Shrinking Branches and Geometric Sums

More pieces, a bounded route

Follow the bright route from the trunk outward. Each segment is half as long as the one before it. The route keeps gaining pieces, yet those pieces become very small. How can an endless sequence of positive lengths fit within a finite total?

Add two more generations. The new highlighted pieces are tiny compared with the trunk, but they are real additions. We will measure the length along this route, including each bend. That is different from the straight-line distance between its starting and ending points.

Return to a shorter route so its parts are easy to compare. We choose the trunk to have length one unit. The unit is a mathematical reference, not a fixed number of pixels. Zooming the picture changes apparent sizes while preserving all these length ratios.

Approach the first child and its parent. The child is half the parent’s length. The same relation holds at the next junction, and at every later junction. A constant child-to-parent ratio is the rule that makes these lengths a geometric sequence.

Add the pieces of one path

Begin with the trunk and one highlighted child. Their lengths are one and one half. Along this route the total is one and a half units. The sibling branch is visible, but we do not add it because we are measuring one route, not the entire tree.

Include the next highlighted child. Half of one half is one quarter. Add that quarter to the previous one and a half, giving one and three quarters, or one point seven five units. Every path reaching this generation has the same length, although its direction can differ.

Include another generation. The next segment has length one eighth, because it is half of one quarter. The route total becomes one point eight seven five. The new contribution is positive, so the total increases, but the increase is smaller than before.

Continue to generation four. The added segment is one sixteenth, and the total is one point nine three seven five. Compare that with two units. The difference is also one sixteenth. The still-missing part can be described as precisely as the part already present.

What remains after the visible route

Return to generation two. We have one plus one half plus one quarter, totaling one and three quarters. The gap to two is one quarter. The next segment would fill half that gap, the following segment half what remains, and so on.

Show generation three again. Adding one eighth reduces the gap from one quarter to one eighth. The construction is not merely getting close in an unspecified way. At each repetition, both the new branch length and the remaining gap are divided by two.

Show generation four. The remaining gap is now one sixteenth. Every finite depth still leaves a positive gap, so the finite sum has not become exactly two. Two describes the limiting total approached as the repetition continues without a final generation.

Extend the route to six generations. Its total is one point nine eight four three seven five, leaving one sixty-fourth to reach two. We can make the remaining amount as small as a chosen positive tolerance by taking enough generations, without claiming that this finite image contains infinitely many branches.

A different ratio gives a different limit

Compare a route whose children are one quarter of their parents. With two repetitions, the lengths are one, one quarter, and one sixteenth. Their sum is one point three one two five. The visible additions shrink much faster than in the half-scale example.

Add another generation at this quarter-scale ratio. The new piece is one sixty-fourth, giving one point three two eight one two five. The infinite sum approaches four thirds. That limit is smaller than two because every child after the trunk is shorter than in the half-scale case.

Now compare children that are three quarters of their parents. At depth two the lengths are one, three quarters, and nine sixteenths. Their sum is two point three one two five. The same number of repetitions already gives a longer route than the half-scale limit.

Continue the three-quarter route to depth four. Its total is three point zero five zero seven eight one two five. The infinite route approaches four units, but this finite route is still noticeably shorter. A ratio closer to one makes convergence slower.

Read the sum from its repeated structure

Return to the half-scale route at depth three. Its lengths are one, one half, one quarter, and one eighth. Multiplying the whole list by one half produces the next shifted list: one half, one quarter, one eighth, and one sixteenth. Most terms are shared.

Keep the complete highlighted route in view. The original finite sum exceeds its half-sized copy by one minus one sixteenth. The middle terms cancel in that comparison. This is a numerical explanation of the sum; the displayed tree is not claiming to animate a separate subtraction diagram.

For a general child ratio r, the finite path through generation n has lengths one, r, r squared, through r to the power n. When r is not one, its sum is one minus r to the power n plus one, all divided by one minus r. The extra exponent includes the root term.

Compare the three-quarter example again. Here one minus r is one quarter. As depth increases, the leftover power of three quarters approaches zero, so the limit becomes one divided by one quarter: four. The formula records the repeated shrinking already visible in the branches.

Separate length, distance, and count

Set the child ratio to one. Every highlighted segment now has the same length as the trunk. With three repetitions there are four unit segments along the route, totaling four. Later generations would keep adding one unit each. There is no finite infinite-route length in this case.

Restore the half-scale ratio and compare a wider branching angle. The route bends differently, but the lengths one, one half, one quarter, and one eighth are unchanged. Its total remains one point eight seven five. Angles affect the endpoint’s position, not the sum of segment lengths.

Compare a narrow angle. The endpoint lies in a different direction, and the route is closer to straight. The straight-line distance from the root’s base to that endpoint cannot exceed the route length. Bending can change this displacement even when the total walking distance stays fixed.

Return to the usual angle and clear the highlight. Many other branches were not part of our sum. A finite bound on every single route does not automatically bound the sum of every branch in the entire tree. The next lesson will show why those questions can have very different answers.

Calculate the remainder after a finite route

At depth three with scale one half, the selected route contains lengths one, one half, one quarter, and one eighth. Its finite sum is one point eight seven five. The corresponding infinite geometric sum is two, so the omitted remainder after this displayed route is one eighth.

Focus on the last displayed segment. Its length is one eighth. The next, undisplayed segment would be one sixteenth, followed by one thirty-second and smaller pieces. Those future pieces sum to one eighth altogether. We are comparing a finite picture with a calculated infinite tail, not claiming infinitely many segments are rendered.

Reveal one additional generation and follow the same first-child route. Its new piece has length one sixteenth, bringing the visible total to one point nine three seven five. The remaining infinite tail is now one sixteenth. Each revealed piece halves what remains to the limit in this particular ratio-one-half example.

Change the scale to one quarter and return to depth three. The route lengths are one, one quarter, one sixteenth, and one sixty-fourth. Their sum is one point three two eight one two five. The infinite limit is four thirds, so the missing tail is one over one hundred ninety-two.

The last visible quarter-scale segment is one sixty-fourth, larger than the omitted tail one over one hundred ninety-two. Equality between the last segment and the remaining tail was special to ratio one half. In general the tail after depth d is the scale raised to d plus one, divided by one minus the scale.

Use the length rule deliberately

Highlight one route once more. Before extending it, predict the next length using the child ratio, then add that contribution to the current path total. With half-scale children at depth two, the next segment will be one eighth. Its length comes from the parent, not from the whole route.

Reveal that next segment. The path total has increased from one point seven five to one point eight seven five. We can now read both the local rule and the accumulated result: halve a branch to obtain its child, and add successive lengths to measure the route.

Choose Interact to compare ratios such as one half and three quarters at the same depth. Follow a selected lineage and distinguish changing the branch ratio from zooming the camera. Resume restores this measured example. Very short outer segments may need a closer view to be seen clearly.

One route can contain arbitrarily many shrinking segments while approaching a finite total length. The finite drawing shows an approximation, and the geometric sum explains its remainder. Keep the root, the ratio, the chosen route, and the distinction between length and endpoint distance explicit.