Circles of how many different radii are needed to construct the following figure?
Show Hint
For complex pattern analysis, leverage symmetry. Since this pattern has 4-fold rotational symmetry, you only need to analyze one quadrant in detail to find all the unique elements.
Step 1: Understanding the Concept:
This is a detailed visual analysis problem. We need to carefully examine the provided image, which is a complex pattern made of circles, and count the number of unique circle sizes (radii) used in its construction. The pattern has four-fold rotational symmetry, so we can analyze just one quadrant to identify all the unique circle types.
Step 2: Key Formula or Approach:
The approach is to systematically identify and categorize each circle by its size. It's helpful to start from the center and work outwards, or start with the largest size and work downwards.
Step 3: Detailed Explanation:
Let's identify the distinct radii by assigning a number to each unique size we find, starting from the largest.
Size 1: The single, largest circle in the absolute center of the figure.
Size 2: The four large circles directly surrounding the central one, arranged in a cross shape.
Size 3: The four circles situated on the diagonal axes, between the Size 2 circles. These are visibly smaller than Size 2.
Size 4: The four circles further out on the primary axes (up, down, left, right). These are visibly smaller than Size 3.
Size 5: The circles located at the very tips of the four main arms of the pattern. These are smaller than Size 4.
Size 6: Look at the clusters of small circles in the "corners" between the arms. There is one circle in each of these clusters that is slightly larger than the others around it. This circle is also smaller than Size 5. There are 8 such circles in total (2 per corner cluster).
Size 7: The smallest circles in the figure, used as fillers in the tightest gaps. These are clearly the smallest radius.
By carefully distinguishing between these groups, we can identify a total of 7 different sizes of circles. The visual difference between some of the smaller sizes is subtle, requiring close inspection.
Step 4: Final Answer:
Based on a systematic visual inspection, there are 7 different circle radii used to construct the figure.
