form a hypothesis regarding the cube and rectangular forms

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11.12: Crystal Forms and the Miller Index

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• Dexter Perkins
• University of North Dakota

The replacement of unknown or variable numbers in a Miller index with h, k, or l allows us to make generalizations. The index ( hk 0) describes the family of faces with their third index equal to zero. A Miller index including a zero describes a face is parallel to one or more axes (Figure 11.66). The family of faces described by ( hk 0) is parallel to the c-axis (Figure 11.65); faces with the Miller index (00 l ) are parallel to both the a-axis and the b-axis (Figure 11.64 c ).

Figure 11.66 a shows an orthorhombic prism with six faces. Some faces cannot be seen, but the Miller indices of all six are (100), (010), (001), ( 1 00), (0 1 0), and (00 1 ). Although this crystal contains three forms, indices for all faces contain the same numbers (two zeros and a one) but the order of numbers and the + or – sign changes.

Figure 11.66 b shows an orthorhombic dipyramid. It contains only one form, and all the faces have the same numbers in their Miller index (332). This is always true for faces that belong to the same form; they always have similar Miller indices. This relationship is especially clear for crystals in the cubic system because high symmetry means that many forms may contain many identical faces.

The six identical faces on the cube in Figure 11.67 a have indices (001), (010), (100), (00 1 ), (0 1 0), and ( 1 00). We symbolize the entire form {100}, and the { } braces indicate the form contains all faces with the numerals 1, 0, and 0 in their Miller index, no matter the order.

The four faces on the tetrahedron in Figure 11.67 b , and the eight faces on the octahedron in Figure 11.67 c are all equilateral triangles. For both, the form is {111}. As Figure 11.67 b and c demonstrate, two crystals of different shapes can have the same form if they belong to different point groups. The tetrahedron in Figure 11.67 b belongs to point group 4 3m; the octahedron in Figure 11.67 c belongs to point group 4 / m 3 2 / m . If we know the point group and the form, we can calculate the orientation of faces. If a crystal contains only one form, we then know the shape of the crystal. Note that the cube, octahedron, and dodecahedron all belong to point group 4 / m 3 2 / m . The cubic form is {100}, the octahedral form is {111}, and the dodecahedral form in Figure 11.67 d is {110}.

Figure 11.67 e shows a crystal containing three forms: cube {100}, octahedron {111}, and dodecahedron {110}. Because they all belong to point group 4 / m 3 2 / m , we know the faces are oriented as shown. However, the crystal in Figure 11.67 f belongs to the same point group and contains the same forms, but the size and shape of corresponding faces are different. We do not know the crystal shape if more than one form is present, unless we know some extra information.

Crystallographers sometimes label faces of the same form with the same letter as we have done in Figure 11.67. For some forms, the letter is just the first letter of the form name. For example, o indicates the octahedral form and d the dodecahedral form in the cubic system. Usually, however, the symbols are less obvious (we normally designate cube faces, for example, by the letter a ); labels also vary from one crystal system to another.

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Biology library

Course: biology library   >   unit 1.

• The scientific method
• Controlled experiments

The scientific method and experimental design

• (Choice A)   The facts collected from an experiment are written in the form of a hypothesis. A The facts collected from an experiment are written in the form of a hypothesis.
• (Choice B)   A hypothesis is the correct answer to a scientific question. B A hypothesis is the correct answer to a scientific question.
• (Choice C)   A hypothesis is a possible, testable explanation for a scientific question. C A hypothesis is a possible, testable explanation for a scientific question.
• (Choice D)   A hypothesis is the process of making careful observations. D A hypothesis is the process of making careful observations.

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Skills to Develop

• Classify the different types of bones in the skeleton
• Explain the role of the different cell types in bone
• Explain how bone forms during development

Bone , or osseous tissue , is a connective tissue that constitutes the endoskeleton. It contains specialized cells and a matrix of mineral salts and collagen fibers.

The mineral salts primarily include hydroxyapatite, a mineral formed from calcium phosphate. Calcification is the process of deposition of mineral salts on the collagen fiber matrix that crystallizes and hardens the tissue. The process of calcification only occurs in the presence of collagen fibers.

The bones of the human skeleton are classified by their shape: long bones, short bones, flat bones, sutural bones, sesamoid bones, and irregular bones (Figure \(\PageIndex{1}\)).

Long bones are longer than they are wide and have a shaft and two ends. The diaphysis , or central shaft, contains bone marrow in a marrow cavity. The rounded ends, the epiphyses , are covered with articular cartilage and are filled with red bone marrow, which produces blood cells (Figure \(\PageIndex{2}\)). Most of the limb bones are long bones—for example, the femur, tibia, ulna, and radius. Exceptions to this include the patella and the bones of the wrist and ankle.

Short bones , or cuboidal bones, are bones that are the same width and length, giving them a cube-like shape. For example, the bones of the wrist (carpals) and ankle (tarsals) are short bones (Figure \(\PageIndex{1}\)).

Flat bones are thin and relatively broad bones that are found where extensive protection of organs is required or where broad surfaces of muscle attachment are required. Examples of flat bones are the sternum (breast bone), ribs, scapulae (shoulder blades), and the roof of the skull (Figure \(\PageIndex{1}\)).

Irregular bones are bones with complex shapes. These bones may have short, flat, notched, or ridged surfaces. Examples of irregular bones are the vertebrae, hip bones, and several skull bones.

Sesamoid bones are small, flat bones and are shaped similarly to a sesame seed. The patellae are sesamoid bones (Figure \(\PageIndex{3}\)). Sesamoid bones develop inside tendons and may be found near joints at the knees, hands, and feet.

Sutural bones are small, flat, irregularly shaped bones. They may be found between the flat bones of the skull. They vary in number, shape, size, and position.

Bone Tissue

Bones are considered organs because they contain various types of tissue, such as blood, connective tissue, nerves, and bone tissue. Osteocytes, the living cells of bone tissue, form the mineral matrix of bones. There are two types of bone tissue: compact and spongy.

Compact Bone Tissue

Compact bone (or cortical bone) forms the hard external layer of all bones and surrounds the medullary cavity, or bone marrow. It provides protection and strength to bones. Compact bone tissue consists of units called osteons or Haversian systems. Osteons are cylindrical structures that contain a mineral matrix and living osteocytes connected by canaliculi, which transport blood. They are aligned parallel to the long axis of the bone. Each osteon consists of lamellae , which are layers of compact matrix that surround a central canal called the Haversian canal. The Haversian canal (osteonic canal) contains the bone’s blood vessels and nerve fibers (Figure \(\PageIndex{4}\)). Osteons in compact bone tissue are aligned in the same direction along lines of stress and help the bone resist bending or fracturing. Therefore, compact bone tissue is prominent in areas of bone at which stresses are applied in only a few directions.

Art Connection

Which of the following statements about bone tissue is false?

• Compact bone tissue is made of cylindrical osteons that are aligned such that they travel the length of the bone.
• Haversian canals contain blood vessels only.
• Haversian canals contain blood vessels and nerve fibers.
• Spongy tissue is found on the interior of the bone, and compact bone tissue is found on the exterior.

Spongy Bone Tissue

Whereas compact bone tissue forms the outer layer of all bones, spongy bone or cancellous bone forms the inner layer of all bones. Spongy bone tissue does not contain osteons that constitute compact bone tissue. Instead, it consists of trabeculae , which are lamellae that are arranged as rods or plates. Red bone marrow is found between the trabuculae. Blood vessels within this tissue deliver nutrients to osteocytes and remove waste. The red bone marrow of the femur and the interior of other large bones, such as the ileum, forms blood cells.

Spongy bone reduces the density of bone and allows the ends of long bones to compress as the result of stresses applied to the bone. Spongy bone is prominent in areas of bones that are not heavily stressed or where stresses arrive from many directions. The epiphyses of bones, such as the neck of the femur, are subject to stress from many directions. Imagine laying a heavy framed picture flat on the floor. You could hold up one side of the picture with a toothpick if the toothpick was perpendicular to the floor and the picture. Now drill a hole and stick the toothpick into the wall to hang up the picture. In this case, the function of the toothpick is to transmit the downward pressure of the picture to the wall. The force on the picture is straight down to the floor, but the force on the toothpick is both the picture wire pulling down and the bottom of the hole in the wall pushing up. The toothpick will break off right at the wall.

The neck of the femur is horizontal like the toothpick in the wall. The weight of the body pushes it down near the joint, but the vertical diaphysis of the femur pushes it up at the other end. The neck of the femur must be strong enough to transfer the downward force of the body weight horizontally to the vertical shaft of the femur (Figure \(\PageIndex{5}\)).

Link to Learning

View micrographs of musculoskeletal tissues as you review the anatomy.

Cell Types in Bones

Bone consists of four types of cells: osteoblasts, osteoclasts, osteocytes, and osteoprogenitor cells. Osteoblasts are bone cells that are responsible for bone formation. Osteoblasts synthesize and secrete the organic part and inorganic part of the extracellular matrix of bone tissue, and collagen fibers. Osteoblasts become trapped in these secretions and differentiate into less active osteocytes. Osteoclasts are large bone cells with up to 50 nuclei. They remove bone structure by releasing lysosomal enzymes and acids that dissolve the bony matrix. These minerals, released from bones into the blood, help regulate calcium concentrations in body fluids. Bone may also be resorbed for remodeling, if the applied stresses have changed. Osteocytes are mature bone cells and are the main cells in bony connective tissue; these cells cannot divide. Osteocytes maintain normal bone structure by recycling the mineral salts in the bony matrix. Osteoprogenitor cells are squamous stem cells that divide to produce daughter cells that differentiate into osteoblasts. Osteoprogenitor cells are important in the repair of fractures.

Development of Bone

Ossification , or osteogenesis, is the process of bone formation by osteoblasts. Ossification is distinct from the process of calcification; whereas calcification takes place during the ossification of bones, it can also occur in other tissues. Ossification begins approximately six weeks after fertilization in an embryo. Before this time, the embryonic skeleton consists entirely of fibrous membranes and hyaline cartilage. The development of bone from fibrous membranes is called intramembranous ossification; development from hyaline cartilage is called endochondral ossification. Bone growth continues until approximately age 25. Bones can grow in thickness throughout life, but after age 25, ossification functions primarily in bone remodeling and repair.

Intramembranous Ossification

Intramembranous ossification is the process of bone development from fibrous membranes. It is involved in the formation of the flat bones of the skull, the mandible, and the clavicles. Ossification begins as mesenchymal cells form a template of the future bone. They then differentiate into osteoblasts at the ossification center. Osteoblasts secrete the extracellular matrix and deposit calcium, which hardens the matrix. The non-mineralized portion of the bone or osteoid continues to form around blood vessels, forming spongy bone. Connective tissue in the matrix differentiates into red bone marrow in the fetus. The spongy bone is remodeled into a thin layer of compact bone on the surface of the spongy bone.

Endochondral Ossification

Endochondral ossification is the process of bone development from hyaline cartilage. All of the bones of the body, except for the flat bones of the skull, mandible, and clavicles, are formed through endochondral ossification.

In long bones, chondrocytes form a template of the hyaline cartilage diaphysis. Responding to complex developmental signals, the matrix begins to calcify. This calcification prevents diffusion of nutrients into the matrix, resulting in chondrocytes dying and the opening up of cavities in the diaphysis cartilage. Blood vessels invade the cavities, and osteoblasts and osteoclasts modify the calcified cartilage matrix into spongy bone. Osteoclasts then break down some of the spongy bone to create a marrow, or medullary, cavity in the center of the diaphysis. Dense, irregular connective tissue forms a sheath (periosteum) around the bones. The periosteum assists in attaching the bone to surrounding tissues, tendons, and ligaments. The bone continues to grow and elongate as the cartilage cells at the epiphyses divide.

In the last stage of prenatal bone development, the centers of the epiphyses begin to calcify. Secondary ossification centers form in the epiphyses as blood vessels and osteoblasts enter these areas and convert hyaline cartilage into spongy bone. Until adolescence, hyaline cartilage persists at the epiphyseal plate (growth plate), which is the region between the diaphysis and epiphysis that is responsible for the lengthwise growth of long bones (Figure \(\PageIndex{6}\)).

Growth of Bone

Long bones continue to lengthen, potentially until adolescence, through the addition of bone tissue at the epiphyseal plate. They also increase in width through appositional growth.

Lengthening of Long Bones

Chondrocytes on the epiphyseal side of the epiphyseal plate divide; one cell remains undifferentiated near the epiphysis, and one cell moves toward the diaphysis. The cells, which are pushed from the epiphysis, mature and are destroyed by calcification. This process replaces cartilage with bone on the diaphyseal side of the plate, resulting in a lengthening of the bone.

Long bones stop growing at around the age of 18 in females and the age of 21 in males in a process called epiphyseal plate closure. During this process, cartilage cells stop dividing and all of the cartilage is replaced by bone. The epiphyseal plate fades, leaving a structure called the epiphyseal line or epiphyseal remnant, and the epiphysis and diaphysis fuse.

Thickening of Long Bones

Appositional growth is the increase in the diameter of bones by the addition of bony tissue at the surface of bones. Osteoblasts at the bone surface secrete bone matrix, and osteoclasts on the inner surface break down bone. The osteoblasts differentiate into osteocytes. A balance between these two processes allows the bone to thicken without becoming too heavy.

Bone Remodeling and Repair

Bone renewal continues after birth into adulthood. Bone remodeling is the replacement of old bone tissue by new bone tissue. It involves the processes of bone deposition by osteoblasts and bone resorption by osteoclasts. Normal bone growth requires vitamins D, C, and A, plus minerals such as calcium, phosphorous, and magnesium. Hormones such as parathyroid hormone, growth hormone, and calcitonin are also required for proper bone growth and maintenance.

Bone turnover rates are quite high, with five to seven percent of bone mass being recycled every week. Differences in turnover rate exist in different areas of the skeleton and in different areas of a bone. For example, the bone in the head of the femur may be fully replaced every six months, whereas the bone along the shaft is altered much more slowly.

Bone remodeling allows bones to adapt to stresses by becoming thicker and stronger when subjected to stress. Bones that are not subject to normal stress, for example when a limb is in a cast, will begin to lose mass. A fractured or broken bone undergoes repair through four stages:

• Blood vessels in the broken bone tear and hemorrhage, resulting in the formation of clotted blood, or a hematoma, at the site of the break. The severed blood vessels at the broken ends of the bone are sealed by the clotting process, and bone cells that are deprived of nutrients begin to die.
• Within days of the fracture, capillaries grow into the hematoma, and phagocytic cells begin to clear away the dead cells. Though fragments of the blood clot may remain, fibroblasts and osteoblasts enter the area and begin to reform bone. Fibroblasts produce collagen fibers that connect the broken bone ends, and osteoblasts start to form spongy bone. The repair tissue between the broken bone ends is called the fibrocartilaginous callus, as it is composed of both hyaline and fibrocartilage (Figure \(\PageIndex{7}\)). Some bone spicules may also appear at this point.
• The fibrocartilaginous callus is converted into a bony callus of spongy bone. It takes about two months for the broken bone ends to be firmly joined together after the fracture. This is similar to the endochondral formation of bone, as cartilage becomes ossified; osteoblasts, osteoclasts, and bone matrix are present.
• The bony callus is then remodelled by osteoclasts and osteoblasts, with excess material on the exterior of the bone and within the medullary cavity being removed. Compact bone is added to create bone tissue that is similar to the original, unbroken bone. This remodeling can take many months, and the bone may remain uneven for years.

Scientific Method Connection: Decalcification of Bones

Question: What effect does the removal of calcium and collagen have on bone structure?

Background: Conduct a literature search on the role of calcium and collagen in maintaining bone structure. Conduct a literature search on diseases in which bone structure is compromised.

Hypothesis: Develop a hypothesis that states predictions of the flexibility, strength, and mass of bones that have had the calcium and collagen components removed. Develop a hypothesis regarding the attempt to add calcium back to decalcified bones.

Test the hypothesis: Test the prediction by removing calcium from chicken bones by placing them in a jar of vinegar for seven days. Test the hypothesis regarding adding calcium back to decalcified bone by placing the decalcified chicken bones into a jar of water with calcium supplements added. Test the prediction by denaturing the collagen from the bones by baking them at 250°C for three hours.

Analyze the data: Create a table showing the changes in bone flexibility, strength, and mass in the three different environments.

Report the results: Under which conditions was the bone most flexible? Under which conditions was the bone the strongest?

Draw a conclusion: Did the results support or refute the hypothesis? How do the results observed in this experiment correspond to diseases that destroy bone tissue?

Bone, or osseous tissue, is connective tissue that includes specialized cells, mineral salts, and collagen fibers. The human skeleton can be divided into long bones, short bones, flat bones, and irregular bones. Compact bone tissue is composed of osteons and forms the external layer of all bones. Spongy bone tissue is composed of trabeculae and forms the inner part of all bones. Four types of cells compose bony tissue: osteocytes, osteoclasts, osteoprogenitor cells, and osteoblasts. Ossification is the process of bone formation by osteoblasts. Intramembranous ossification is the process of bone development from fibrous membranes. Endochondral ossification is the process of bone development from hyaline cartilage. Long bones lengthen as chondrocytes divide and secrete hyaline cartilage. Osteoblasts replace cartilage with bone. Appositional growth is the increase in the diameter of bones by the addition of bone tissue at the surface of bones. Bone remodeling involves the processes of bone deposition by osteoblasts and bone resorption by osteoclasts. Bone repair occurs in four stages and can take several months.

Art Exercise

Figure \(\PageIndex{4}\): Which of the following statements about bone tissue is false?

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2.4: Developing a Hypothesis

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Learning Objectives

• Distinguish between a theory and a hypothesis.
• Discover how theories are used to generate hypotheses and how the results of studies can be used to further inform theories.
• Understand the characteristics of a good hypothesis.

Theories and Hypotheses

Before describing how to develop a hypothesis it is important to distinguish between a theory and a hypothesis. A theory is a coherent explanation or interpretation of one or more phenomena. Although theories can take a variety of forms, one thing they have in common is that they go beyond the phenomena they explain by including variables, structures, processes, functions, or organizing principles that have not been observed directly. Consider, for example, Zajonc’s theory of social facilitation and social inhibition. He proposed that being watched by others while performing a task creates a general state of physiological arousal, which increases the likelihood of the dominant (most likely) response. So for highly practiced tasks, being watched increases the tendency to make correct responses, but for relatively unpracticed tasks, being watched increases the tendency to make incorrect responses. Notice that this theory—which has come to be called drive theory—provides an explanation of both social facilitation and social inhibition that goes beyond the phenomena themselves by including concepts such as “arousal” and “dominant response,” along with processes such as the effect of arousal on the dominant response.

Outside of science, referring to an idea as a theory often implies that it is untested—perhaps no more than a wild guess. In science, however, the term theory has no such implication. A theory is simply an explanation or interpretation of a set of phenomena. It can be untested, but it can also be extensively tested, well supported, and accepted as an accurate description of the world by the scientific community. The theory of evolution by natural selection, for example, is a theory because it is an explanation of the diversity of life on earth—not because it is untested or unsupported by scientific research. On the contrary, the evidence for this theory is overwhelmingly positive and nearly all scientists accept its basic assumptions as accurate. Similarly, the “germ theory” of disease is a theory because it is an explanation of the origin of various diseases, not because there is any doubt that many diseases are caused by microorganisms that infect the body.

A hypothesis , on the other hand, is a specific prediction about a new phenomenon that should be observed if a particular theory is accurate. It is an explanation that relies on just a few key concepts. Hypotheses are often specific predictions about what will happen in a particular study. They are developed by considering existing evidence and using reasoning to infer what will happen in the specific context of interest. Hypotheses are often but not always derived from theories. So a hypothesis is often a prediction based on a theory but some hypotheses are a-theoretical and only after a set of observations have been made, is a theory developed. This is because theories are broad in nature and they explain larger bodies of data. So if our research question is really original then we may need to collect some data and make some observation before we can develop a broader theory.

Theories and hypotheses always have this if-then relationship. “ If drive theory is correct, then cockroaches should run through a straight runway faster, and a branching runway more slowly, when other cockroaches are present.” Although hypotheses are usually expressed as statements, they can always be rephrased as questions. “Do cockroaches run through a straight runway faster when other cockroaches are present?” Thus deriving hypotheses from theories is an excellent way of generating interesting research questions.

But how do researchers derive hypotheses from theories? One way is to generate a research question using the techniques discussed in this chapter and then ask whether any theory implies an answer to that question. For example, you might wonder whether expressive writing about positive experiences improves health as much as expressive writing about traumatic experiences. Although this question is an interesting one on its own, you might then ask whether the habituation theory—the idea that expressive writing causes people to habituate to negative thoughts and feelings—implies an answer. In this case, it seems clear that if the habituation theory is correct, then expressive writing about positive experiences should not be effective because it would not cause people to habituate to negative thoughts and feelings. A second way to derive hypotheses from theories is to focus on some component of the theory that has not yet been directly observed. For example, a researcher could focus on the process of habituation—perhaps hypothesizing that people should show fewer signs of emotional distress with each new writing session.

Among the very best hypotheses are those that distinguish between competing theories. For example, Norbert Schwarz and his colleagues considered two theories of how people make judgments about themselves, such as how assertive they are (Schwarz et al., 1991) [1] . Both theories held that such judgments are based on relevant examples that people bring to mind. However, one theory was that people base their judgments on the number of examples they bring to mind and the other was that people base their judgments on how easily they bring those examples to mind. To test these theories, the researchers asked people to recall either six times when they were assertive (which is easy for most people) or 12 times (which is difficult for most people). Then they asked them to judge their own assertiveness. Note that the number-of-examples theory implies that people who recalled 12 examples should judge themselves to be more assertive because they recalled more examples, but the ease-of-examples theory implies that participants who recalled six examples should judge themselves as more assertive because recalling the examples was easier. Thus the two theories made opposite predictions so that only one of the predictions could be confirmed. The surprising result was that participants who recalled fewer examples judged themselves to be more assertive—providing particularly convincing evidence in favor of the ease-of-retrieval theory over the number-of-examples theory.

Theory Testing

The primary way that scientific researchers use theories is sometimes called the hypothetico-deductive method (although this term is much more likely to be used by philosophers of science than by scientists themselves). A researcher begins with a set of phenomena and either constructs a theory to explain or interpret them or chooses an existing theory to work with. He or she then makes a prediction about some new phenomenon that should be observed if the theory is correct. Again, this prediction is called a hypothesis. The researcher then conducts an empirical study to test the hypothesis. Finally, he or she reevaluates the theory in light of the new results and revises it if necessary. This process is usually conceptualized as a cycle because the researcher can then derive a new hypothesis from the revised theory, conduct a new empirical study to test the hypothesis, and so on. As Figure 2.2 shows, this approach meshes nicely with the model of scientific research in psychology presented earlier in the textbook—creating a more detailed model of “theoretically motivated” or “theory-driven” research.

As an example, let us consider Zajonc’s research on social facilitation and inhibition. He started with a somewhat contradictory pattern of results from the research literature. He then constructed his drive theory, according to which being watched by others while performing a task causes physiological arousal, which increases an organism’s tendency to make the dominant response. This theory predicts social facilitation for well-learned tasks and social inhibition for poorly learned tasks. He now had a theory that organized previous results in a meaningful way—but he still needed to test it. He hypothesized that if his theory was correct, he should observe that the presence of others improves performance in a simple laboratory task but inhibits performance in a difficult version of the very same laboratory task. To test this hypothesis, one of the studies he conducted used cockroaches as subjects (Zajonc, Heingartner, & Herman, 1969) [2] . The cockroaches ran either down a straight runway (an easy task for a cockroach) or through a cross-shaped maze (a difficult task for a cockroach) to escape into a dark chamber when a light was shined on them. They did this either while alone or in the presence of other cockroaches in clear plastic “audience boxes.” Zajonc found that cockroaches in the straight runway reached their goal more quickly in the presence of other cockroaches, but cockroaches in the cross-shaped maze reached their goal more slowly when they were in the presence of other cockroaches. Thus he confirmed his hypothesis and provided support for his drive theory. (Zajonc also showed that drive theory existed in humans (Zajonc & Sales, 1966) [3] in many other studies afterward).

Incorporating Theory into Your Research

When you write your research report or plan your presentation, be aware that there are two basic ways that researchers usually include theory. The first is to raise a research question, answer that question by conducting a new study, and then offer one or more theories (usually more) to explain or interpret the results. This format works well for applied research questions and for research questions that existing theories do not address. The second way is to describe one or more existing theories, derive a hypothesis from one of those theories, test the hypothesis in a new study, and finally reevaluate the theory. This format works well when there is an existing theory that addresses the research question—especially if the resulting hypothesis is surprising or conflicts with a hypothesis derived from a different theory.

To use theories in your research will not only give you guidance in coming up with experiment ideas and possible projects, but it lends legitimacy to your work. Psychologists have been interested in a variety of human behaviors and have developed many theories along the way. Using established theories will help you break new ground as a researcher, not limit you from developing your own ideas.

Characteristics of a Good Hypothesis

There are three general characteristics of a good hypothesis. First, a good hypothesis must be testable and falsifiable . We must be able to test the hypothesis using the methods of science and if you’ll recall Popper’s falsifiability criterion, it must be possible to gather evidence that will disconfirm the hypothesis if it is indeed false. Second, a good hypothesis must be logical. As described above, hypotheses are more than just a random guess. Hypotheses should be informed by previous theories or observations and logical reasoning. Typically, we begin with a broad and general theory and use deductive reasoning to generate a more specific hypothesis to test based on that theory. Occasionally, however, when there is no theory to inform our hypothesis, we use inductive reasoning which involves using specific observations or research findings to form a more general hypothesis. Finally, the hypothesis should be positive. That is, the hypothesis should make a positive statement about the existence of a relationship or effect, rather than a statement that a relationship or effect does not exist. As scientists, we don’t set out to show that relationships do not exist or that effects do not occur so our hypotheses should not be worded in a way to suggest that an effect or relationship does not exist. The nature of science is to assume that something does not exist and then seek to find evidence to prove this wrong, to show that really it does exist. That may seem backward to you but that is the nature of the scientific method. The underlying reason for this is beyond the scope of this chapter but it has to do with statistical theory.

Key Takeaways

• A theory is broad in nature and explains larger bodies of data. A hypothesis is more specific and makes a prediction about the outcome of a particular study.
• Working with theories is not “icing on the cake.” It is a basic ingredient of psychological research.
• Like other scientists, psychologists use the hypothetico-deductive method. They construct theories to explain or interpret phenomena (or work with existing theories), derive hypotheses from their theories, test the hypotheses, and then reevaluate the theories in light of the new results.
• Practice: Find a recent empirical research report in a professional journal. Read the introduction and highlight in different colors descriptions of theories and hypotheses.
• Schwarz, N., Bless, H., Strack, F., Klumpp, G., Rittenauer-Schatka, H., & Simons, A. (1991). Ease of retrieval as information: Another look at the availability heuristic. Journal of Personality and Social Psychology, 61 , 195–202.
• Zajonc, R. B., Heingartner, A., & Herman, E. M. (1969). Social enhancement and impairment of performance in the cockroach. Journal of Personality and Social Psychology, 13 , 83–92.
• Zajonc, R.B. & Sales, S.M. (1966). Social facilitation of dominant and subordinate responses. Journal of Experimental Social Psychology, 2 , 160-168.

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A right rectangular prism is a box-shaped object, that is, a 3-dimensional solid with six rectangular faces.

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A right rectangular prism is also called a cuboid, box, or rectangular hexahedron. Moreover, the names "rectangular prism" and " right rectangular prisms" are often used interchangeably.

The most common math problems related to this solid are of the type right rectangular prism calc find V or find A , where the letters stand for the V olume and A rea , respectively. Let's see the necessary rectangular prism formula and learn how to solve t these problems quickly and easily.

We calculate the volume of a rectangular prism with the following formula:

volume = h × w × l ,

where h is the height of the prism, w is its width, and l is its length. To calculate the volume of a cardboard box:

• Find the length of the box , say 18 in.
• Determine its width - 12 in.
• Find out the height of the rectangular prism - 15 in.
• Calculate the volume of the cuboid . Using the rectangular prism volume formula, we get: volume = (18 × 12 × 15) in³ volume = 3240 in³ .

The surface area of the cuboid consists of 6 faces - three pairs of parallel rectangles. To find the rectangular prism surface area, add the areas of all faces:

surface_area = 2 × (h × w) + 2 × (h × l) + 2 × (l × w) = 2 × (h × w + h × l + l × w) ,

where h is prism height, w is its width, and l is its length.

Let's see an example of how to solve the right rectangular prism calc - find A problem. We'll come back to our example with the box and calculate its surface area:

• Calculate the rectangular prism surface area . First rectangle area is 15in × 12in = 180in² , second 15in × 18in = 270in² and third one 18in × 12in = 216in² . Add all three rectangles' areas - it's equal to 666 in² ( what a number! ) - and finally multiply by 2. The surface area of our cardboard box is 1332in².
• Or save yourself some time and use our rectangular prism calculator .

Finally, let's attack the right rectangular prism calc find d (that is, the diagonal) type of problem.

To determine the diagonal of a rectangular prism, follow these steps:

Apply the formula:

diagonal = √(l² + h² + w²)

where h is the height of the prism, w is its width, and l is its length.

Substitute the values.

Do you have the feeling that you saw the formula before? Yes, that's possible because this equation resembles the famous one from the Pythagorean theorem.

That rectangular prism was a piece of cake! If you are amazed at how easily you can calculate the volume with our tool, try out the other volume calculators:

• triangular prism calculator
• cylinder volume calculator
• sphere volume calculator
• cone volume calculator
• pyramid volume calculator

Be sure to check out the volume calculator - the volume of basic 3D solids, all in one place!

How many edges does a rectangular prism have?

The answer is twelve edges . A rectangular prism has:

• Eight vertices (or corners); and
• Twelve edges.

If you are not sure about the result, you can try to draw a rectangular prism and count its faces, vertices, and edges.

How do I calculate the volume of a rectangular prism with only its length?

You cannot calculate the volume of a rectangular prism, knowing only its length . You need to know its length, width, and height to calculate it. Once you have these parameters, you can use the equation:

volume = h × w × l .

What is the volume of a box with all sides equal?

Assuming that all sides are equal to 20 in, the volume is 8,000 in³ . We arrive at this answer by following these steps:

• Get the length, width, and height of the box.
• Using the rectangular prism volume formula, we get: volume = (20 × 20 × 20) in³ volume = 8,000 in³ .

How do I find the perimeter of a rectangular prism?

In a 3D solid, you find the surface area of the solid instead of the perimeter. For the rectangular prism, you can find its surface area using the formula:

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Understanding Architectural Form

• Updated: February 12, 2024

The bread and butter of the architecture profession centers around the form of the structures we create as designers. It is through these elements that we shape the space around us to form new places of meaning and function.

In this article, we discuss the principles, elements, and functions of architectural form to help you understand the intricacies of this fundamental architectural cornerstone, with the intention to further develop your conceptual development process.

What is an architecture form?

Architecture is a three-dimensional medium, expressed in forms that envelop the space around us. Form is used to describe the elements of the building that define its overall shape, size, proportions, and profile. It refers to the appearance of a building as a three-dimensional volume, and it can apply to both exterior and interior spaces.

Buildings can be interpreted as a collection of shapes, masses, lines and other elements, all coming together in a particular space.

Importance of form in architecture

Architectural form is one of the most revered aspects of the profession. It is often what makes buildings memorable, iconic, and most importantly more meaningful for its occupiers.

As early as the year 27 BC, Vitruvius coined the Latin terms for the three main principles of architecture; Firmitas , Utilitas , and Venustas . These three aspects continue to be the essential properties of architectural design. Firmitas means strength or stability, utilitas means function and use, and venustas refers to form and beauty.

Form is a means for expression and the translation of concepts . It is also what ultimately fills and divides the space that we inhabit. Form in architecture can hold a great amount of symbolic and cultural significance, and it can transform areas for better or for worse.

How is form used to affect and influence architecture design?

The below list summarizes the various types and categories of architectural form, and how they can be used to influence design decisions to create both internal and external spaces.

Primary elements

The primary elements are the core shapes and forms that can be found when more complex shapes are simplified. These are the basic geometric elements that shape the world around us.

Primary shapes and solids

The simplest of forms are known as primary shapes. The primary shapes include circles, triangles, and squares. Every shape can be created by a combination of the primary shapes. When rotated or extended, the primary shapes can create three-dimensional volumes known as the primary solids.

These basic solids are distinct due to their simple form.

Circles are centralized and introduce stability to a composition. They are naturally self-centering, and help centralize a plane or space. When positioned alongside straight lines and angled intersections, circles can give the impression of rotation and movement.

Being round in nature, circles add curves and smooth edges to a form, and can be used to soften the rigidity of straight lines and pointed angles.

The triangle is known as one of the most stable shapes in terms of structural form. When resting on one of its sides, the shape is stable, but when it is rotated onto one of its vertices, the triangle’s state of equilibrium becomes more dynamic.

When balanced on one of its angled points, a triangle has a physical tendency of falling to one side. Even if the triangle is fixed or anchored to a surface, it can give the appearance of falling over even when in place.

A square is a quadrilateral with four equal sides. Squares often represent balance, purity, and mental fortitude. They are neutral by nature, with no distinct direction.

Similar to the triangle, the square is stable when resting on one of its four sides, but is made more dynamic when placed on one of its corners.

Regular and irregular forms

Regular vs irregular forms – When solids are described with defined and consistent geometric properties, these solids are called regular forms. Regular forms are generally symmetrical, with right angles and similar sides all around. They appear more stable, with consistent shapes and angles.

If similarly shaped sides are not congruent, or the axis of extrusion is not vertical or horizontal, the form can be described as irregular. Irregular forms are composed of varying parts that relate to one another inconsistently. These irregular attributes do not change the solid from its type of form, but they cause it to deviate from the variables embodied in a regular solid.

In architecture, space is designed in consideration of both solid mass and the spatial voids within them. Buildings can contain both regular and irregular forms, and each can be enclosed or intertwined with the other.

A sphere is a type of solid created by rotating a circle 180 degrees, or by a semicircle rotating 360 degrees. A regular sphere has a surface that is equal distance from the center point all the way around, making it a perfectly round solid.

Spheres share many attributes with their base shape, the circle. They are self-centering and appear stable on a level surface.

Cylinders are generated by extruding a circle along a central axis, or by rotating a rectangle along a single direction. A cylinder has a round surface wrapping around its sides, with two circular faces at each end.

It is stable when resting on one of its flat, circular ends, but it can become less stable when positioned at an angle or along its round sides.

A cone is a solid created by the revolution of a right triangle with its vertical side in place. The vertical side becomes the center point connecting the circular base to the end point of its tapering surface.

When the circular face is used as the base, a cone is stable and balanced, but when tilted to the side it can become more dynamic and prone to motion.

A pyramid consists of a polygonal base whose vertices lead to edges that converge at a single point on the opposite end. Its tapering form creates triangular sides that enclose the space between the base polygon and the end vertex.

All sides of a pyramid are flat, making it generally stable regardless of which face it is positioned on. A pyramid only becomes unstable when made to stand on a vertex, edge, or corner.

Cubes are made from the extrusion of a square in a perpendicular direction from the square’s face. It is composed of six equal sides, each side a square. A cube has no particular direction, making it static in form and stable on any side.

With equal dimensions throughout, it lacks dynamic movement, and maintains a similar appearance from all angles. 

Transformation of form

From the primary elements, all other forms can be understood as a transformation of solids from their primary form to other shapes and volumes. This transformation can occur through the manipulation of its dimensions or by the addition or subtraction of elements.

Subtractive forms

Subtractive transformation involves removing one or more parts of a form’s volume to achieve a new form. This new form can transform the solid into another family, or it may remain a part of its original solid family depending on the extent of the transformation.

If, for example, a regular pyramid were cut horizontally near its base, it would still be a regular pyramid. However, if the pointed tip is subtracted from the pyramid’s form, it would become a truncated pyramid.

Subtractive forms are common in architecture. They can be used to create fenestrations, or recesses for things like entrances, windows, and courtyards. They can also be used to create dynamic volumes that introduce sunlight and wind to internal spaces, or to provide protection from the natural elements.

Additive forms

Additive forms refer to a transformation through the addition of elements. The extent of additive transformation depends on the number of added parts, as well as their placement and size. These additive forms can alter the original solid, or cause the form to change its profile completely.

Unlike subtractive forms, where parts are removed, additive forms introduce new pieces to the volume that may be smaller or larger than the original solid. There are many different ways in which these forms can be grouped. Here are some of the most well-known relationships of additive forms:

Spatial tension

In spatial tension, forms and parts are interrelated by close proximity, forming a larger group that represents the whole form. This type of relationship does not require direct attachment. As Aristotle once said, “The whole is greater than the sum of its parts.” Spatial tension embraces the group in its entirety, with each additive piece playing an integral part.

Edge to edge

Edge-to-edge relationships are forms that share a common edge as their point of linear attachment. The volumes attached may be simple or complex, as long as one or more edges meet or are shared between them.

Face to face

A face-to-face relationship involves two or more forms that make contact with one another along planar surfaces. These surfaces come in contact parallel to one another, with or without alignment of their adjacent edges.

Interlocking

An interlocking relationship involves the clashing or converging of forms. Interlocking forms are volumes that overlap or penetrate one another, creating a mass of intersecting solids.

Centralized

A centralized form consists of multiple secondary forms arranged around a central parent form. The central form becomes the centerpiece, and is usually the emphasis of the composition. Spheres, cones, and other circular solids work well as the central form due to their natural self-centering properties.

Linear relationships are achieved by arranging volumes in rows or columns, creating linear sequences through the solid mass, void space, or the patterns that emerge. Sequential changes in dimension or size can also create linear relationships.

Radial form refers to a series of objects arranged around a centralized core element in a radial manner. The forms extend outwards in a revolving pattern to create a larger composition that typically converges at the center.

A cluster is a group of forms that create a composition through proximity, size, shape, or function. Unlike other relationships that involve an organized structure, a cluster can be grouped and organized without geometric alignment or pattern.

A grid layout is a way of organizing forms into linear rows and columns. A grid can be three-dimensional, and is often based on a series of squares. This relationship allows forms to be arranged with consistent space and distance. Grids can also be used to analyze or break down a group or surface.

Collisions of geometry

It is common for two or more geometries to collide, resulting in a new composite form. These colliding shapes or solids can be of equal size and shape, or have different attributes altogether. There are many reasons for the collision of geometry, including:

• To create an internal space within an existing form
• For symbolic or conceptual significance
• To satisfy the functional requirements of the form
• As a means to direct space to desired locations of the site
• To maintain or disrupt symmetry in the structure
• As a response to site conditions and context

Mass and scale

Mass in architecture refers to the physical size or bulk of a building. It can be interpreted as a building’s actual size by measurement, or its relative size by context. Mass, combined with shape, defines form.

Scale is a relative perception of size. In architecture, it refers to the size of a building as compared to other contextual elements. These elements may be familiar architectural features, surrounding buildings and landmarks, or most commonly, the human figure.

Human scale is frequently used as a standard for scale and measurement, to ensure buildings are considerate of anthropomorphic and ergonomic functionality. The standards for human scale often vary based on region, culture, and the target users of the facility.

Architects can use scale to make a building appear larger or smaller than its actual size. A single design can also contain several different scales, to achieve a more complex visual and spatial composition.

In an architectural composition, proportion refers to the physical and spatial relationships of one element to the other elements present, and to the building as a whole.

Over centuries of art and architecture, several different proportioning systems have been developed to help organize and unify the parts of a building. The most well-known systems are arithmetic, geometric, harmonic, material, or structural.

Proportions that follow an arithmetic system use mathematics and numerical functions to determine the patterns and restraints of the design. Arithmetic systems are prevalent in Ancient Greek architecture, with clear functions, ratios, and numerical sequences used in many of their most iconic structures.

A geometric system uses shapes and geometric values to determine the size and scale of architectural features. In Classical architecture, the dimensions of a building were often measured by the diameter of a classical column.

Likewise, many Renaissance facades made use of regular shapes and lines to create orderly designs based on squares, circles, and triangles. One of the most widespread uses of geometric proportion is The Golden Section.

Also expressed as the Golden Mean, it is both a geometric and arithmetic proportioning system that can be found in architecture, product design, and nature.

Harmonic proportions are inspired by the ancient discoveries of repeating harmonies in music. It involves using repeating ratios such as 1:2, 2:3, or 3:4 in buildings and spaces to create what many believe are harmonious designs.

This can be observed in Roman Renaissance architecture, or later on from the works of Palladio and Venetian musical theorists, who created a more complex system based on the major and minor third, with a ratio of 4:5 or 5:6.

In consideration of construction and cost-effectiveness, many buildings are designed in appropriate proportions based on the materials being used. With the majority of building materials following industry standard unit sizes, their dimensions create another unit of measurement for the building.

Materials such as wood planks, concrete masonry units, or bricks are produced and sold in conventional sizes, and these sizes can form an additional level of proportion for the overall design.

With structural members playing a critical role in architectural design, they can have a significant impact on the proportions of the building. Structural proportions are largely dependent on the load bearing capacity and structural requirements of each member.

These members are applied as necessary, and can contribute to or disrupt the proportioning system.

A series of recurring architectural elements or shapes can be described as rhythm. This repetition may be regular or complex. Rhythm can commonly be observed with repeating windows, arches, columns, or moldings.

Articulation

The way in which building surfaces come together to define form is commonly known as articulation. This includes the treatment of corners, edges, solids and voids, all contributing to the articulation of the building’s form.

It can also include the texture appearance of a building, its visual weight, and its overall resemblance to something else.

Texture is an attribute mainly determined by the building materials, but it can also be used to describe the appearance and surface qualities of different architectural compositions. Materials like stone can be made to appear smooth or rough, and it can also be carved to add more depth and relief.

Similarly, a building with many angular protrusions on its facade may appear rough or jagged, while a round organic structure may seem smooth, but both can have openings or fenestration’s that add depth and character to their original form.

Light refers to the way in which a form is being illuminated. Form can be perceived in multiple different ways depending on the light conditions present at the time of viewing. As such, light and shadow play an integral role in making forms visible to the human eye.

Edges and corners

Edges and corners are formed at the perimeter of planar surfaces. They can often be found at the extents of the building, as well as on many elements of the facade . More intricate geometry typically translates to more edges and corners on the building envelope, and these elements are often carefully articulated to achieve a desired look for the design.

Corners are present at the convergence of two planes. They may be distinct corners, with the planes physically connected, or they may be implied, with one or more faces set back from the other.

The change in direction for corners results in a contrast of light, and this can be used to the design’s advantage to explore the interplay of shade and shadows. They may also be fitted with different materials or architectural features to highlight the change of plane.

In cases wherein a defined corner is not desired, they can be rounded off to create a smooth transition between the adjoining planes. This can either soften the sharp edge, or create the appearance of a continuous surface, depending on the radius of the rounded curve.

How is all this used to articulate form?

It’s all very well listing all these various types of form, but what are its practical uses and how can its understanding make us better designers?

Example of architectural form

Scale – metropolis museum in amsterdam designed by renzo piano workshop.

Also known as the NEMO Science and Technology Museum, this ship-like building sits surrounded by water, clad in pre-oxidized copper with dramatic curves and stunning facilities.

Its design utilizes scale and texture to give a distinct impression of an aged ship in the water. The building’s sheer size allows for scenic roof decks that complete its maritime-inspired experience.

Surface – Guggenheim Museum in Bilbao, Spain designed by Frank Gehry Architects

The iconic Guggenheim Museum in Bilbao helped to establish Frank Gehry as one of the most prominent architects of the era. It also exemplified Gehry’s mastery of using surfaces as the building’s defining features.

The museum’s exterior is wrapped in sweeping metallic forms that gently curve and intertwine. No two angles appear the same, but the structure maintains unity and consistency on all sides.

Thanks to its reflective materials, light and color bounce around between each surface, casting subtle hues on the facade throughout the day.

Mass – Church San Paolo Apostolo by Studio Fuksas

The church of San Paolo Apostolo is located in Foligno, Italy. It is also known as the St. Paolo Parish Complex, and it was designed as a symbol of rebirth after the Umbria-Marche earthquakes of 1997.

Its primary form is tall and boxy, with abstract voids cutting through its massive concrete facade. These voids create deep windows that allow light to penetrate into the church’s spacious interiors.

The building’s imposing presence makes for a decisive landmark in the area, one of the key goals of the original competition brief.

Distortion – JustK by Amunt Architekten und Nagel Tehissen

JustK is a residential project located in Germany. This single family home is known for its unique form that resulted from adjusting to the site context, specific building regulations, interior requirements, and views of the nearby castle.

The exterior walls and roof share the same colors and materials, making the house appear as a distorted solid with dynamic angles and spaces.

Inclusion – Overkapping Commandantswoning by Oving Architekten

This powerful project involved encasing an old Nazi commander’s concentration camp home in a glass and steel ecnclosure. The glass vitrine serves as a World War II memorial for victims of the holocaust, and it works to preserve the building for educational events and history.

The external structure maintains a minimally invasive form to accentuate the contents inside, and to keep the house as the focal point of historical significance. This manner of additive form is known as inclusion.

Oving Architekten designed a large transparent prism around the house, effectively stopping the effects of weather on the site without obstructing views of its original state.

Link – Nelson Atkins Museum by Steven Holl Architects

This contemporary addition to the 1933 classical “Temple of Art” helps to create an interaction between the old and the new. The design provides a sequence of five “lenses” intended to bring visitors on a historical and cultural journey through the museum.

The threaded link between each lens allows the new building to be woven into the surrounding landscape in harmony around the original structure.

Open/Closed – Summer Pavilion by SANNA

SANNA’s Summer Pavilion features large canopies with thin roofing and numerous slim steel columns for support. The columns wrap around under its curving eaves to create a boundary without enclosure, the feeling of a wall without barrier.

This results in an environment that blurs the lines between the outside and the indoor, the open space from the sheltered and shaded.

Embed – Aloni by Deca Architecture

The Aloni house in Greece is a fine example of landscape integration in architecture. The building is tucked into a saddle where two slopes meet, with its linear form embedded into the natural topography of the site.

The surrounding landscape continues over the building’s roof, bridged by a plateau of grass and vegetation that further conceals the house’s mass. Its architecture is revealed by stone walls and four courtyards carved into the rolling hills.

The design pays homage to the site’s agricultural roots, and builds a strong relationship between the built and natural conditions of its setting.

To conclude…

Form is at the very core of architectural design, and it carries with it endless possibilities in space and mass.

Understanding the fundamentals of form can help you maximize your creativity with three-dimensional volumes, resulting in more complex and meaningful architectural designs.

Whether you’re a student, a young professional, or a seasoned architect in the field, there are always new ways to explore form in design.

Every design project begins with site analysis … start it with confidence for free!.

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Science & Technology

Plato was right. Earth is made, on average, of cubes

The ancient greek philosopher was on to something, the school of arts & sciences’ douglas jerolmack and colleagues found..

Plato, the Greek philosopher who lived in the 5th century B.C.E., believed that the universe was made of five types of matter: earth, air, fire, water, and cosmos. Each was described with a particular geometry, a platonic shape. For earth, that shape was the cube.

Science has steadily moved beyond Plato’s conjectures, looking instead to the atom as the building block of the universe. Yet Plato seems to have been onto something, researchers have found.

In a new paper in the Proceedings of the National Academy of Sciences , a team from the University of Pennsylvania , Budapest University of Technology and Economics , and University of Debrecen uses math, geology, and physics to demonstrate that the average shape of rocks on Earth is a cube. 

“Plato is widely recognized as the first person to develop the concept of an atom, the idea that matter is composed of some indivisible component at the smallest scale,” says Douglas Jerolmack , a geophysicist in Penn’s School of Arts & Sciences ’ Department of Earth and Environmental Science  and in the School of Engineering and Applied Sciences ' Department of Mechanical Engineering and Applied Mechanics . “But that understanding was only conceptual; nothing about our modern understanding of atoms derives from what Plato told us. 

“The interesting thing here is that what we find with rock, or earth, is that there is more than a conceptual lineage back to Plato. It turns out that Plato’s conception about the element earth being made up of cubes is, literally, the statistical average model for real earth. And that is just mind-blowing.”

The group’s finding began with geometric models developed by mathematician Gábor Domokos of the Budapest University of Technology and Economics, whose work predicted that natural rocks would fragment into cubic shapes. 

“This paper is the result of three years of serious thinking and work, but it comes back to one core idea,” says Domokos. “If you take a three-dimensional polyhedral shape, slice it randomly into two fragments and then slice these fragments again and again, you get a vast number of different polyhedral shapes. But in an average sense, the resulting shape of the fragments is a cube.”

Domokos pulled two Hungarian theoretical physicists into the loop: Ferenc Kun, an expert on fragmentation, and János Török, an expert on statistical and computational models. After discussing the potential of the discovery, Jerolmack says, the Hungarian researchers took their finding to Jerolmack to work together on the geophysical questions; in other words, “How does nature let this happen?”

“When we took this to Doug, he said, ‘This is either a mistake, or this is big,’” Domokos recalls. “We worked backward to understand the physics that results in these shapes.”

Fundamentally, the question they answered is what shapes are created when rocks break into pieces. Remarkably, they found that the core mathematical conjecture unites geological processes not only on Earth but around the solar system as well. 

“Fragmentation is this ubiquitous process that is grinding down planetary materials,” Jerolmack says. “The solar system is littered with ice and rocks that are ceaselessly smashing apart. This work gives us a signature of that process that we’ve never seen before.”

Part of this understanding is that the components that break out of a formerly solid object must fit together without any gaps, like a dropped dish on the verge of breaking. As it turns out, the only one of the so-called platonic forms—polyhedra with sides of equal length—that fit together without gaps are cubes. 

“One thing we’ve speculated in our group is that, quite possibly Plato looked at a rock outcrop and after processing or analyzing the image subconsciously in his mind, he conjectured that the average shape is something like a cube,” Jerolmack says.

“Plato was very sensitive to geometry,” Domokos adds. According to lore, the phrase “Let no one ignorant of geometry enter” was engraved at the door to Plato’s Academy. “His intuitions, backed by his broad thinking about science, may have led him to this idea about cubes,” says Domokos.

To test whether their mathematical models held true in nature, the team measured a wide variety of rocks, hundreds that they collected and thousands more from previously collected datasets. No matter whether the rocks had naturally weathered from a large outcropping or been dynamited out by humans, the team found a good fit to the cubic average.

However, special rock formations exist that appear to break the cubic “rule.” The Giant’s Causeway in Northern Ireland, with its soaring vertical columns, is one example, formed by the unusual process of cooling basalt. These formations, though rare, are still encompassed by the team’s mathematical conception of fragmentation; they are just explained by out-of-the-ordinary processes at work.

“The world is a messy place,” says Jerolmack. “Nine times out of 10, if a rock gets pulled apart or squeezed or sheared—and usually these forces are happening together—you end up with fragments which are, on average, cubic shapes. It’s only if you have a very special stress condition that you get something else. The earth just doesn’t do this often.”

The researchers also explored fragmentation in two dimensions, or on thin surfaces that function as two-dimensional shapes, with a depth that is significantly smaller than the width and length. There, the fracture patterns are different, though the central concept of splitting polygons and arriving at predictable average shapes still holds.

“It turns out in two dimensions you’re about equally likely to get either a rectangle or a hexagon in nature,” Jerolmack says. “They’re not true hexagons, but they’re the statistical equivalent in a geometric sense. You can think of it like paint cracking; a force is acting to pull the paint apart equally from different sides, creating a hexagonal shape when it cracks.”

In nature, examples of these two-dimensional fracture patterns can be found in ice sheets, drying mud, or even the earth’s crust, the depth of which is far outstripped by its lateral extent, allowing it to function as a de facto two-dimensional material. It was previously known that the earth’s crust fractured in this way, but the group’s observations support the idea that the fragmentation pattern results from plate tectonics.

Identifying these patterns in rock may help in predicting phenomena such as rock fall hazards or the likelihood and location of fluid flows, such as oil or water, in rocks. 

For the researchers, finding what appears to be a fundamental rule of nature emerging from millennia-old insights has been an intense but satisfying experience.

“There are a lot of sand grains, pebbles, and asteroids out there, and all of them evolve by chipping in a universal manner,” says Domokos, who is also co-inventor of the Gömböc , the first known convex shape with the minimal number—just two—of static balance points. Chipping by collisions gradually eliminates balance points, but shapes stop short of becoming a Gömböc; the latter appears as an unattainable end point of this natural process. 

The current result shows that the starting point may be a similarly iconic geometric shape: the cube with its 26 balance points. “The fact that pure geometry provides these brackets for a ubiquitous natural process, gives me happiness,” he says.

“When you pick up a rock in nature, it’s not a perfect cube, but each one is a kind of statistical shadow of a cube,” adds Jerolmack. “It calls to mind Plato’s allegory of the cave. He posited an idealized form that was essential for understanding the universe, but all we see are distorted shadows of that perfect form.”

Douglas Jerolmack is a professor in the Department of Earth and Environmental Science in the School of Arts & Sciences and in the Department of Mechanical Engineering and Applied Mechanics in the School of Engineering and Applied Science at the University of Pennsylvania.

Gábor Domokos is a professor and director of the MTA-BME Morphodynamics Research Group at the Budapest University of Technology and Economics.

Ferenc Kun is a professor in the Department of Theoretical Physics at the University of Debrecen.

János Török is an associate professor in the Department of Theoretical Physics at the Budapest University of Technology and Economics.

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Penn celebrates operation and benefits of largest solar power project in Pennsylvania

Solar production has begun at the Great Cove I and II facilities in central Pennsylvania, the equivalent of powering 70% of the electricity demand from Penn’s academic campus and health system in the Philadelphia area.

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The Empowerment Through Education Scholarship Program at Penn’s Graduate School of Education is helping to prepare and retain teachers and educational leaders.

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‘The Illuminated Body’ fuses color, light, and sound

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The iconic sculpture by pop artist Robert Indiana arrived on campus in 1999 and soon became a natural place to come together.