Thursday, October 13, 2011

The Ultimate High Jump

In any top-notch athletics meeting, one of the blue riband events has always been the high jump and the world record for this event has been of great and constant interest to the general public.

In 1912 the record was 2.00 m and this progressed steadily over the following decades, finishing up with 2.45 m in 1993, a progress of nearly half a metre in 100 years. The fact that the 1993 record still stands may well be an indication that we are approaching the height limit that a human being can jump and it's an interesting problem to try and estimate this limit.

And a limit seems to be intuitively likely – we cannot jump 5 m so therefore a limit exists somewhere between the current world record of 2.45 m and 5 m that places a ceiling on how high a human can jump.

Since 1920 several areas of progress have emerged in this event. The type of jump itself has evolved into more efficient styles that have enabled greater heights to be reached.

The old scissor jump was replaced by the so-called "western roll" in around 1912 and this was followed by the "straddle" jump over the following two decades. In his jump the athlete rotated over the bar whilst in a horizontal position and the US athlete Charles Dumas became the first human to jump 7 feet (2.13 m) in 1956 using this technique.

Above: A jumper clears the bar using the straddle technique (Image from Wikipedia Commons (Click to enlarge)

The Soviet jumper Valery Brumel was the great straddle expert of history and he took the record up to 2.28 m in 1964. Some great footage of Brumel in action can be seen at

Brumel was an extraordinary athlete in every way, and he regularly preformed a “party trick” that was truly amazing. He could leap upwards and touch the ring of a basketball hoop 10 feet off the floor with his foot! (I wonder how many could do this today?)

For an image showing this incredible feat see

The next style breakthrough came through a young American, Dick Fosbury, who in the late 1960’s pioneered a technique in which the jumper cleared the bar backwards and looking upwards – a sort of half back somersault. This style was made possible by improvements in the landing pit that was softer and raised so that an athlete could land on his back without risking serious injury. This jump became known as the “Fosbury Flop” and enabled jumpers to reach significantly greater heights.

Above - The Fosbury Flop
(Image from Wikipedia Commons - click to enlarge)

Dick Fosbury used his technique to win the high jump gold medal at the 1968 Olympic Games in Mexico City. His winning leap was 2.24 metres (7 feet 4.25 inches) which was also a new Olympic record. This turning point in high jump history can be seen here:

It was used with great effect by jumpers over the following four decades, with the present record holder Javier Sotomayor using it to establish the present world record of 2.45 m in 1993.

A complete list of the high jump world records can be found at

but for the sake of our discussion here we will consider only the following four:

2.03 m in 1924 by Harold Osborn (USA)
2.28 m in 1963 by Valery Brumel (URS)
2.41 m in 1985 by Rudolf Povarnitsyn (URS)
2.45 m in 1993 by Javier Sotomayor (CUB)

We can try and simulate this record progression with mathematics and it turns out that by using a device called the geometric series, we can come up with a reasonable simulation. In doing so I remark that there is no physical reason whatsoever for adopting this method but what I set out to do was

(a) Use a mathematical process that reproduced the past
(b) Had an accuracy of greater than 95% in doing this
(c) Tended to a limit that was realistic

It works like this:

The world record in 1925 was 2.03 metres and we write this as

0.53+1+0.5 = 2.03

40 years later, in 1965, the record was 2.28 m, which is

0.53+1+0.5 + 0.25 = 2.28, or

0.53+1+(1/2)+(1/4) = 2.28

20 years later, in 1985, it was 2.41 m, which is close to

0.53+1+0.5 + 0.25 + 0.125 = 2.405, or

0.53+1+(1/2)+(1/4)+(1/8) = 2.405

10 years later, in 1995, it was 2.45 m, which is close to

0.53+1+0.5 + 0.25 + 0.125 + 0.0625 = 2.4675, or

0.53+1+(1/2)+(1/4)+(1/8)+(1/16) = 2.4675

Our prediction therefore produces an error of less than 2%.

From this we would predict that 5 years later, in 2000, the record would be

0.53+1+(1/2)+(1/4)+(1/8)+(1/16)+(1/32) = 2.49875,

but in fact it remained at 2.45m, as the record set in 1993 still stood.

This represents an error of 4.9%.

Mathematicians will recognise what is called a geometric series emerging here, which is

1+(1/2)+(1/4)+(1/8)+(1/16)+(1/32)+(1/64)+(1/128) + …..,

and the interesting fact about this series is that it converges to a limit, which is 2. When we add our 0.53 that we started with to set our figure at the 1925 record, this predicts that the theoretical height limit to which a human being can jump is 2.53 metres.

If we assume an error of 5% on top of this, it means that its likely the world high jump record will peak somewhere between 2.53 and 2.66 metres.

However humans can seldom be described by mathematics. Perhaps somewhere, sometime out there, a boy will be born with special genetic gifts. As he reaches manhood he will have the spring of a Brumel, the finesse of a Fosbury and the strength of a Sotomayor. He will defy the geometric series and astound the world. It is only a question of when.

Stop press: Since writing this article I discovered an interview with Dick Fosbury located at

A quote from this says: "I have predicted in the past (1978) that I believe the peak in my lifetime would be about 2.50 m based on my experience and observations. I'm still watching".

This figure agrees well with our 2.53 m prediction through the geometric series.


  1. oMG. O_o I'm doing a mathematical I.B. I.A. portfolio based on Olympic men's high jumps and I find this...
    Is there an exact formula you apply to come up with our theory?

  2. Hello Eve;

    The so called geometric series is used here - a special case of this is

    1+(1/2)+(1/4)+(1/8)+(1/16)+(1/32)+(1/64)+(1/128) + ….. =2.

    This is only my theory - not necessarily correct!

    All the best with your project.

  3. All this thing is just theory. How we can put it in practice?

  4. Hi Dunkdunk - there's only one way, and answer is here:

  5. Who would have thought that maths plays such a big part in high jumping

  6. I would love if a person will do this and give us an honest review about the experience. Though in theory, this seems interesting especially for people who want to jump high

  7. I found so many informative content here. I like to say thanks for sharing these great article with us. You wrote something that people could understand and made the subject intriguing for everyone. Really, great blog you have got here.Como hacer backflip