xy Constraint: 2x+y = 2400ResearchNevertheless, many design problems are more sophisticated. In this research a case study of a space planning project for an extension of an existing out-patient hospital is conducted. The goal of this case study is to find mathematical models for achieving optimal solutions of (1) overall travel distances in one day and (2) numbers and locations of exits. A patient's travel distance is defined as the total distance traveled by the patient inside the hospital from the entrance to the exit. This research began by looking at the importance of the center, shapes of a building and locations of entrances and exits. The objective was to understand how decisions on those design elements affect the total travel distance, and then to identify applicable mathematical formulas to minimize travel distance in one day. In order to apply any mathematical formula the designer first needs to establish the constraints, constants and variables. In this research, the project requirements such as numbers of visitors, numbers of rooms, functions of rooms and hierarchy of spaces and adjacencies are based on similar projects and are hypothetically established. There are two levels in this outpatient hospital. This research focuses on the ground level, in which the designer needs to do space planning to accommodate Internal Medicine, Emergency, Radiology and Imaging, Pharmacy and Intravenous Therapy Departments. Preliminary research revealed that Emergency should have its own entrance that should be close to the parking lot for easy access. Pharmacy, Radiology and Imaging, and Intravenous Therapy are shared functions of other medical departments—especially Pharmacy, which 90% of patients use just before leaving the hospital. 10% of patients from Internal Medicine will need to do an x-ray, CT or MRI with the Imaging Department, while 35% of patients from Emergency will need to use the Imaging Department. The question is where each department should be located to minimize travel. The existing site conditions such as the location of the new entrance, parking location and city traffic are arbitrarily decided to support the research. The project requirements have not themselves been verified for optimization. The arbitrary decisions may not reflect true human needs, but in any case they affect the calculated results. However, they would not affect the validity of this research because it depends on inherent mathematical logic. It is worth noting that the focus of this research is purely on functionality and individual topics. The goal of this research is to offer insight into how simple mathematical formulas could help solve design problems. However, these formulas can never offer a fully-fledged creative design solution, because all the problems are infinitely more sophisticated than a controlled scenario, especially when aesthetics are concerned.ConclusionDesign is often considered to be an applied art rather than an applied science in today's world. In recent history, designers were separated from engineers and builders. When a beautiful freehanded line can communicate everything on paper, the need of knowing the radius and tangency became less important. In addition, the knowledge required to understand mathematical formulas is often beyond designers’ reach. For example, many designers may not be able to fully comprehend Milutin Milankovitch's arch theory of masonry construction, even though it has significance in civil engineering and architecture (Foce 2008). Designers often are disengaged with rapid developments in science and leave them to civil engineers or mechanics. Nonetheless, this disengagement isolates designers from the rest of the team and makes them less effective. In order to rediscover design, new approaches are needed. This research reestablishes the connection between mathematic and design through space planning, and seeks to offer some different perspectives of what design really is.ReferencesEvans, James. 1998. The History and Practice of Ancient Astronomy. Oxford University Press.Foce, Federico. 2008. “Milankovitch's Theorie Der Druckkurven: Good Mechanics for Masonry Architecture.” In Nexus Network Journal, edited by Kim Williams, 9,2:185–209. Nexus Network Journal. Birkhäuser Basel. http://dx.doi.org/10.1007/978-3-7643-8699-3_3.Tsishchanka, Kiryl. 2010. “Optimization Problems.” Courant Institute of Mathematical Science of NYU. Accessed October 04, 2015. https://cims.nyu.edu/ ∼ kiryl/Calculus/Section_4.5–Optimization%20Problems/Optimization_Problems.pdf